A Brief Manifesto

The Peierls-Onsager Effective Hamiltonian in a complete gauge covariant setting: Description of the spectrum. Viorel Iftimie∗ and Radu Purice†‡ arXiv:1304.2496v2 [math-ph] 9 Oct 2013 July 19, 2018 Abstract Using the procedures in [9] and [14] and the magnetic pseudodifferential calculus we have developped in [25, 27, 19, 20] we construct an effective Hamitonian that describes the spectrum in any compact subset of the real axis for a large class of periodic pseudodifferential Hamiltonians in a bounded smooth magnetic field, in a completely gauge covariant setting, without any restrictions on the vector potential and without any adiabaticity hypothesis. 1 Introduction In this paper we consider once again the construction of an effective Hamiltonian for a particle described by a periodic Hamiltonian and subject also to a magnetic field that will be considered bounded and smooth but neither periodic nor slowly varying. Our aim is to use some of the ideas in [9, 14] in conjunction with the magnetic pseudodifferential calculus developed in [25, 19, 20, 27] and obtain the following improvements: 1. cover also the case of pseudodifferential operators, as for example the relativistic Schrödinger operators with principal symbol < η >; 2. consider magnetic fields that are neither constant nor slowly variable, and thus working in a manifestly covariant form and obtain results that clearly depend only on the magnetic field; 3. give up the adiabatic hypothesis (slowly variable fields) and consider only the intensity of the magnetic field as a small parameter; 4. consider hypothesis formulated only in terms of the magnetic field and not of the vector potential one uses. Let us point out from the beginning, that as in [14] we construct an effective Hamiltonian associated to any compact interval of the energy spectrum but its significance concerns only the description of the real spectrum as a subset of R. In a forthcoming paper our covariant magnetic pseudodifferential calculus will be used in order to construct an effective dynamics associated to any spectral band of the periodic Hamiltonian. Let us mention here that the magnetic pseudodifferential calculus has been used in the Peierls-Onsager problem in [12] where some improvements of the results in [31] are obtained but still in an adiabatic setting. In fact in our following paper we intend to extend the results in [12] and construct a more natural framework for the definition of the Peierls-Onsager effective dynamics associated to a spectral band. Finally let us also point out here that an essential ingredient in the method elaborated in [14] is a necessary and sufficient criterion for a tempered distribution to belong to some given Hilbert spaces (Propositions 3.2 and 3.6 in [14]). In our ’magnetic’ setting some similar criteria have to be proved and this obliges us to some different formulations that allows to avoid a gap in the original proof given in [14, 13]. 1.1 The problem We shall constantly use the notation X ≡ Rd , its dual X ∗ being cannonically isomorphic to Rd ; let < ., . >: X ∗ × X → R denote the duality relation. We shall always denote by Ξ := X × X ∗ (considered as a symplectic space with the canonical symplectic form σ(X, Y ) := hξ, yi − hη, xi); we shall denote by Ξ := X ∗ × X . ∗ Institute of Mathematics Simion Stoilow of the Romanian Academy, P.O. Box 1-764, Bucharest, RO-70700, Romania. of Mathematics Simion Stoilow of the Romanian Academy, P.O. Box 1-764, Bucharest, RO-70700, Romania. ‡ Laboratoire Européen Associé CNRS Franco-Roumain Math-Mode † Institute 1 We shall consider a discrete abelian localy compact subgroup Γ ⊂ X . It is isomorphic to Zd and we can view it as a lattice Γ := ⊕dj=1 Zej , with {e1 , . . . , ed } an algebraic basis of Rd (that we shall call the basis of Γ). We consider the quotient group Rd /Γ that is canonically isomorphic to the d-dimensional torus T ≡ TΓ ≡ T and let us denote by p : Rd ∋ x 7→ x̂ ∈ T the canonical projection onto the quotient. Let us consider an elementary cell :   d   X tj ej ∈ Rd | 0 ≤ tj < 1, ∀j ∈ {1, . . . , d} , EΓ = y =   j=1 d having the interior (as subset of R ) locally homeomorphic to its projection on T. The dual lattice of Γ is then its polar set in X ∗ defined as Γ∗ := {γ ∗ ∈ X ∗ | < γ ∗ , γ >∈ (2π)Z, ∀γ ∈ Γ} . Considering the dual basis {e∗1 , . . . , e∗d } ⊂ X ∗ of {e1 , . . . , ed }, defined by < e∗j , ek >= (2π)δjk , we have evidently that Γ∗ := ⊕dj=1 Ze∗j . By definition, we have that Γ∗ ⊂ X ∗ is the polar of Γ ⊂ X . We define TΓ∗ := X ∗ /Γ∗ ≡ T∗ ≡ T∗ and EΓ∗ and notice that TΓ∗ is isomorphic to the dual group of Γ (in the sense of abelian localy compact groups). We evidently have the following gourp isomorphisms X ∼ = Γ∗ × TΓ∗ that are not topological = Γ × TΓ , X ∗ ∼ isomorphisms. We shall consider the following free Hamiltonian: H0,V := −∆ + V (y), V ∈ BC ∞ (X , R), Γ − periodic, (1.1) that describes the evolution of an electron in a periodic crystal without external fields. The above operator has a self-adjoint extension in L2 (X ) that commutes with the translations τγ for any γ ∈ Γ. We can thus apply the Floquet-Bloch theory. For any ξ ∈ X ∗ we can define the operator H0,V (ξ) := Dy + ξ
2 + V (y) that has a self-adjoint extension in L2 (T) that has compact resolvent. Thus its spectrum consists in a growing sequence of finite multiplicity eigenvalues λ1 (ξ) ≤ λ2 (ξ) ≤ ... that are continuous and Γ∗ -periodic functions of ξ. Thus, if we denote by Jk := λk (T∗ ), we can write that σ H0,V
= ∞ ∪ Jk k=1 (1.2) and it follows that this spectrum is absolutely continuous. The above analysis implies the following statement that can be considered as the spectral form of the OnsagerPeierls substitution in a trivial situation (with 0 magnetic field):

⇐⇒ ∃k ≥ 1 0 ∈ σ λ − λk (D) , (1.3) λ ∈ σ H0,V where λk (D) is the image of the multiplication operator with the function λk (ξ) on L2 (X ∗ ) under the conjugation with the Fourier transform (i.e. the Weyl quantization of the symbol λk ) and thus defines a bounded self-adjoint operator on L2 (X ). The problem we are interested in, consists in superposing a magnetic field B in the above situation; let us first consider a constant magnetic field B = (Bjk )1≤j,k≤d with Bjk = −Bkj . Let us recall that using the transversal gauge one can define the following vector potential A = (Aj )1≤j≤d Aj (x) := − 1 X Bjk xk . 2 1≤k≤d We are considering A as a differential 1-form on X so that B is the 2-form given by the exterior differential of A. Then the associated magnetic Hamiltonian is defined as
2 HA,V := D + A + V (y), (1.4) that has also a self-adjoint extension in L2 (X ). The structure of the spectrum of this operator may be very different
of the structure of σ H0,V (for example it may be pure point with infinite multiplicity!), but one expects that modulo some small correction (depending on |B|), for small |B| the property 1.3 with D replaced by D + A should still be true. More precisely it is conjectured that there exists a symbol rk (x, ξ; B, λ) (in fact a BC ∞ (Ξ) function) such that lim rk (x, ξ; B, λ) = 0 in BC ∞ (Ξ) (1.5) |B|→0 2 and for λ in a compact neighborhood of Jk and for small |B| we have that

⇐⇒ 0 ∈ σ λ − λk (D + A(x)) + rk (x, D + A(x); B, λ) , λ ∈ σ H0A,V (1.6) where rk (x, D + A(x); B, λ) is the Weyl quantization of rk (x, ξ + A(x); B, λ). The first rigorous proof of such a result appeared in [30] for a simple spectral band (i.e. λk (ξ) is a nondegenerated eigenvalue of H0,V (ξ) for any ξ ∈ X ∗ and Jk ∩ Jl = ∅, ∀l 6= k). In [17] the authors study this case of a simple spectral band but also the general case, by using Wannier functions. In these references the operator  N appearing on the right hand side of the equivalence 1.6 is considered to act in the Hilbert space l2 (Γ) (with N = 1 for the simple spectral band). In fact we shall prove that for a simple spectral band one can replace l2 (Γ) with L2 (X ). Let us also notice that if one would like to consider also non-constant magnetic fields, then the above Weyl quantization of A(x)-dependent symbols gives operators that are not gauge covariant and thus unsuitable for a physical interpretation. 1.2 The result by Gerard, Martinez and Sjöstrand In [14] the above three authors consider the evolution of an electron (ignoring the spin) in a periodic crystal under the action of exterior non-constant, slowly varying, magnetic and electric fields. More precisely the magnetic field B is defined as B = dA with a vector potential A = (A1 , . . . , Ad ), Aj ∈ C ∞ (X ; R), ∂ α Aj ∈ BC ∞ (X ) ∀|α| ≥ 1, (1.7) and the electric potential is described by φ ∈ BC ∞ (X ; R). (1.8) The Hamiltonian is taken to be PA,φ = X 1≤j≤d
2 Dyj + Aj (ǫy) + V (y) + φ(ǫy), (1.9) with |ǫ| small enough; this defines also a self-adjoint operator in L2 (X ). In this situation, in order to define an effective Hamiltonian, the authors apply an idea of Buslaev [9] (see also [17]); this idea consists in “doubling” the number of variables and separating the periodic part (that is also “rapidly varying”) from the non-periodic part (that is also “slowly varying”). One defines the following operator acting on X 2 : X
2 (1.10) ǫDxj + Dyj + Aj (x) + V (y) + φ(x). PeA,φ := 1≤j≤d Let us point out the very interesting connection between the operators PA,φ and PeA,φ . If we consider the following change of variables: πǫ : X 2 → X 2 , πǫ (x, y) := (x − ǫy, y), then for any tempered distribution F ∈ S ′ (X ) we have that:

PeA,φ ◦ πǫ∗ (δ0 ⊗ F ) = πǫ∗ δ0 ⊗ (PA,φ F ) . (1.11) Buslaev considers the operator PeA,φ as a semi-classical operator valued pseudodifferential operator on X and uses the above remark in order to obtain asymptotic solutions for the equation PA,φ u = λu. Let us develop a little bit this idea in the frame of our previous magnetic pseudodifferential calculus ([25, 19, 20]), presenting some arguments that will be useful in our proofs. We recall that given a symbol a(y, η) defined on Ξ and a potential vector A defined on X , one can define two ’candidates’ for the semi-classical ’magnetic’ quantization of the symbol a: ZZ
x + y

x+y , hη + A u(y)dy dη, ¯ ∀u ∈ S (X ), (1.12) OpA,h (a)u (x) := ei<η,x−y> a 2 2 Ξ that is used in [14] but is not gauge covariant, and ZZ

x+y A , hη u(y)dy dη, ¯ ∀u ∈ S (X ), (1.13) ei<η,x−y> ωh−1 A (x, y)a Oph (a)u (x) := 2 Ξ R with ωA (x, y) := exp{−i [x,y] A}, that has been introduced in [25] and is gauge covariant. In both the above formulae h is a strictly positive parameter. 3 For A = 0 the two quantizations above coincide with the semi-classical Weyl quantization of a denoted by Oph (a). For h = 1 we use the notations OpA (a), OpA (a) and Op(a). Let us come back now to the operators PA,φ and PeA,φ and consider the following notations Aǫ (x) := A(ǫx), We evidently have that   p(x, y, η) pe(x, y, ξ, η)  ◦ pǫ (y, η) := |η|2 + V (y) + φ(x) := |ξ + η|2 + V (y) + φ(x) = p(x, y, ξ + η) := p(ǫy, y, η). ◦ PA,φ = OpAǫ (pǫ ), (1.14) (1.15) while PeA,φ may be thought as being obtained through the following procedure: one computes the Weyl quantization of pe considered as a symbol in the variables (y, η) ∈ Ξ and obtains an operator valued symbol in the variables (x, ξ) ∈ Ξ, that is then quantized by OpA,ǫ :  q(x, ξ) := PeA,φ := Op(e p(x, ., ξ, .) OpA,ǫ (q). (1.16) The rather strange presence of the parameter ǫ in 1.9 has to be considered as a reflection of the semi-classical quantization used in 1.16 and of the formula 1.11. In order to deal with a more natural class of perturbations (thus to eliminate the slow variation hypothesis!) one has to give up the semi-classical quantization in the second step and insert the parameter ǫ in the symbol (like in 1.15). Let us briefly review now the main steps of the argument in [14]. As previously remarked, they consider a magnetic field described by a vector potential satisfying 1.7. They propose to consider the following generalization of PA,φ . • The starting point is a symbol p(x, y, η) that is polynomial in the variable η ∈ X ∗ and satisfies the following relations: P aα (x, y)η α , aα ∈ BC ∞ (X × X ; R), m ∈ N∗ , i) p(x, y, η) = |α|≤m ii) aα (x, y + γ) = aα (x, y), ∀|α| ≤ m, P∀γ ∈ Γ, α aα (x, y)η ≥ c|η|m , ∀(x, y) ∈ X × X , ∀η ∈ X ∗ i.e. p is an elliptic iii) ∃c > 0 such that pm (x, y, η) := |α|=m symbol (let us notice that this condition implies that m is even). • They introduce then the symbols: ◦ pǫ (y, η) := p(ǫy, y, η), ∀ǫ > 0; pe(x, y, ξ, η) := p(x, y, ξ + η). • The interest is focused on the self-adjoint operator in L2 (X ) defined by: ◦ Pǫ := OpAǫ (pǫ ). (1.17) • The auxiliary operator (obtained by doubling the variables) is the self-adjoint operator in L2 (X 2 ) defined by: Peǫ := OpA,ǫ (q),
q(x, ξ) := Op pe(x, ., ξ, .) . • One verifies that a relation similar to 1.11 is still verified:


Peǫ ◦ πǫ∗ δ0 ⊗ F = πǫ∗ δ0 ⊗ (Pǫ F ) , ∀F ∈ S ′ (X ). (1.18) (1.19) In order to define an effective Hamiltonian to describe the spectrum of Pǫ , in [14] the authors bring together three important ideas from the literature on the subject. 1. First, the idea introduced in [9, 15] of “doubling the variables” and considering the operator Peǫ . 2. Then, the use of an operator valued pseudodifferential calculus, idea introduced in [9] and having a rigorous development in [5]. 3. The formulation of a Grushin type problem, as proposed in [17]. 4 In the following we shall discuss the use of the Grushin type problem in our spectral problem. The ideas are the following. First one fixes some compact interval I ⊂ R and some ǫ0 > 0 small enough. Then one has to take into account that T is compact and thus any elliptic self-adjoint operator in L2 (T) is Fredholm and becomes bijective on specific finite co-dimension subspaces. Thus one can find N ∈ N∗ and N functions φj ∈ C ∞ (X × X × X ∗ ) (with 1 ≤ j ≤ N ) that are Γ-periodic in the second variable and such that the following statement is true: Proposition 1.1. If we define: • the operator valued symbols 
 R : Ξ → B L2 (T); CN ,   +
   R− : Ξ → B CN ; L2 (T) , o n R+ (x, ξ)u := hu, φj (x, ., ξ)iL2 (T) 1≤j≤N R− (x, ξ)c := P 1≤j≤N ∈ CN , ∀u ∈ L2 (T), cj φj (x, ., ξ) ∈ L2 (T), ∀c := (cj )1≤j≤N ∈ CN , (1.20) • the associated operators obtained by a semi-classical (non-covariant) quantization:
R± (ǫ) := OpA,ǫ R± . • Then, for λ ∈ I, ǫ ∈ (0, ǫ0 ], the operator: Pǫ :=  Peǫ − λ R− (ǫ) R+ (ǫ) 0 (1.21)  (1.22)

is self-adjoint in L2 X × T ⊕ L2 X ; CN and has an inverse:   E(ǫ, λ) E + (ǫ, λ) E(ǫ, λ) := . E − (ǫ, λ) E −+ (ǫ, λ) (1.23)

−+ −+ Moreover one can prove then that E −+ (ǫ, λ) = OpA,ǫ Eǫλ with Eǫλ ∈ BC ∞ Ξ; B(CN ) uniformly for (ǫ, λ) ∈ (0, ǫ0 ] × I. Then it is easy to prove that:
Proposition 1.2. The operator E −+ (ǫ, λ) is bounded and self-adjoint in L2 X ; CN and we have the following equivalence: λ ∈ σ(Peǫ ) ⇐⇒ 0 ∈ σ(E −+ (ǫ, λ)). (1.24) Finally, in order to pass from Peǫ to Pǫ one makes use of 1.19 and of some unitary transforms of the spaces l2 (Γ) and L2 (X ). More precisely the following two explicit Hilbert spaces are considered in [14]: • V0,ǫ :=    F ∈ S ′ (X ) | F = X fγ δǫγ , ∀(fγ )γ∈Γ ∈ l2 (Γ) γ∈Γ that is evidently unitarily equivalent to l2 (Γ); • L0,ǫ :=    F ∈ S ′ (X 2 ) | F (x, y) = X    , kF kV0,ǫ := kf kl2 (Γ) , v(x)δ0 (x − ǫ(y − γ)), ∀v ∈ L2 (X ) γ∈Γ that is evidently unitarily equivalent to L2 (X ).    , (1.25) kF kL := kvkL2 (X ) , (1.26) The following step is to extend the operators Peǫ and R± (ǫ), considered as pseudodifferential operators, to continuous operators from S ′ (X 2 ) to S ′ (X 2 ) and respectively from S ′ (X ) to S ′ (X ). Then we can directly restrict them to the subspaces L0,ǫ and respectively VN 0,ǫ and due to the Γ-periodicity of the initial symbol p the authors prove as in [14] that they leave these spaces invariant and that the matrix operator Pǫ still defines an invertible selfadjoint operator in L0,ǫ ⊕ VN 0,ǫ having as inverse the coresponding restriction of E(ǫ, λ); moreover the restriction of E −+ (ǫ, λ) is a bounded self-adjoint operator in VN 0,ǫ and we still have the property 1.24. The remark that allows 2 one to end the analysis is that Pǫ acting in L (X ) is unitarily equivalent with the transformed operator Peǫ acting on L0,ǫ and thus 1.24 implies directly the following equivalence: λ ∈ σ(Pǫ ) ⇐⇒ 0 ∈ σ(E −+ (ǫ, λ)), (1.27) with E −+ (ǫ, λ) bounded self-adjoint operator on VN 0,ǫ that can be evidently identified with a bounded self-adjoint 2 N operator on [l (Γ)] . 5 1.3 Summary of our results Let us briefly comment upon our hypothesis. First the magnetic field Bǫ := 2−1 P 1≤j,k≤d 2-form valued smooth function (Bǫ,jk = −Bǫ,kj ) Bǫ,jk dxj ∧ dxk is a closed H.1 For any pair of indices (j, k) between 1 and d we are given a function [−ǫ0 , ǫ0 ] ∋ ǫ 7→ Bǫ,jk ∈ BC ∞ (X ; R) such that lim Bǫ,jk = 0 in BC ∞ (X ; R), for some ǫ0 > 0. ǫ→0 Using the transversal gauge we can define a vector potential Aǫ (described by a 1-form valued smooth function defined on [−ǫ0 , ǫ0 ]) such that Bǫ = dAǫ : Aǫ,j (x) := − X xk 1≤k≤d Z 1 Bǫ,jk (sx)sds. (1.28) 0 We shall not suppose that our vector potential Aǫ satisfy (1.7), but we shall always suppose the following behavior (that results from our hypothesis (H.1) and (1.28)): lim < x >−1 Aǫ,j (x) = 0 in BC ∞ (X ; R). ǫ→0 (1.29) The symbols we are considering are also considered as functions [−ǫ0 , ǫ0 ] ∋ ǫ 7→ pǫ ∈ C ∞ X × X × X ′ satisfying conditions of type S1m with m > 0 uniformly in ǫ ∈ [−ǫ0 , ǫ0 ]:
H.2 ∃m > 0, such that ∀(e α, β) ∈ N2d × Nd , ∃Cαeβ > 0 such that   α e ∂x,y ∂ηβ pǫ (x, y, η) ≤ Cαeβ < η >m−|β|, ∀(x, y, η) ∈ X × X × X ′ , ∀ǫ ∈ [−ǫ0 , ǫ0 ], H.3 lim pǫ = p0 in S1m (X ), ǫ→0
H.4 ∀α ∈ Nd with |α| ≥ 1 we have lim ∂xα pǫ = 0 in S1m (X ), ǫ→0 H.5 pǫ is an elliptic symbol uniformly in ǫ ∈ [−ǫ0 , ǫ0 ], i.e. ∃C > 0, ∃R > 0 such that pǫ (x, y, η) ≥ C|η|m , ∀(x, y, η) ∈ X × X × X ′ with |η| ≥ R, ∀ǫ ∈ [−ǫ0 , ǫ0 ], H.6 pǫ is Γ-periodic with respect to the second variable, i.e. ∀γ ∈ Γ, ∀(x, y, η) ∈ X × X × X ′ , ∀ǫ ∈ [−ǫ0 , ǫ0 ]. pǫ (x, y + γ, η) = pǫ (x, y, η), Let us remark here that the hypothesis (H.3) and (H.4) imply that the limit p0 only depends on the second and third variables ((y, η) ∈ Ξ) and thus we can write pǫ (x, y, η) := p0 (y, η) + rǫ (x, y, η), lim rǫ (x, y, η) = 0 in S1m (X × Ξ). ǫ→0 Let us also notice that our hypothesis (H.3) is not satisfied if we consider a perturbation of the form (adiabatic electric field) φ(ǫy) but is verified for a perturbation of the form ǫφ(y). One can consider a weaker hypothesis, allowing also for the adiabatic electric field perturbation, without losing the general construction of the effective Hamiltonian, but some consequences that we shall prove would no longer be true. We associate to our symbols the two types of symbols proposed in [14]: ◦ pǫ (y, η) := pǫ (y, y, η), The operator we want to study is The auxiliary operator is defined as: peǫ (x, y, ξ, η) := pǫ (x, y, ξ + η). ◦
Pǫ := OpAǫ pǫ . Peǫ := OpAǫ (qǫ ),
qǫ (x, ξ) := Op peǫ (x, ., ξ, .) . 6 (1.30) (1.31) Let us notice that in particular all the above hypothesis are satisfied if we take Bǫ := ǫB with B a magnetic field with components of class BC ∞ (X ), Aǫ = ǫA with A a potential vector associated to B by 1.28 and Pǫ one of the following possible Schrödinger operators: X
2 Pǫ = (1.32) Dyj + ǫAj (y) + V (y) + ǫφ(y), 1≤j≤d Pǫ = OpǫA (< η >) + V (y) + ǫφ(y), q Pǫ = OpǫA (|η|2 ) + 1 + V (y) + ǫφ(y), (1.33) (1.34) where V and φ satisfy 1.1 and 1.8. In the Proposition 9.29 of the Appendix we prove that the difference between 1.33 and 1.34 is of the form OpǫA (qǫ ) with lim qǫ = 0 in S10 (Ξ). ǫ→0 The connection between the operators 1.30 and 1.31 is the following:


Peǫ ◦ π1∗ δ0 ⊗ F = π1∗ δ0 ⊗ (Pǫ F ) , ∀F ∈ S ′ (X ). (1.35) In order to define an effective Hamiltonian for Pǫ we shall apply the same ideas as in [14] with the important remark that the operator valued pseudodifferential calculus we use is not a semi classical calculus but the ’magnetic’ calculus so that all our constructions are explicitly gauge covariant. This fact obliges us to a lot of new technical lemmas in order to deal with this new calculus. Our main result is the following: Theorem 1.3. We assume the Hypothesis [H.1]-[H.6]. For any compact interval I ⊂ R there exists ǫ0 > 0 and
−+ N ∈ N∗ such that ∀λ ∈ I and ∀ǫ ∈ [−ǫ0 , ǫ0 ] there exists a bounded self-adjoint operator E −+ (ǫ, λ) := OpAǫ Eǫ,λ
−+ acting in [V0,1 ]N , where Eǫ,λ ∈ BC ∞ Ξ; B(CN ) uniformly in (ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I and is Γ∗ -periodic in the ∗ variable ξ ∈ X for which the following equivalence is true:
λ ∈ σ(Pǫ ) ⇐⇒ 0 ∈ σ E −+ (ǫ, λ) . (1.36) A direct consequence of the above theorem is a stability property for the spectral gaps of the operator Pǫ of the same type as that obtained in [3, 29, 2] for the Schrödinger operator. Corollary 1.4. Under the hypothesis [H.1]-[H.6], for any compact interval K ⊂ R disjoint from σ(P0 ), there exists ǫ0 > 0 such that ∀ǫ ∈ [−ǫ0 , ǫ0 ] the interval K is disjoint from σ(Pǫ ). In fact we obtain a much stronger result, giving the optimal regularity property but only for ǫ = 0 i.e. at vanishing magnetic field. Proposition 1.5. We denote by dH (F1 , F2 ) the Hausdorff distance between the two closed subsets F1 and F2 of R. Then, under the Hypothesis H.1 - H.7 and I.1 - I.3, there exists a strictly positive constant C such that


∀ǫ ∈ [−ǫ0 , ǫ0 ]. (1.37) dH σ Pǫ ∩ I , σ P0 ∩ I ≤ Cǫ Let us consider now the case of a simple spectral band and generalize the result we discuss previously in this case. By hypothesis we have that τγ P0 = P0 τγ , ∀γ ∈ Γ and we can apply the
Floquet-Bloch theory. We denote by λ1 (ξ) ≤ λ2 (ξ) ≤ . . . the eigenvalues of the operators P0,ξ := Op p0 (·, ξ + ·) that are self-adjoint in L2 (T); they are continuous functions on the torus T∗d := X ∗ /Γ∗ (and they are even C ∞ in the case of a simple spectral band). d Thus σ(P0 ) = ∪ Jj with Jj := λj (T∗d ). Let us consider now the following new Hypothesis: j=1 H.7 There exists k ≥ 1 such that Jk is a simple spectral band for P0 , i.e. ∀ξ ∈ T∗d we have that λk (ξ) is a non-degenerate eigenvalue of P0 and for any l 6= k we have that Jl ∩ Jk = ∅. Proposition 1.6. Assume the hypothesis [H.1]-[H.7] are true and that moreover we have that p0 (y, −η) = p0 (y, η), ∀(y, η) ∈ Ξ. Let I ⊂ R be a compact neighborhood of Jk disjoint from ∪ Jl . Then there exists ǫ0 > 0 such that l6=k ∀(ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I in Theorem 1.3 we can take N = 1 and −+ Eǫ,λ (x, ξ) = λ − λk (ξ) + rǫ,λ (x, ξ), with lim rǫ,λ = 0, in BC ∞ (Ξ), uniformly in λ ∈ I. ǫ→0 (1.38) In the case of a constant magnetic field, under some more assumptions on the symbol pǫ we can have even more information concerning the operator E −+ (ǫ, λ). Proposition 1.7. Assume that the hypothesis [H.1]-[H.7] are true and that Bǫ are constant magnetic fields (for any ǫ) and that the symbols pǫ do not depend on the first variable (x ∈ X ). Then we can complete the conclusion of Theorem 1.3 with the following statements: 1. E −+ (ǫ, λ) is a bounded self-adjoint operator in [L2 (X )]N . −+ 2. The symbol Eǫ,λ is independent of the first variable (y ∈ X ) and is Γ∗ -periodic in the second variable ∗ (ξ ∈ X ). 7 1.4 Overview of the paper The first Section analysis the properties of the auxiliary operator, its self-adjointness and its connection with our main Hamiltonian. The second Section recalls some facts about the Floquet-Bloch theory and the connection between the spectra of the auxiliary operator in L2 (X × X ) and L2 (X × T). In Section 3 we introduce a Grushin type problem and define the principal part of the symbol of the effective Hamiltonian. In Section 4 we use a perturbative method to construct the Peierls effective Hamiltonian and study the connection between its spectrum and the spectrum of the auxiliary operator acting in L2 (X × T). Section 5 is devoted to the definition and study of some auxiliary Hilbert spaces of tempered distributions V0 and L0 . In Section 6 we make a rigorous study of the Peierls Hamiltonian reducing the local study of its spectrum to a spectral problem of the effective Hamiltonian acting in [V0 ]N (the proof of Theorem 1.3). We also consider an application to the stability of spectral gaps (Corollary 1.4). The Lipschitz regularity of the boundaries of the spectral bands at vanishing magnetic field is also obtained as an application of the main Theorem. The 7-th Section is devoted to the case of a simple spectral band (Proposition 1.6). In Section 8 we consider the particular case of a constant magnetic field (Proposition 1.7). An Appendix is gathering a number of facts concerning operator valued symbols and their associated magnetic pseudodifferential operators and some spaces of periodic distributions. Some of the notations and definitions we use in the paper are introduced in this Appendix. 2 The auxiliary operator in L2(X × X ) Let us consider given a family {Bǫ }ǫ∈[−ǫ0 ,ǫ0 ] of magnetic fields on X satisfying Hypothesis H.1 and {Aǫ }ǫ∈[−ǫ0 ,ǫ0 ] an associated family of vector potentials (we shall always work with the vector potentials given by formula (1.23)). Let us also consider a given family of symbols {pǫ }ǫ∈[−ǫ0 ,ǫ0 ] that satisfy the Hypothesis H.2 - H.6. We shall use the following convention: if f is a function defined on X × Ξ, we denote by f ◦ the function defined on Ξ by taking the restriction of f at the subset ∆X ×X × X ∗ where ∆X ×X := {(x, x) ∈ X × X | x ∈ X } is the diagonal of the Cartesian product and we denote by fe the function on Ξ × Ξ defined by the formula fe(X, Y ) ≡ fe(x, ξ, y, η) := f (x, y, ξ + η). It is evident that with the above hypothesis and notations we have that p◦ǫ ∈ S1m (Ξ) and is elliptic, both properties being uniform in ǫ ∈ [−ǫ0 , ǫ0 ]. Then the operator Pǫ := OpAǫ (p◦ǫ ), the main operator we are interested m in, is self-adjoint and lower semi-bounded in L2 (X ) having the domain HA (X ) (the magnetic Sobolev space of ǫ order m defined in Definition 2.4 below); moreover, with the choice of vector potential that we made, it is essentially self-adjoint on the space of Schwartz test functions S (X ). m (X × Ξ) so that by defining qǫ (X) := Taking into account the example 9.9 it follows that {e pǫ }|ǫ|≤ǫ0 ∈ S1,ǫ
Op peǫ (X, .) , we have that for any s ∈ R the following is true

0 {qǫ }|ǫ|≤ǫ0 ∈ S0,ǫ Ξ; B H•s+m (X ); H•s (X ) with the notations introduced at the begining of subsection 9.1 of the Appendix. We can then define the auxiliary
operator Peǫ := OpAǫ qǫ that will play a very important role in our arguments. In the following Proposition we collect the properties of the operator Peǫ that result from Proposition 9.8 and exemple 9.9. Lemma 2.1. With the above notations and under the above Hypothesis H.1 - H.6 we have that 1. For any s ∈ R:





Peǫ ∈ B S X ; Hs+m (X ) ; S X ; Hs (X ) ∩ B S ′ X ; Hs+m (X ) ; S ′ X ; Hs (X ) , uniformly in ǫ ∈ [−ǫ0 , ǫ0 ].





2. Peǫ ∈ B S X 2 ; S X 2 ∩ B S ′ X 2 ; S ′ X 2 , uniformly in ǫ ∈ [−ǫ0 , ǫ0 ]. 3. Peǫ considered as unbounded operator in L2 (X 2 ) with domain S (X 2 ) is symetric for any ǫ ∈ [−ǫ0 , ǫ0 ]. Let us now discuss some different forms of the operator Peǫ that we shall use further. We consider first the isomorphisms: ψ : X 2 → X 2 , ψ(x, y) := (x, x − y); ψ −1 = ψ; (2.1) ∗ χ : X 2 → X 2, ∗ χ(x, y) := (x + y, y); 2 ∗ χ−1 (x, y) = (x − y, y). (2.2) The operators ψ and χ that they induce on L (X ) (ψ u := u ◦ ψ) are evidently unitary. Lemma 2.2. For any u ∈ S (X 2 ), the image Peǫ u may be written in any of the following three equivalent forms: Z Z

−d/2 e Pǫ u (x, y) = (2π) ei<η,y−ỹ> ωAǫ (x, x+ỹ−y) pǫ x−y+(y+ỹ)/2, (y+ỹ)/2, η u(x+ỹ−y, ỹ) dỹ dη, (2.3) X X∗ 8 i  h
◦ 
u(., y) (x), ψ ∗ Peǫ ψ ∗ u (x, y) = OpAǫ (idl ⊗ τy ⊗ idl)pǫ i  h
◦ 
u(x, .) (y). χ∗ Peǫ (χ∗ )−1 u (x, y) = Op(τ−x Aǫ ) (τx ⊗ idl ⊗ idl)pǫ (2.4) (2.5) Proof. Let us fix u ∈ S (X 2 ). Starting from the definitions of Peǫ and qǫ and using oscillating integral techniques, we get Z Z

  Peǫ u (x, y) = (2π)−d/2 ei<ξ,x−x̃> ωAǫ (x, x̃) qǫ (x + x̃)/2, ξ u(x̃, .) (y) dx̃ dξ = −d = (2π) X = (2π)−d e i<ξ,x−x̃> X∗ Z Z Z X X∗ X Z Z X X∗ Z Z ωAǫ (x, x̃) X e i<η,y−ỹ>
 pǫ (x + x̃)/2, (y + ỹ)/2, ξ + η u(x̃, ỹ) dỹ dη dx̃ dξ = X∗ ei<η,y−ỹ> ωAǫ (x, x̃) pǫ (x + x̃)/2, (y + ỹ)/2, η
Z X∗  ei<ξ,x−x̃−y+ỹ> dξ u(x̃, ỹ) dx̃ dỹ dη. By the Fourier inversion theorem the inner oscillating integral is in fact (2π)d/2 δ0 (x−x̃−y+ỹ) = (2π)d/2 [τx−y+ỹ δ0 ](x̃) and we can eliminate the integrals over ξ ∈ X ∗ and over x̃ ∈ X in order to obtain formula (2.3). In order to prove (2.4), we apply (2.3) to ψ ∗ u, that meaning to replace in (2.3) u(x+ỹ−y, ỹ) with u(x+ỹ−y, x−y) and we finally replace y with x − y, obtaining Z Z

ψ ∗ Peǫ ψ ∗ u (x, y) = (2π)−d/2 ei<η,x−y−ỹ> ωAǫ (x, ỹ +y) pǫ y +(x−y + ỹ)/2, (x−y + ỹ)/2, η u(ỹ +y, y) dỹ dη. X X∗ Changing the integration variable ỹ to ỹ − y we obtain Z Z

∗e ∗ −d/2 ψ Pǫ ψ u (x, y) = (2π) ei<η,x−ỹ> ωAǫ (x, ỹ) pǫ (x + ỹ)/2, (x + ỹ)/2 − y, η u(ỹ, y) dỹ dη, X X∗ i.e. (2.4)
−1 The formula (2.5) can be easily obtained in a similar way, starting with (2.3) applied to χ∗ u, i.e. replacing in (2.3) u(x + ỹ − y, ỹ) with u(x − y, ỹ) and finally replacing the variable x ∈ X with x + y; this gives us the result Z Z

∗e ∗ −1 −d/2 χ Pǫ (χ ) u (x, y) = (2π) ei<η,y−ỹ> ωAǫ (x + y, x + ỹ) pǫ x + (y + ỹ)/2, (y + ỹ)/2, η u(x, ỹ) dỹ dη. X X∗ We end the proof of (2.5) by noticing that the following equalities are true (see also (9.30)): ωAǫ (x + y, x + ỹ) = exp{−i   Z = exp i y − ỹ, 0 1 Z Aǫ [x+y,x+ỹ]   Z } = exp i y − ỹ, 0
Aǫ x + (1 − s)y + sỹ ds = exp{−i Z  1
Aǫ (1 − s)(x + y) + s(x + ỹ) ds   Z = exp i y − ỹ, 1 0
τ−x Aǫ } = ω(τ−x Aǫ ) (y, ỹ). 

τ−x Aǫ (1 − s)y + sỹ ds =  = [y,ỹ] The following Corollary is a direct consequence of Lemma 2.2. Corollary 2.3. We have the following two relations between the operators Peǫ and Pǫ : 1. For any v ∈ S ′ (X )

χ∗ Peǫ (χ∗ )−1 (δ0 ⊗ v) = δ0 ⊗ Pǫ v ,

ψ ∗ Peǫ ψ ∗ (v ⊗ δ0 ) = Pǫ v ⊗ δ0 . (2.6) (2.7) 2. If pǫ does not depend on its second variable y ∈ X , then the following equality is true: ψ ∗ Peǫ ψ ∗ = Pǫ ⊗ idl. 9 (2.8) Proof. It is enough to prove (2.6) for any v ∈ S (X ). We choose ϕ ∈ C0∞ (X ) so that {ϕn }n∈N∗ with ϕn (x) := nd ϕ(nx) is a δ0 -sequence. We apply now (2.5) to u := ϕn ⊗ v in order to obtain h 

◦  i χ∗ Peǫ (χ∗ )−1 (ϕn ⊗ v) (x, y) = ϕn (x) Op(τ−x Aǫ ) (τx ⊗ idl ⊗ idl)pǫ v (y). We end the proof of (2.6)
◦ by taking the limit
n◦ → ∞ and noticing that for x = 0 we obtain in the second factor above: (τ0 ⊗ idl ⊗ idl)pǫ = (idl ⊗ idl ⊗ idl)pǫ = pǫ . For (2.7) we use exactly the same preocedure starting from (2.4). The equality (2.8) follows directly from (2.4) under the given assumption on pǫ that implies that (idl ⊗ τy ⊗
◦ idl)pǫ = p◦ǫ . In order to study the continuity and the self-adjointness of Peǫ we need some more function spaces related to the magnetic Sobolev spaces; in order to define these spaces we shall need the family of operator valued symbols {qs,ǫ }(s,ǫ)∈R×[−ǫ0,ǫ0 ] introduced in Remark 9.26 and the associated operators:
Qs,ǫ := OpAǫ qs,ǫ , Q′s,ǫ := Qs,ǫ ⊗ idl. e s,ǫ := ψ ∗ Q′ ψ ∗ with ψ from (2.1) and let us notice that due to Corollary 2.3 (2) the Let us still denote by Q s,ǫ e s,ǫ and Qs,ǫ are in the same relation as the pair Peǫ and Pǫ . operators Q Definition 2.4. For magnetic fields {Bǫ }ǫ∈[−ǫ0 ,ǫ0 ] verifying Hypothesis H.1 and for choices of vector potentials given by (1.28) we define the following spaces. 1. The magnetic Sobolev space of order s ∈ R (as defined in [19]) is  s (X ) := u ∈ S ′ (X ) | Qs,ǫ u ∈ L2 (X ) , HA ǫ (2.9) s (X ) ∀u ∈ HA ǫ (2.10) endowed with the following natural quadratic norm kukHsA ǫ (X ) := kQs,ǫ ukL2 (X ) , that makes it a Hilbert space containing S (X ) as a dense subspace. T s ∞ HAǫ (X ) with the projective limit topology. 2. We shall define also HA (X ) := ǫ s∈R 3. For s ∈ R we consider also the spaces s eA (X 2 ) := H ǫ n e s,ǫ u ∈ L2 (X 2 ) u ∈ S ′ (X 2 ) | Q , (2.11) s eA (X 2 ) ∀u ∈ H ǫ (2.12) endowed with the following natural quadratic norm kukHe s Aǫ (X := 2) e s,ǫ u Q L2 (X 2 ) , o that makes it a Hilbert space containing S (X 2 ) as a dense subspace. e s (X 2 ) onto Hs (X ) ⊗ L2 (X ). Remark 2.5. ψ ∗ is a unitary operator from H Aǫ Aǫ Proof. Let us choose some u ∈ S ′ (X 2 ) and notice that s eA (X 2 ) ⇔ ψ ∗ Q′s,ǫ ψ ∗ u ∈ L2 (X 2 ) ⇔ u∈H ǫ and evidently we have that kψ ∗ ukHs
s (X ) ⊗ L2 (X ) Qs,ǫ ⊗ idl ψ ∗ u ∈ L2 (X 2 ) ⇔ ψ ∗ u ∈ HA ǫ Aǫ (X )⊗L 2 (X ) = kukH es Aǫ (X 2) . We can prove now a continuity property of the operator Peǫ on the spaces defined by (2.11).
e s (X 2 ) uniformly for ǫ ∈ [−ǫ0 , ǫ0 ]. e s+m (X 2 ); H Lemma 2.6. For any s ∈ R we have that Peǫ ∈ B H Aǫ Aǫ 10 Proof. Due to (2.4) and Remark 2.5 it is enough to prove that the application i  h
◦  u(., y) (x) ∈ S (X 2 ) S (X 2 ) ∋ u 7→ OpAǫ (idl ⊗ τy ⊗ idl)pǫ s+m s (X ) ⊗ L2 (X ) to HA has a continuous extension from HA (X ) ⊗ L2 (X ) uniformly for ǫ ∈ [−ǫ0 , ǫ0 ]. But ǫ ǫ
◦ (idl ⊗ τy ⊗ idl)pǫ (x, ξ) = pǫ (x, x − y, ξ) and the family {pǫ (x, x − y, ξ)}(y,ǫ)∈X ×[−ǫ0,ǫ0 ] of symbols (in the variables (x, ξ) ∈ Ξ) is a bounded subset of S1m (Ξ) (due to our Hypothesis). Applying the continuity properties of magnetic pseudodifferential operators in magnetic Sobolev spaces proved in [19] we conclude that there exists a constant C > 0 such that  2
◦  u(., y) s ≤ C 2 ku(, .y)k2Hs+m (X ) OpAǫ (idl ⊗ τy ⊗ idl)pǫ HAǫ (X ) Aǫ for any u ∈ S (X 2 ), ∀y ∈ X and ∀ǫ ∈ [−ǫ0 , ǫ0 ]. We end then the proof by integrating the above inequality with respect to y ∈ X . In order to prove the self-adjointness of Peǫ in L2 (X 2 ) we use the following Remark. Remark 2.7. Suppose given r ∈ S1m (X × Ξ). Then evidently r(., y, .) ∈ S1m (Ξ) uniformly for y ∈ X . If B is a magnetic field on X with components of class BC ∞ (X ) and A an associated vector potential having components ∞ of class Cpol (X ) we define the magnetic pseudodifferential operator with parameter y ∈ X Z Z

Ru (x, y) := (2π)−d/2 ei<ξ,x−x̃> ωA (x, x̃) r (x + x̃)/2, y, ξ u(x̃, y) dx̃ dξ, ∀u ∈ S (X 2 ), ∀(x, y) ∈ X 2 . X X∗ (2.13) A straightforward modification of the arguments from [19], and denoting by Rǫ the operator defined as above in (2.13) with a vector potential Aǫ , allows to prove that

s+m s Rǫ ∈ B S (X 2 ); S (X 2 ) ∩ B HA (X ) ⊗ L2 (X ) , ∀s ∈ R. (2.14) (X ) ⊗ L2 (X ); HA ǫ ǫ Moreover, if r is elliptic, then for any u ∈ L2 (X 2 ) and any s ∈ R we have the equivalence relation: s+m s u ∈ HA (X ) ⊗ L2 (X ). (X ) ⊗ L2 (X ) ⇐⇒ Rǫ u ∈ HA ǫ ǫ (2.15) e m (X 2 ). It is essentially self-adjoint on Proposition 2.8. Peǫ is a self-adjoint operator in L2 (X 2 ) with domain H Aǫ 2 S (X ). e m (X 2 ) is well defined in L2 (X 2 ). Moreover we know Proof. Following Lemma 2.6 the operator Peǫ with domain H Aǫ e m (X 2 ) for its own norm-topology by Lemma 2.1 (3) that Peǫ is symmetric when defined on S (X 2 ) that is dense in H Aǫ e m (X 2 ). so that we conclude that Peǫ is symmetric as defined on H Aǫ Considering now equation (2.4) and Remark 2.5 it follows that Peǫ as considered in the hypothesis of the Proposition is self-adjoint if and only if the operator Rǫ defined by (2.13) with a symbol rǫ (x, y, ξ) := pǫ (x, x − y, ξ) m is self-adjoint in L2 (X 2 ) with domain HA (X ) ⊗ L2 (X ). Using the symmetry of Peǫ and its unitary equivalence with ǫ Rǫ we conclude that Rǫ is also symmetric on its domain. Let us fix some v ∈ D(R∗ǫ ); thus we know that v ∈ L2 (X 2 ) and there exists some f ∈ L2 (X 2 ) such that (Rǫ u, v)L2 (X 2 ) = (u, f )L2 (X 2 ) , ∀u ∈ S (X 2 ). s We conclude that Rǫ v = f in S ′ (X 2 ) and due to (2.15) we have that v ∈ HA (X ) ⊗ L2 (X ). In conclusion we get ǫ ∗ that Rǫ = Rǫ . e m (X 2 ) and the continuity property The last statement of the Proposition follows from the density of S (X 2 ) in H Aǫ proved in Lemma 2.6. 3 The auxiliary operator in L2 X × T
We begin with some elements concerning the Floquet-Bloch theory. We use the notations from the beginning of subsection 9.2 of the Appendix.
Definition 3.1. SΓ′ X 2 × X ∗ :=
 := v ∈ S ′ X 2 × X ∗ | v(x, y + γ, θ) = ei<θ,γ>v(x, y, θ) ∀γ ∈ Γ, v(x, y, θ + γ ∗ ) = v(x, y, θ) ∀γ ∗ ∈ Γ∗ , 11 
endowed with the topology induced from S ′ X 2 ×
X ∗ .


Definition 3.2. F0 X 2 × X ∗ := SΓ′ X 2 × X ∗ ∩ L2loc X 2 × X ∗ ∩ L2 X × E × E ∗ endowed with the quadratic norm s Z Z Z
(3.1) |E ∗ |−1 |v(x, y, θ)|2 dx dy dθ, ∀v ∈ F0 X 2 × X ∗ , kvkF0 := X
∗ E∗ E that makes F0 X 2 × X into a Hilbert space.

We evidently have a continuous embedding of F
0 X 2 × X ∗ into S ′ X 2 × X ∗ . Lemma 3.3. The following map defined on S X 2 : X

UΓ u (x, y, θ) := ei<θ,γ>u(x, y − γ), ∀(x, y) ∈ X 2 , ∀θ ∈ X ∗ , ∀u ∈ S X 2 , (3.2) γ∈Γ

extends as a unitary operator UΓ : L2 X 2 → F0 X 2 × X ∗ . The inverse of the above operator is explicitely given by Z

WΓ v (x, y) := |E ∗ |−1 v(x, y, θ)dθ, ∀(x, y) ∈ X 2 , ∀v ∈ F0 X 2 × X ∗ . (3.3) E∗

Proof. Let us notice that for u ∈ S X 2 the series in (3.2) converges in E X 2 × X ∗
and its sum, that we denoted
by UΓ u evidently verifies the two relations that characterize the subspace SΓ′ X 2 ×X ∗ as subspace of S ′ X 2 ×X ∗ :

UΓ u (x, y, θ + γ ∗ ) = UΓ u (x, y, θ), ∀(x, y, θ) ∈ X 2 × X ∗ , ∀γ ∗ ∈ Γ∗ , (3.4) X

UΓ u (x, y + α, θ) = ei<θ,γ>u(x, y + α − γ) = ei<θ,α> UΓ u (x, y, θ), ∀(x, y, θ) ∈ X 2 × X ∗ , ∀α ∈ Γ. (3.5) γ∈Γ

In particular, considering a fixed pair (x, y) ∈ X 2 , we obtain an element UΓ u (x, y, .) ∈ S T∗,d and due to
Remark 9.11 we can compute its Fourier series that converges in S T∗,d : Z X



∗ −1 i<θ,γ> d UΓ u (x, y, θ) = U u (x, y, γ) = |E | Ud u (x, y, γ)e , e−i<θ,γ> UΓ u (x, y, θ) dθ. (3.6) Γ Γ E∗ γ∈Γ We also have the Parseval equality X γ∈Γ
Ud Γ u (x, y, γ) 2 ∗ −1 = |E | Z
2 UΓ u (x, y, θ) dθ. E∗ (3.7)
Comparing (3.2) and (3.6) we conclude that Ud Γ u (x, y, γ) = u(x, y − γ); replacing then in (3.7) we get the equality Z X
2 2 |u(x, y − γ)| . (3.8) UΓ u (x, y, θ) dθ = |E ∗ |−1 E∗ γ∈Γ If we integrate the above equality over X × E we obtain that s sZ Z Z Z Z
2 |E ∗ |−1 UΓ u (x, y, θ) dx dy dθ = |u(x, y)|2 dx dy. X E E∗ X (3.9) X


We conclude that UΓ u ∈ F0 X 2 × X ∗ and UΓ extends to an isometry UΓ : L2 X 2 → F0 X 2 × X ∗ . Thus, to end
our proof it is enough to prove that the operator WΓ is an inverse for UΓ . Let us consider some v ∈ F0 X 2 × X ∗ , then for almost every x ∈ X and y ∈ E we can write that X v(x, y, θ) = v̂γ (x, y)ei<θ,γ> , in L2 (E ∗ ), (3.10) γ∈Γ with ∗ −1 v̂γ (x, y) = |E | Z E∗ e −i<θ,γ> ∗ −1 v(x, y, θ)dθ = |E | Z E∗ v(x, y − γ, θ)dθ = v̂0 (x, y − γ). (3.11) Using the above identity (3.11) and the Parseval equality we notice that we
have the following equalities that finally imply that WΓ v ∈ L2 (X 2 ) and the fact that the map WΓ : F0 X 2 × X ∗ → L2 (X 2 ) is an isometry.     Z Z Z Z XZ XZ   |v̂0 (x, y)|2 dx dy = kWΓ vk2L2 (X 2 ) = |v̂0 (x, y − γ)|2 dy  dx = |v̂γ (x, y)|2 dy  dx = X X X γ∈Γ E X 12 γ∈Γ E = Z X   Z X E γ∈Γ  |v̂γ (x, y)| dy  dx = 2 Z Z ∗ −1 E X |E | Z E∗ 2 |v(x, y, θ)| dθ dy dx = kvk2 Moreover, for any u ∈ S (X 2 ) we have that   Z Z X
∗ −1 i<θ,γ> ∗ −1   WΓ UΓ u (x, y) = |E | e u(x, y − γ) dθ = |E | E∗  F0 X 2 ×X ∗ ∀(x, y) ∈ X 2 . u(x, y)dθ = u(x, y), E∗ γ∈Γ
. Lemma 3.4. With the above definitions for the operators UΓ and WΓ we have that 1. UΓ admits a continuous extension to S ′ (X 2 ) with values in SΓ′ (X 2 × X ∗ ). 2. WΓ admits a continuous extension to SΓ′ (X 2 × X ∗ ) with values in S ′ (X 2 ). 3. We have the equalities: UΓ WΓ = idlSΓ′ (X 2 ×X ∗ ) , WΓ UΓ = idlS ′ (X 2 ) . Proof. Let us consider a tempered distribution u ∈ S ′ (X 2 ); in order to prove the convergence of the series (3.2) in the sense of tempered distributions on X 2 × X ∗ ) we choose a test function ϕ ∈ S (X 2 × X ∗ ) and notice that for any ν ∈ N there exist a seminorm k.kν on ϕ ∈ S (X 2 × X ∗ ) and a seminorm k.k′ν on S ′ (X 2 ) such that the following is true:   Z ei<θ,γ> ϕ(x, y + γ, θ)dθ ≤ kuk′ν kϕkν < γ >−ν . (3.12) ei<θ,γ>u(x, y − γ), ϕ(x, y, θ) = u(x, y), X∗ This last inequality implies the convergence of the series (3.2) in the sense of tempered distributions on X 2 × X ∗ ) and the fact that the map UΓ : S ′ (X 2 ) → S ′ (X 2 × X ∗ ) is continuous. The fact that UΓ u belongs to SΓ′ (X 2 × X ∗ ) results either by a direct calculus or by approximating with test functions. Let us fix now some v ∈ SΓ′ (X 2 × X ∗ ); then, for any test function ϑ ∈ S (X 2 ), the application vϑ : S (X ) → C, ∀ϕ ∈ S (X ∗ ) hvϑ , ϕi := hv, ϑ ⊗ ϕi, (3.13)
is a Γ∗ -periodic tempered distribution that can be canonically identified with an element from S ′ T . We define WΓ v by hWΓ v, ϑi := |E ∗ |−1 hvϑ , 1iT∗,d , ∀ϑ ∈ S (X 2 ). (3.14) ′ 2 ∗ ′ 2 It is straightforward to verify that WΓ v ∈ S ′ (X 2 ) and that the application W
Γ : SΓ (X × X ) → S (X ) is linear 2 ∗ and continuous. Moreover it is evident that for the case v ∈ F0 X × X , the definition (3.14) coincides with (3.3). The equality WΓ UΓ = idlS ′ (X 2 ) results from the one valid on the test functions by the density of S (X 2 ) in ′ ′ 2 ∗ 2 S (X 2 ). In order to prove the other equality we notice that for
any v ∈ SΓ (X × X ) and any ϑ ∈ S (X ) the ′ ∗,d tempered distribution vϑ defined in (3.13) belongs to
S T and thus may be written as the sum of a Fourier series converging as tempered distribution in S ′ X ∗ : vϑ = |E ∗ |−1 X vϑ , e−i<.,γ> T∗,d ei<.,γ> . (3.15) γ∈Γ But, from (3.2) we have that X

ei<θ,γ> WΓ v (x, y − γ), UΓ WΓ v (x, y, θ) = γ∈Γ in S ′ (X 2 × X ∗ ). Let us notice that due to (3.14) we can write

WΓ v (x, y − γ), ϑ(x, y) = WΓ v (x, y), ϑ(x, y + γ) = |E ∗ |−1 v(idl⊗τ−γ )ϑ , 1 On the other hand, from (3.13) we deduce that ∀ϕ ∈ S (X ∗ ) one has that 

v(idl⊗τ−γ )ϑ , ϕ = v, idl ⊗ τ−γ ϑ ⊗ ϕ = idl ⊗ τγ ⊗ idl v, ϑ ⊗ ϕ =
= v, ϑ ⊗ e−i<.,γ> ϕ = vϑ , e−i<.,γ> ϕ . 13 T∗,d . (3.16) Let us recall the relation between a Γ∗ -periodic tempered distribution and the on the torus P distribution it induces (as described in the Remark 9.10): if ψ ∈ C0∞ (X ∗ ) is choosen such that τγ ∗ ψ = 1 on X ∗ (such a choice is γ ∗ ∈Γ∗ evidently possible), then for any ρ ∈ S (T∗,d ) we have that v(idl⊗τ−γ )ϑ , ρ T∗,d = v(idl⊗τ−γ )ϑ , ψρ X , hvϑ , ρiT∗,d = hvϑ , ψρiX . Thus it follows that (3.16) implies that v(idl⊗τ−γ )ϑ , 1 T∗,d = v(idl⊗τ−γ )ϑ , ψ We conclude that U Γ WΓ v
ϑ = |E ∗ |−1 X X = vϑ , e−i<.,γ>ψ vϑ , e−i<.,γ> T∗,d X = vϑ , e−i<.,γ> T∗,d . ei<.,γ> = vϑ , γ∈Γ so that finally we obtain that UΓ WΓ = idlSΓ′ (X 2 ×X ∗ ) . Lemma 3.5. Let Peǫ be the operator defined in Section 1 and Peǫ,Γ := Peǫ ⊗ idl.
1. Peǫ,Γ is a linear continuous operator in SΓ′ X 2 × X ∗ .
2. UΓ Peǫ = Peǫ,Γ UΓ on S ′ X 2 .



Proof. We evidently have that Peǫ,Γ : S X 2 × X ∗ → S X 2 × X ∗ and Pe
ǫ,Γ : S ′ X 2 × X ∗ → S ′ X 2 × X ∗ are linear and continuous. It is thus sufficient to prove that ∀v ∈ S X 2 × X ∗ we have that: h  i h i  Peǫ ⊗ idl v (x, y, θ + γ ∗ ) = Peǫ ⊗ idl (idl ⊗ idl ⊗ τ−γ ∗ )v (x, y, θ), ∀(x, y, θ) ∈ X 2 × X ∗ , ∀γ ∗ ∈ Γ∗ (3.17) and h  i h  i Peǫ ⊗ idl v (x, y + γ, θ) = Peǫ ⊗ idl (idl ⊗ τ−γ ⊗ idl)v (x, y, θ), ∀(x, y, θ) ∈ X 2 × X ∗ , ∀γ ∈ Γ. (3.18) While the equality (3.17) is obvious, for the equality (3.18) we use (2.3) (with ỹ replaced by ỹ + γ) and the Γ-periodicity of pǫ with respect to the second variable. For the second point of the Lemma we use (3.2) and (3.18) and notice that for any u ∈ S (X 2 ) and for any (x, y, θ) ∈ X 2 × X ∗ we have that  h    X X
i
ei<θ,γ> Peǫ u (x, y − γ) = UΓ Peǫ u (x, y, θ) Peǫ,Γ UΓ u (x, y, θ) = ei<θ,γ> Peǫ idl ⊗ τγ u (x, y) = γ∈Γ γ∈Γ We shall study the self-adjointness of Peǫ acting in some new spaces of functions that are periodic in one argument. Let us first consider the operator Peǫ,Γ . e s,ǫ from Definition 2.4 (2) we define the operator Q e s,ǫ,Γ := Q e s,ǫ ⊗ idl. Definition 3.6. Recalling the operator Q e As we have already noticed about the Definition 2.4 (2), the operator Qs,ǫ is obtained from Qs,ǫ by ”doubling the variable“ in the same way as Peǫ is associated to Pǫ . Then we have a result similar to Lemma 3.5 and deduce
e s,ǫ,Γ is a linear continuous operator in S ′ X 2 × X ∗ . that Q Γ Definition 3.7. For any s ∈ R we define n
o

e s,ǫ,Γ v ∈ F0 X 2 × X ∗ Fs,ǫ X 2 × X ∗ := v ∈ SΓ′ X 2 × X ∗ | Q that is evidently a Hilbert space for the quadratic norm kvkFs,ǫ := e s,ǫ,Γ v Q F0
∀v ∈ Fs,ǫ X 2 × X ∗ . (3.19)


e s X 2 → Fs,ǫ X 2 × X ∗ is e s X 2 be the Hilbert space defined in (2.11). Then UΓ : H Lemma 3.8. Let H Aǫ Aǫ unitary. 14 

e s,ǫ u ∈ L2 (X 2 ). We denote by v := UΓ u ∈ e s X 2 ; thus we know that u ∈ S ′ X 2 and Q Proof. Let us pick u ∈ H Aǫ

e s,ǫ = Q e s,ǫ,Γ UΓ on S ′ X 2 , and from Lemma 3.3 we have SΓ′ X 2 × X ∗ . From Lemma 3.5 we deduce that UΓ Q


e s,ǫ,Γ v ∈ F0 X 2 × X ∗ so that v ∈ Fs,ǫ X 2 × X ∗ . Moreover, e s,ǫ u ∈ F0 X 2 × X ∗ . We conclude that Q that UΓ Q e s,ǫ,Γ v kUΓ ukFs,ǫ = kvkFs,ǫ = Q F0 e s,ǫ u = UΓ Q F0 e s,ǫ u = Q L2 (X 2 ) = kukHe s , Aǫ

e s X 2 → Fs,ǫ X 2 × X ∗ is isometric. In order to prove its surjectivity we consider v ∈ implying that UΓ : H Aǫ


e s,ǫ,Γ v ∈ F0 X 2 × X ∗ . Let us define u := WΓ v ∈ S ′ (X 2 ) Fs,ǫ X 2 × X ∗ ; then v ∈ SΓ′ X 2 × X ∗ and also Q
e s,ǫ,Γ = Q e s,ǫ WΓ on S ′ X 2 × X ∗ , so that we have (Lemma 3.4). Then we apply Lemma 3.5 and deduce that WΓ Q Γ
e s,ǫ,Γ v ∈ L2 (X 2 ) and we conclude that u ∈ H e s X 2 and UΓ u = v. e s,ǫ u = WΓ Q Q Aǫ
Lemma 3.9.
The operator Peǫ,Γ defined on Fm,ǫ X 2 × X ∗ is self-adjoint as operator acting in the Hilbert space F0 X 2 × X ∗ .
e m X 2 . By Lemma 3.8 Proof. By Proposition 2.8, Peǫ,Γ is self-adjoint as operator acting in L2 (X 2 ), with domain H Aǫ


e m X 2 → Fm,ǫ X 2 × X ∗ are unitary.Finally, by Lemma the operators UΓ : L2 (X 2 ) → F0 X 2 × X ∗ and UΓ : H Aǫ
e m X 2 , so that Peǫ,Γ is unitarily equivalent with Peǫ . 3.5 we have that Peǫ,Γ UΓ = UΓ Peǫ on H Aǫ We shall need some more function spaces in order to come back to the operator Peǫ . Definition 3.10. Let θ ∈ X ∗ and s ∈ R.


1. Sθ′ X 2
:= u ∈ S ′ X 2 | idl ⊗ τ−γ u = ei<θ,γ>u ∀γ ∈ Γ endowed with the topology induced from S ′ X2 . n


o s e s,ǫ u ∈ L2 X × E 2. Hθ,ǫ endowed with the following quadratic norm X 2 := u ∈ Sθ′ X 2 | Q e s,ǫ u kukHsθ,ǫ := Q

s X2 . 3. Kǫs X 2 := H0,ǫ L2 (X ×E) ,
s X2 . ∀u ∈ Hθ,ǫ (3.20)
As we already noticed in the proof of Lemma 3.5, for any u ∈ S ′ X 2 the following equality holds:

idl ⊗ τ−γ Peǫ u = Peǫ idl ⊗ τ−γ u, ∀γ ∈ Γ.

e s,ǫ leave the space S ′ X 2 invariant. We shall use the notation S0′ X 2 ≡ It follows that the operators Peǫ and Q θ
SΓ′ X 2 . Let us also
on ǫ and will be
notice that for s = 0 the
spaces defined in (2) and (3) above do not depend denoted by Hθ X 2 and respectively by K X 2 ; this last one may be identified with L2 X × T . Lemma Let
us consider by
(2.1). Then for any s ∈ R the adjoint ψ ∗ is a unitary operator
the map ψ defined

3.11. 2 s s 2 s Kǫ X → HAǫ X ⊗ L T . In particular Kǫ X 2 is a Hilbert space for the norm (3.20) having S X × T as a dense subspace.
∗ ′ 2 Proof. The
case s = 0 is straightforward since the map ψ leaves invariant the space SΓ X and for any u ∈ S X × T we have that   Z Z Z Z 2 kψ ∗ ukL2 (X ×T) = |u(x, x − y)|2 dy dx = |u(x, x + y)|2 dy dx = X = Z Z X E X 2 x−E  |u(x, y)| dy dx = Z Z X E −E  |u(x, y)| dy dx = kuk2L2 (X ×T) .
For any s ∈ R \ {0} we fix some u ∈ SΓ′ X 2 and notice that: 2

e s,ǫ u ∈ L2 (X × T) ⇔ ψ ∗ Q′ ψ ∗ u ∈ L2 (X × T) ⇔ Qs,ǫ ⊗ idl ψ ∗ u ∈ L2 (X × T) ⇔ u ∈ Kǫs X 2 ⇔ Q s,ǫ

s ⇔ ψ ∗ u ∈ HA X ⊗ L2 T ǫ and we also have that kψ ∗ ukHs Aǫ (X )⊗L 2 (T) = kukKsǫ .


s X ⊗ L2 T is a Hilbert space with S X × T a dense The last statement becomes obvious noticing that HA ǫ subspace in it that is invariant under the map ψ. 15 

Lemma 3.12. For any θ ∈ X ∗ and s ∈ R we have that the operator Tθ : S X 2 → S X 2 defined by
Tθ u (x, y) := ei<θ,x−y> u(x, y),


s s induces a unitary operator Hθ,ǫ X 2 is a Hilbert space containing X 2 → Kǫs X 2 . In particular we have that Hθ,ǫ

 S X 2 := Tθ−1 S X × T as a dense subspace. Proof. Let us prove first that for any θ ∈ X ∗ we have the equality:
on S ′ X 2 . Peǫ Tθ = Tθ Peǫ , (3.21)
It is clearly enough to prove it on S X 2 ; but in this case it results directly from (2.3) because (x+ ỹ−y)− ỹ = x−y. Then the following equality also follows
e s,ǫ , e s,ǫ Tθ = Tθ Q on S ′ X 2 . (3.22) Q

e s,ǫ leaves invariant We notice further that Tθ takes the space Sθ′ X 2 into the space SΓ′ X 2 , while the operator Q

′ 2 ′ 2 both spaces Sθ X
and SΓ X . For u ∈ Sθ′ X 2 we have the equivalence relations:
s e s,ǫ u ∈ L2 (X × T) ⇔ Q e s,ǫ Tθ u ∈ L2 (X × T) ⇔ e s,ǫ u ∈ L2 (X × E) ⇔ Tθ Q X2 ⇔ Q u ∈ Hθ,ǫ ⇔ Tθ u ∈ Kǫs X 2
and the equality kTθ ukKsǫ = kukHsθ,ǫ .

The last statement is obvious since Lemma 3.11 implies that Kǫs X 2 is a Hilbert space having S X × T as a dense subspace. 

 Lemma 3.13. For any s ∈ R we have that Peǫ ∈ B Kǫs+m X 2 ; Kǫs X 2 uniformly for ǫ ∈ [−ǫ0 , ǫ0 ]. Proof. We have seen that:

• S X × T is a dense subspace of Kǫs+m X 2 ,


s X ⊗ L2 (T) is a unitary operator that leaves S X × T invariant. • ψ ∗ : Kǫs X 2 → HA ǫ It is thus enough to prove that ∀s ∈ R, ∃Cs > 0 such that: ψ ∗ Peǫ ψ ∗ u
HsAǫ X ⊗L2 (T) ≤ Cs kuk
Hs+m X ⊗L2 (T) Aǫ ,
∀u ∈ S X × T , ∀ǫ ∈ [−ǫ0 , ǫ0 ]. (3.23) Formula (2.4) in Lemma 2.2 implies the equality: Z Z  x + ỹ x + ỹ   
∗e ∗ −d ψ Pǫ ψ u (x, y) = (2π) , − y, η u(ỹ, y) dỹ dη, ∀u ∈ S X 2 . ei<η,x−ỹ> ωAǫ (x, ỹ) pǫ 2 2 ∗ X X (3.24)
But let us notice that the integral in (3.24) is well defined for any u ∈ S X × T so that we can extend it to such functions
(considered as periodic smooth functions in the second variable) either by duality and a computation

in S ′ X 2 or by approximating with functions from S X 2 with respect to the topology induced from S ′ X 2 . By the same time, the properties of the oscillating integral defining the right side of (3.24) allow to conclude that

 ∗e ∗ ψ Pǫ ψ ∈ B S X × T ; S X × T . Considering now y ∈ X in (3.24) as a parameter, the usual properties of magnetic pseudodifferential operators (see [19]) imply that ∀ǫ ∈ [−ǫ0 , ǫ0 ], ∃Cs > 0 such that: ψ ∗ Peǫ ψ ∗ u(., y) 2 HsAǫ X
≤ Cs2 ku(., y)k2 s+m HAǫ X
,
∀(y, ǫ) ∈ X × [−ǫ0 , ǫ0 ], ∀u ∈ S X × T . (3.25) Integrating the above inequality for y ∈ T we obtain (3.23).

Proposition 3.14. Peǫ is
a self-adjoint operator in K X 2 ≡ L2 (X × T) with domain Kǫm X 2 ; it is essentially self-adjoint on S X × T . 16 

s Proof. Considering Lemma 3.11 that implies that for any s ∈ R the operator ψ ∗ : Kǫs X 2 → HA X ⊗ L2 (T) ǫ
is unitary and leaves invariant the subspace S X × T , it will be enough to prove
that ψ ∗ Peǫ ψ ∗ is self-adjoint in
m X ⊗ L2 (T) and essentially self-adjoint on S X × T . L2 (X × T) with domain HA ǫ Due to the
arguments in the proof of Lemma 3.13 we know that ψ ∗ Peǫ ψ ∗ is well defined in L2 (X × T) with
m X ⊗ L2 (T) and on S X × T is defined by the equality (3.24). A straightforward check using (3.24) domain HA ǫ

m X ⊗ L2 (T). As we shows that the operator ψ ∗ Peǫ ψ ∗ is symmetric on S X × T , that is a dense subspace of HA ǫ

m X ⊗ L2 (T); L2 (X × T) , it follows that ψ ∗ Peǫ ψ ∗ is symmetric on its domain too. In know that ψ ∗ Peǫ ψ ∗ ∈ B HA ǫ
order to prove its self-adjointness let us fix some v ∈ D [ψ ∗ Peǫ ψ ∗ ]∗ ; it follows that v ∈ L2 (X × T) and it exists some f ∈ L2 (X × T) such that we have the equality  
= (u, f )L2 (X ×T), ∀u ∈ S X × T . ψ ∗ Peǫ ψ ∗ u, v 2 L (X ×T)

Thus ψ ∗ Peǫ ψ ∗ v = f as elements of S ′ X × T ≡ SΓ′ X 2 . We notice that the Remark 2.7 remains true if we
m X ⊗ L2 (T) and thus v belongs to the domain of ψ ∗ Peǫ ψ ∗ . replace X 2 by X × T and thus we have that v ∈ HA ǫ The last statement clearly follows from the above results. We shall present now a connection between the operators defined
in the Propositions 2.8 and 3.14. Proposition 3.15. Considering Peǫ as operator acting in S ′ X 2 we shall denote by Peǫ′ the self-adjoint operator
e m X 2 (as in Proposition 2.8) and by Pe ′′ the self-adjoint operator that it that it induces in L2 (X 2 ) with domain H ǫ Aǫ
induces in L2 (X × T) with domain Kǫm X 2 (as in Proposition 3.14). Then we have the equality: σ Peǫ′

= σ Peǫ′′ . (3.26) e ′ −1 e e Proof. From the arguments in the proof of Lemma 3.9 we deduce

that UΓ Pǫ UΓ = Pǫ,Γ := Pǫ ⊗ idl, that is a 2 ∗ 2 ∗ self-adjoint operator in F0 X × X with domain Fm,ǫ X × X . On the other side from Lemma 3.12 (and the arguments in its proof) we deduce that for any θ ∈ X ∗ the operator

m Peǫ,θ := Tθ−1 Peǫ′′ Tθ is the self-adjoint operator associated to Peǫ in Hθ X 2 , having the domain Hθ,ǫ X2 .

We shall consider the spaces F0 X 2 × X ∗ and Fm,ǫ X 2 × X ∗ as direct integrals of Hilbert spaces over the dual torus; more precisely: Z ⊕ Z ⊕



m F0 X 2 × X ∗ ∼ Hθ X 2 dθ, Fm,ǫ X 2 × X ∗ ∼ X 2 dθ. Hθ,ǫ = = T∗,d T∗,d Taking into account that:
Peǫ,Γ u (x, y, θ) = and the function: T∗,d ∋ θ 7→ Peǫ,θ + i
−1
(Peǫ,θ u)(., ., θ) (x, y),
∀u ∈ Fm,ǫ X 2 × X ∗ ,
∈ B Hθ ; Hθ is measurable, we can write: Peǫ,Γ = Z ⊕ T∗,d Peǫ,θ dθ. We can now apply Theorem XIII.85 (d) from [32] in order to conclude that we have the equivalence: o n

λ ∈ σ Peǫ,Γ ⇐⇒ ∀δ > 0, θ ∈ T∗,d | σ Peǫ,θ ∩ (λ − δ, λ + δ) 6= ∅ > 0. (3.27) (3.28)


Let us notice that σ Peǫ,θ is independent of θ ∈ T∗,d and deduce that σ Peǫ,Γ = σ Peǫ′′ . But the conclusion of the

first paragraph in this proof implies that σ Peǫ,Γ = σ Peǫ′ and we finish the proof.

We shall end up this section with a result giving a connection between the spaces: Kǫs X 2 , S X ; Hs (T) and
S ′ X ; Hs (T) . We start with a technical Lemma. Lemma 3.16. Let B be a magnetic field with components of class BC ∞ (X ) and A an associated vector ∞ potential with components of class Cpol . Let us consider a symbol q ∈ S1s (Ξ) for some s ∈ R. We dee := ψ ∗ Q′ ψ ∗ , where ψ is defined by (2.1). Then we have that note by Q := OpA (q), Q′ := Q ⊗ idl and Q


e ∈ B S X ; Hs (T) ; S X ; L2 (T) uniformly for q varying in bounded subsets of S1s (Ξ) and for B varying in Q bounded subsets of BC ∞ (X ). 17 
Proof. On S X ; Hs (T) we shall use the following family of seminorms: |u|s,l := sup |α|≤l Z <x> 2l ∂xα u X
(x, .) 2 Hs (T) 1/2 dx ,
l ∈ N, u ∈ S X ; Hs (T) . (3.29) We have to prove that for any k ∈ N there exist l ∈ N and C > 0 such that: e Qu 0,k
∀u ∈ S X ; Hs (T) , ≤ C|u|s,l , (3.30) uniformly for q varying in bounded subsets of S1s (Ξ) and for B varying in bounded subsets of BC
∞ (X ). Using (2.3) and (2.8), or a straightforward computation, we obtain that for any u ∈ S X × T : Z Z 
ỹ − y  −d e , η u(x − y + ỹ, ỹ) dỹ dη. Qu (x, y) = (2π) ei<η,y−ỹ> ωA (x, x − y + ỹ) q x + 2 X X∗
e ∈S X ×T . In particular we obtain that Qu For x, y, ỹ and η fixed in X ∗ , we consider the following function of the argument t ∈ X :  t−y  , η u(x − y + t, ỹ), Φ(t) := ωA (x, x − y + t) q x + 2 (3.31) (3.32) and notice that its value for t = ỹ is exactly the factor that multiplies the exponential ei<η,y−ỹ> under the integral in (3.31); let us consider its Taylor expansion in t ∈ X around t = y with integral rest of order n > d + s: X X
α
α gα (x, y, ỹ, η) ỹ − y , (3.33) fα (x, y, ỹ, η) ỹ − y + Φ(ỹ) = |α|=n |α|<n where fα (x, y, ỹ, η) := X
fαβ (x) qαβ (x, η) ∂xβ u (x, ỹ), ∞ fαβ ∈ Cpol (X ), qαβ ∈ S1s (Ξ) β≤α and gα (x, y, ỹ, η) := XZ β≤α 1 0 hτ,α,β (x, y − ỹ) qαβ x + (1 − τ ) (3.34) ỹ − y
β
, η ∂x u (x − (1 − τ )(y − ỹ), ỹ) dτ 2 (3.35) ∞ where hτ,α,β ∈ Cpol (X × X ) uniformly for τ ∈ [0, 1] and qαβ ∈ S1s (Ξ). We use the relations (3.33)-(3.35) in (3.31) and eliminate the monomials (ỹ − y)α through partial integrations using the identity
α (ỹ − y)α ei<η,ỹ−y> = − Dη ei<η,ỹ−y> . Finally we obtain: X X X XZ

e Qu (x, y) = fαβ (x) Tαβ u (x, y) + |α|<n β≤α where (2π)−d
Tαβ u (x, y) := (2π)−d Z Z X X∗ |α|=n β≤α Z Z X X∗ 0 1
Rαβ (τ )u (x, y) dτ,
ei<η,y−ỹ> tαβ (x, η) ∂xβ u (x, ỹ) dỹ dη,
Rαβ (τ )u (x, y) := ei<η,y−ỹ> hτ,α,β (x, y − ỹ) rαβ x+ (1 − τ ) s−|α| tαβ ∈ S1 (3.36) (Ξ),
ỹ − y
β
, η ∂x u x− (1 − τ )(y − ỹ), ỹ dỹ dη, 2 (3.37) (3.38) rαβ ∈ S1s−n (Ξ). We begin by estimating the term Tαβ u, by using Lemma 9.20; Starting from (3.37) and considering x ∈ X as a parameter we conclude that there exists a semi-norm cαβ (q) of q ∈ S1s (Ξ) such that
Tαβ u (x, .) 2 L2 (T) ≤ cαβ (q)2
∂xβ u (x, .) 2 Hs (T) , ∀x ∈ X , ∀u ∈ S (X × T). (3.39) Let us consider now the term Rαβ (τ )u. We begin by noticing that due to our hypothesis there exists a constant C(B) (bounded when the components of the magnetic field B take values in bounded subsets of BC ∞ (X )) and there exists an entire number a ∈ Z such that |hτ,α,β (x, y − ỹ)| ≤ C(B) < x >a < y − ỹ >a , 18 ∀(x, y, ỹ) ∈ X 3 , ∀τ ∈ [0, 1]. (3.40) We integrate by parts in (3.38), using the identity ei<η,y−ỹ> = < y − ỹ >−2N 1 − ∆η
N ei<η,y−ỹ> . This allows us to conclude that there exists a seminorm c′α,β,N (p) of the symbol p ∈ S1s (Ξ) for which we have the inequality: Z Z


Rαβ (τ )u (x, y) ≤ C(B)c′α,β,N (p) < x >a < z >a−2N ∂xβ u x − (1 − τ )z, y − z dz < η >s−n dη X∗ X (3.41) for any (x, y) ∈ X 2 and any τ ∈ [0, 1]. We recall our choice s − n < −d, we choose further 2N ≥ a + 2d and we estimate the last integral by using the Cauchy-Schwartz inequality. We take the square of the inequality (3.41) and integrate with respect to y ∈ E concluding that there exists a constant C0 > 0 such that Z
2 Rαβ (τ )u (x, y) dy ≤ E ≤ C0 C(B)2 c′α,β,N (p)2 <x> 2a Z <z> X −2d Z E ∂xα u
x − (1 − τ )z, y − z
2 dy dz, For any Γ-periodic function v ∈ L2loc (X ) and for any z ∈ X we have that Z Z Z |v(y)|2 dy |v(y)|2 dy = |v(y − z)|2 dy = ∀x ∈ X , ∀τ ∈ [0, 1]. E τz E E  so that for any k ∈ N there exists Ck > 0 such that for any τ ∈ [0, 1] we have that Z Z

2 2k 2 ′ 2 <x> Rαβ (τ )u (x, .) L2 (T) ≤ Ck C(B) cα,β,N (p) < x >2a+2k ∂xα u (x, .) X X 2 dx. L2 (T) (3.42)

For the derivatives ∂xµ Tαβ u (x, .) and ∂xµ Rαβ (τ )u (x, .) (for any µ ∈ Nd ) we obtain in a similar way estimations of the same form (3.39) and (3.42) and using (3.36) we obtain (3.30). Lemma 3.17. The following topological embeddings are true (uniformly in ǫ ∈ [−ǫ0 , ǫ0 ]):

S X ; Hm (T) ֒→ Kǫm (X × X ) ֒→ S ′ X ; Hm (T) . (3.43)

Proof. In order to prove the first embedding we take into account the density of S X × T into S X ; Hm (T) m and the Definition
3.10 (c) of the space Kǫ (X × X ). It is thus enough to prove that there exists a seminorm |.|m,l m on S X ; H (T) such that e m,ǫ u Q L2 (X ×T) ≤ C|u|m,l ,
∀u ∈ S X × T . (3.44) But this fact has been proved in Lemma 3.16 (inequality (3.30)).

For the second embedding let us notice that the canonical sesquilinear map on S ′ X ; Hm (T) × S X ; Hm (T) (associated to the duality map) is just a continuous extension of the scalar product Z


(u, v)m := u(x, .), v(x, .) Hm (T) dx, ∀(u, v) ∈ S X ; Hm (T) × S X ; Hm (T) . (3.45) X
Due to the density of S X × T into Kǫm (X × X ), this amounts to prove that it exists a continuous seminorm |.|m,l
on S X ; Hm (T) such that we have that

|(u, v)m | ≤ kukKm · |v|m,l ∀(u, v) ∈ S X ; Hm (T) × S X ; Hm (T) , (3.46) ǫ e m,ǫ u where kukKm = Q ǫ (u, v)m = L2 (X ×T) . Let us notice that 


 e m,ǫ u , Q e −m,ǫ 1⊗ < DΓ >2m v u, 1⊗ < DΓ >2m v L2 (X ×T) = Q

We denote by vΓ := 1⊗ < DΓ >2m v ∈ S X × T and notice that we have the inequality |(u, v)m | ≤ e m,ǫ u Q L2 (X ×T) 19 e −m,ǫ vΓ Q L2 (X ×T) . L2 (X ×T) . (3.47) We conclude
thus that the inequality (3.46) follows if we can prove that there exists a seminorm |.|m,l on S X ; Hm (T) such that we have e −m,ǫ vΓ Q
∀v ∈ S X × T . ≤ C|v|m,l , L2 (X ×T) (3.48)
From Lemma3.16 (inequality (3.30)) we know that there exists a seminorm |.|−m,l on S X × T such that we have e −m,ǫ vΓ Q L2 (X ×T) ≤ C|vΓ |−m,l , Now (3.48) follows from (3.49) once we notice that
∀v ∈ S X × T . (3.49) |vΓ |−m,l = |v|m,l . 4 The Grushin Problem Suppose given a symbol p satisfying the assumptions of Lemma 9.22, i.e. p ∈ S1m (T) real and elliptic, with m > 0. The operator P := Op(p) has a self-adjoint realisation in L2 (X ) having domain Hm (X ) and being lower semibounded and a self-adjoint realisation PΓ in L2 (T) with domain Km,0 being also lower semibounded. Lemma 4.1. There exists N ∈ N∗ , C > 0 and the linear independent family {φ1 , . . . , φN } ⊂ S (T), such that the following inequality is true: (PΓ u, u)L2 (T) ≥ C −1 kuk2Km/2,0 − C N   X u, φj j=1 2 L2 (T) ∀u ∈ Km,0 . , (4.1) Proof. The manifold T being a compact manifold without border, Km,0 is compactly embedded in L2 (T) and the operator PΓ has compact resolvent. Let us fix some λ ∈ R and let us denote by Eλ the spectral projection of PΓ for the semiaxis (−∞, λ]. We choose an orthonormal basis {φ1 , . . . , φN } for the subspace Ran(Eλ ) of L2 (T). Then Ran(1l − Eλ ) is the orthogonal complement of the space Sp{φ1 , . . . , φN } generated by {φ1 , . . . , φN } in L2 (T); moreover, for any v ∈ D(PΓ ) ∩ Ran(1l − Eλ ), one has that (PΓ v, v)L2 (T) ≥ λkvk2L2 (T) . In conclusion: (PΓ v, v)L2 (T) ≥ λkvk2L2 (T) , If u ∈ Km,0 we have that v := u − have that (PΓ v, v)L2 (T) ≥ λ u − N P j=1 ⊥ ∀v ∈ Km,0 ∩ [Sp{φ1 , . . . , φN }] . (4.2) (u, φj )L2 (T) φj belongs to the subset of vectors verifying (4.2) and thus we N X  2 (u, φj )L2 (T) φj j=1 L2 (T) = λ kuk2L2(T) − N X j=1 2 (u, φj )L2 (T)  . On the other side, if we know that PΓ φj = λj φj for any 1 ≤ j ≤ N , then we have that   N N X X (u, φk )L2 (T) φk  (u, φj )L2 (T) PΓ φj , u − (PΓ v, v)L2 (T) = PΓ u − j=1 = (PΓ u, u)L2 (T) − N X k=1 + k=1 (u, φk )L2 (T) (u, PΓ φk )L2 (T) − N X N X (u, φj )L2 (T) (PΓ φj , u)L2 (T) + j=1 (u, φj )L2 (T) (u, φk )L2 (T) (PΓ φj , φk )L2 (T) = N X j=1 20 = L2 (T) j,k=1 = (PΓ u, u)L2 (T) −  2 λj (u, φj )L2 (T) . (4.3) If we compare this inequality with (4.3) we conclude that (PΓ u, u)L2 (T) ≥ λkuk2L2 (T) − N X j=1 (λ − λj ) (u, φj )L2 (T) 2 . Finaly we obtain that − kuk2L2 (T)  N  X 1 λj ≥ − (PΓ u, u)L2 (T) − 1− (u, φj )L2 (T) λ λ j=1 2 ∀u ∈ Km,0 . , (4.4) In order to prove (4.1) we put together (4.4) with the Gårding inequality (9.36) and conclude that it exists C0 > 0 such that (PΓ u, u)L2 (T) ≥ C0−1 kuk2Km/2,0 − C0 kuk2L2(T) , ∀u ∈ Km,0 . Remark 4.2. From Remark 9.23 we know that for any ξ ∈ X ∗ the operator PΓ,ξ is self-adjoint and lower m semibounded in L2 (T) on the domain Km,ξ . If we identify Km,ξ with Hloc (X ) ∩ SΓ′ (X ) endowed with the norm m 2 k < D + ξ > ukL2 (E) , we deduce that the operator Pξ is self-adjoint in Lloc (X ) ∩ SΓ′ (X ) with the domain Km,ξ . Noticing that P = σξ Pξ σ−ξ and σξ : Ks,ξ → Fs,ξ is a unitary operator for any s ∈ R and any ξ ∈ X ∗ , it follows that P generates in F0,ξ a self-adjoint lower semibounded operator on the domain Fm,ξ . Lemma 4.3. Suppose given a compact interval I ⊂ R; it exists a constant C > 0, a natural integer N ∈ N and the family of functions {ψ1 , . . . , ψN } having the following properties: a) ψj ∈ C ∞ (Ξ). b) ψj (y, η + γ ∗ ) = ψj (y, η), ∀(y, η) ∈ Ξ, ∀γ ∗ ∈ Γ∗ , 1 ≤ j ≤ N . c) {ψj (., ξ)}1≤j≤N is an orthonormal system in F0,ξ for any ξ ∈ X ∗ . We denote by Tξ the complex linear space generated by the family {ψj (., ξ)}1≤j≤N in F0,ξ and by Tξ⊥ its orthogonal complement in the same Hilbert space. d) The following inequality is true:

P − λ u, u F 0,ξ ≥ Ckuk2F0,ξ , ∀u ∈ Fm,ξ ∩ Tξ⊥ , ∀ξ ∈ X ∗ , ∀λ ∈ I. (4.5) Proof. It is evidently enough to prove (4.5) for λ = λ0 := sup I. We apply Lemma 4.1 to the operator PΓ,ξ0 − λ0 with ξ0 ∈ X ∗ to be considered fixed. We deduce that there exists C0 > 0, N0 ∈ N∗ and a family of functions {ψe1 (·, ξ0 ), . . . , ψeN0 (·, ξ0 )} from S (T) such that the following inequality is true:

PΓ,ξ0 − λ0 v, v L2 (T) ≥ C0−1 kvk2Km/2,ξ 0 − C0 N0 X
v, ψej (., ξ0 ) L2 (T) j=1 2 , ∀v ∈ Km,ξ0 . (4.6)
Taking into account the result in Example 9.21, we notice that the map X ∗ ∋ ξ 7→ PΓ,ξ ∈ B Ks+m,ξ0 ; Ks,ξ0 is continuous for any s ∈ R (it is even smooth); it follows that

PΓ,ξ0 − PΓ,ξ v, v L2 (T) ≤ kPΓ,ξ0 − PΓ,ξ kB(Km/2,ξ 0 ;K−m/2,ξ0 ) kvk2Km/2,ξ , 0 ∀v ∈ Km,ξ0 . We conclude that for some smaller constant C0 , the inequality (4.6) is true with PΓ,ξ in place of PΓ,ξ0 on the left side, for ξ ∈ V0 some small neighborhood of ξ0 in X ∗ . Let us define now the family of functions {ψ1 , . . . , ψN }. Let us first notice that ψj′ (., ξ0 ) := ei<ξ0 ,.> ψej (., ξ0 ) ∈ C ∞ (X ) ∩ F0,ξ0 . ◦ ◦ ◦ Then let us also notice that for any δ > 0 we can find functions ψ j ∈ C0∞ (E), with E the interior set of E, such that ◦ ψj′ (., ξ0 ) − ψ j Then we define ψj (x, ξ0 ) := X γ∈Γ L2 (E) ◦ ≤ δ, 1 ≤ j ≤ N0 . ψ j (x − γ)ei<ξ0 ,γ> , 21 1 ≤ j ≤ N0 ◦ ◦ and we notice that ψj (., ξ0 ) ∈ C ∞ (X ) ∩ F0,ξ0 and ψj (., ξ0 ) = ψ j on E. Thus we can finally define X ψj (x, ξ) := γ∈Γ ◦ ψ j (x − γ)ei<ξ,γ> , 1 ≤ j ≤ N0 , ∀(x, ξ) ∈ Ξ. (4.7) These functions evidently verify the properties (a) and (b) in the statement of the Lemma. It is also clear that for ◦ ◦ any ξ ∈ X ∗ we have that ψj (., ξ) ∈ F0,ξ . Moreover we have that ψj (., ξ) = ψ j = ψj (., ξ0 ) on E, so that ψj′ (., ξ0 ) − ψj (., ξ) ≤ δ, L2 (E) ∀ξ ∈ X ∗ , 1 ≤ j ≤ N0 . From this estimation we may conclude that ∀κ > 0 we can reduce if necessary the neighborhood V0 fixed above, such that ∀ξ ∈ V0 and ∀v ∈ Km,ξ0 we have that for 1 ≤ j ≤ N0 : Z Z
v(y)ψej (y, ξ0 )dy = v, ψej (., ξ0 ) L2 (T) = ei<ξ0 ,y> v(y)ψj′ (y, ξ0 )dy ≤ E ≤ ≤ Z ei<ξ0 ,y> v(y)ψj (y, ξ)dy + E Z E Z   ei<ξ0 ,y> v(y) ψj′ (y, ξ0 ) − ψj (y, ξ) dy ≤ E ei<ξ,y> v(y)ψj (y, ξ)dy + E + Z E E ≤ Z Z   ei<ξ0 ,y> − ei<ξ,y> v(y)ψj (y, ξ)dy +   ei<ξ0 ,y> v(y) ψj′ (y, ξ0 ) − ψj (y, ξ) dy ≤ ei<ξ,y> v(y)ψj (y, ξ)dy + κkvkL2 (E) . E From this estimation we deduce that, by reducing
if necessary the constant C0 > 0 we can replace
in the right hand side of (4.6) the scalar products v, ψej (., ξ0 ) L2 (T) with the scalar products ei<ξ,.> v, ψj (., ξ) F for ξ ∈ V0 . 0,ξ Let us consider a vector u ∈ Fm,ξ and associate to it the vector v := e−i<ξ,.> u that belongs to Km,ξ and taking into account the equality ei<ξ,.> Pξ e−i<ξ,.> u = P u we deduce from (4.6) and the above arguments that we have

P − λ0 u, u F 0,ξ ≥ C0−1 kuk2F0,ξ − C0 N0 X (u, ψj (., ξ))F0,ξ j=1 2 , ∀u ∈ Fm,ξ , ∀ξ ∈ V0 . (4.8) Taking into account that Fs,ξ+γ ∗ = Fs,ξ for any s ∈ R, any ξ ∈ X ∗ and any γ ∗ ∈ Γ∗ and the fact that the functions ψj (x, .) are Γ∗ -periodic (for 1 ≤ j ≤ N0 ), we conclude that it is enough to prove (4.5) for ξ ∈ T∗ . Being compact, T∗ can be covered by a finite number of neighborhoods of type V0 (as defined in the argument above). In this way, repeating the procedure explained above we can find a finite family of functions {ψ1 , . . . , ψÑ } (with some quite larger Ñ in principle) that will satisfy the properties (a), (b) and (d) from the statement of the Lemma. We select now out of this family a maximal linearly independent subfamily of N functions {ψ1 , . . . , ψN } (it can ◦ ◦ ◦ be characterized by the property that the functions {ψ 1 , . . . , ψ N } is a linearly independent system in C0∞ (E)). Let us notice that this last step (the choice of the maximal linearly independent subfamily) does not change the subspace Tξ that they generate. Finally we may use the Gram-Schmidt procedure in order to obtain a family of N orthonormal functions from F0,ξ . Lemma 4.4. Under the assumptions of Lemma 4.3 we denote by Πξ the orthogonal projection on Tξ in the Hilbert space F0,ξ and by S(ξ, λ) the unbounded operator in Tξ⊥ defined on the domain Fm,ξ ∩ Tξ⊥ by the action

of 1l − Πξ P − λ . Then the following statements are true:
a) The operator S(ξ, λ) is self-adjoint and invertible and S(ξ, λ)−1 ∈ B Tξ⊥ ; Tξ⊥ uniformly with respect to (ξ, λ) ∈ Td∗ × I.
b) The operator S(ξ, λ)−1 also belongs to B Tξ⊥ ; Fm,ξ uniformly with respect to (ξ, λ) ∈ Td∗ × I. 22 Proof. The operator S(ξ, λ) is densly defined by definition and is symmetric on its domain because for any couple  2 (u, v) ∈ Fm,ξ ∩ Tξ⊥ we can write that:





= = u, P − λ 1l − Πξ v F (S(ξ, λ)u, v)F0,ξ = 1l − Πξ P − λ u, v F 0,ξ 0,ξ


1l − Πξ u, P − λ v F = = (u, S(ξ, λ)v)F0,ξ . 0,ξ
In order to prove now the self-adjointness of the operator S(ξ, λ) let us fix some vector v ∈ D
S(ξ, λ)∗ ; thus
we deduce first that v ∈ Tξ⊥ and secondly that there exists a vector f ∈ Tξ⊥ such that S(ξ, λ)u, v F = u, f F 0,ξ 0,ξ
for any u ∈ Fm,ξ ∩ Tξ⊥ . For any vector w ∈ Fm,ξ we can write that w = Πξ w + 1l − Πξ w and Πξ w = N X j=1
w, ψj (., ξ) F 0,ξ Thus we have that


P − λ 1l − Πξ w, v F 0,ξ

P − λ Πξ w, v F 0,ξ where f0 (., ξ) := N P j=1



1l − Πξ P − λ 1l − Πξ w, v F = 0,ξ

1l − Πξ w, f F = and =
1l − Πξ w ∈ Fm,ξ ∩ Tξ⊥ . ψj (., ξ) ∈ Fm,ξ ∩ Tξ , 0,ξ N X j=1

P − λ ψj (., ξ), v F

P − λ w, v F 0,ξ 0,ξ 0,ξ 0,ξ = = (w, f )F0,ξ ,

P − λ ψj (., ξ), v F
w, ψj (., ξ) F 0,ξ

= S(ξ, λ) 1l − Πξ w, v F = (w, f0 (., ξ))F0,ξ ψj (., ξ) ∈ Tξ . In conclusion we have that for anny w ∈ Fm,ξ : = (w, f + f0 (., ξ))F0,ξ , f + f0 (., ξ) ∈ F0,ξ .
Recalling that P − λ is self-adjoint in F0,ξ with domain Fm,ξ we conclude that v ∈ Fm,ξ ∩ Tξ⊥ = D S(ξ, λ) . The invertibility of S(ξ, λ) follows from (4.5) noticing that (S(ξ, λ)u, u)F0,ξ ≥ Ckuk2F0,ξ ∗d ∀u ∈ Fm,ξ Tξ⊥ , ∀ξ ∈ T , ∀λ ∈ I. (4.9)
From this last inequality follows also that S(ξ, λ)−1 ∈ B Tξ⊥ ; Fm,ξ uniformly with respect to (ξ, λ) ∈ T∗ × I. b) For any fixed ξ0 ∈ T∗ and λ0 ∈ I we know that PΓ,ξ0 − λ0 is self-adjoint in K0,ξ0 on the domain Km,ξ0 and that the Hilbert norm on Km,ξ0 is equivalent with the graph-norm of PΓ,ξ0 − λ0 . It exists thus a constant C0 > 0 such that  
kvkKm,ξ0 ≤ C0 kvkK0,ξ0 + PΓ,ξ0 − λ0 v K (4.10) , ∀v ∈ Km,ξ0 . 0,ξ0
Taking into account the Example 9.21 we know that the application X ∗ ∋ ξ 7→ PΓ,ξ ∈ B Km,0 ; K0,0 is of class C ∞ . Noticing that

≤ C kPΓ,ξ0 − PΓ,ξ kB(Km,0 ;K0,0 ) kvkKm,ξ0 + |λ − λ0 | kvkK0,ξ0 , PΓ,ξ0 − λ0 v − PΓ,ξ − λ v K 0,ξ0 for any (ξ, ξ0 ) ∈ T∗ × T∗ , any (λ, λ0 ) ∈ I × I and any v ∈ Km,ξ0 , we deduce that there exist a constant C0′ > 0 and a neighborhood V0 of ξ0 ∈ T∗ such that  
, ∀(ξ, λ) ∈ V0 × I, ∀v ∈ Km,ξ . (4.11) kvkKm,ξ ≤ C0′ kvkK0,ξ + PΓ,ξ − λ v K 0,ξ The manifold T∗ being compact we can find a finite cover with neighborhoods of type V0 as above and we conclude that (4.11) is true for any ξ ∈ T∗ with a suitable constant C0′ . Considering now some vector u ∈ Fm,ξ and denoting by v := σ−ξ u ∈ Km,ξ we deduce from (4.11) that  
, ∀(ξ, λ) ∈ T∗ × I, ∀u ∈ Fm,ξ . (4.12) kukFm,ξ ≤ C0′ kukF0,ξ + P − λ u F 0,ξ Considering now u ∈ Fm,ξ ∩ Tξ⊥
= D S(ξ, λ) we can write that N X



u, P − λ ψj (., ξ) F P − λ u = S(ξ, λ)u + Πξ P − λ u = S(ξ, λ)u + 0,ξ j=1 23 ψj (., ξ), and we know that the norm in F0,ξ of the second term on the right (the finite sum over j) can be bounded by a constant that does not depend on ξ ∈ Td∗ multiplied by kukF0,ξ . Using (4.12) we deduce that there exists a constant C > 0 such that   ∀(ξ, λ) ∈ T∗ × I, ∀u ∈ Fm,ξ ∩ Tξ⊥ . kukFm,ξ ≤ C kukF0,ξ + kS(ξ, λ)ukF0,ξ This inequality clearly implies point (b) of the Lemma. Let us define now the following operators associated to the family of functions {ψj }1≤j≤N introduced above. For any ξ ∈ T∗ we define: o n
e+ (ξ)u := ∀u ∈ F0,ξ , R ∈ CN ; (4.13) u, ψj (., ξ) F 0,ξ e− (ξ)u := R ∀u := {u1 , . . . , uN } ∈ CN , N X j=1 1≤j≤N uj ψj (., ξ) ∈ F0,ξ . (4.14)

e+ (ξ) ∈ B F0,ξ ; CN and R e− (ξ) ∈ B CN ; F0,ξ . Evidently we have that ∀ξ ∈ T∗ , R We can define now the Grushin type operator associated to our operator P : ! e− (ξ) P −λ R Q(ξ, λ) := , e+ (ξ) R 0 (4.15)
that due to our previous results belongs to B Fm,ξ × CN ; F0,ξ × CN uniformly with respect to (ξ, λ) ∈ T∗ × I. Lemma 4.5. For any values of (ξ, λ) ∈ T∗ × I the operator Q(ξ, λ) acting as an unbounded linear operator in the Hilbert space F0,ξ × CN is self-adjoint on the domain Fm,ξ × CN . e+ (ξ)∗ = R e− (ξ) so that Proof. We know that P is self-adjoint in F0,ξ with domain Fm,ξ and it is easy to see that R N we conclude that Q(ξ, λ) is symmetric on Fm,ξ × C
. Let us consider now a pair (v, v) ∈ D Q(ξ, λ)∗ ; this means that v ∈ F0,ξ , v ∈ CN and there exists a pair (f, f ) ∈ F0,ξ × CN such that we have          f v u u . = Q(ξ, λ) f , , v u u F ×CN F ×CN 0,ξ Considering the case u = 0 we get that


= u, f F P − λ u, v F 0,ξ 0,ξ 0,ξ e+ (ξ)u, v − R
CN = e− (ξ)v u, f − R
F0,ξ , ∀u ∈ Fm,ξ . Taking into account the self-adjointness of the operator P − λ in F0,ξ on the domain Fm,ξ we may deduce that in
fact v belongs to Fm,ξ and thus (v, v) ∈ D Q(ξ, λ) . Lemma 4.6. The operator Q(ξ, λ) defined in (4.15) is bijective and has an inverse Q(ξ, λ)−1 ∈ B F0,ξ ×CN ; Fm,ξ ×
CN uniformly with respect to (ξ, λ) ∈ T∗ × I. Proof. Let us first prove the injectivity. Let us choose u ∈ Fm,ξ and u ∈ CN verifying the following equations: 
e− (ξ)u = 0,  P −λ u + R (4.16)  e R+ (ξ)u = 0. e− (ξ)u ∈ Tξ , the first equality The second equality in (4.16) implies that u ∈ Tξ⊥ . As by definition we have that R

in (4.16) implies that 1l − Πξ P − λ u = 0, or equivalently that S(ξ, λ)u = 0. Taking now into account Lemma e− (ξ)u = 0; but the linear 4.4 it follows that u = 0. Now, the first equality in (4.16) implies that we also have R e− (ξ) is injective and thus we independence of the system of functions {ψj (., ξ)}1≤j≤N implies that the operator R deduce that we also have u = 0. Let us consider now the surjectivity of the operator Q(ξ, λ). Thus let us choose an arbitrary pair (v, v) ∈ F0,ξ × CN and let us search for a pair (u, u) ∈ Fm,ξ × CN such that the following equalities are true: 
e− (ξ)u = v,  P −λ u + R (4.17)  e R+ (ξ)u = v. 24 Let us denote by u1 := N P j=1 v j ψj (., ξ) so that by definition we have that e+ (ξ)u1 = v, R u1 ∈ Fm,ξ ∩ Tξ , (4.18) ku1 kFm,ξ ≤ CkvkCN , ∀ξ ∈ T∗ .   Thus we have to search now for a pair (u2 , u) ∈ Fm,ξ ∩ Tξ⊥ × CN that should verify the equality:

e− (ξ)u = v − P − λ u1 ∈ F0,ξ . P − λ u2 + R (4.19) (4.20) e+ (ξ)u2 = 0, the relations (4.18) and (4.20) imply directly that u = u1 + u2 In fact, as we have by definition that R and u is the solution we looked for. Let us project the equation (4.20) both on Tξ and on its complement Tξ⊥ to obtain S(ξ, λ)u2 =


1l − Πξ v − P − λ u1 ,

e− (ξ)u = Πξ v − P − λ (u1 + u2 ) . R (4.21) (4.22) Fm,ξ ∩ Tξ⊥ Taking into account Lemma 4.4, it follows that equation (4.21) has a unique solution u2 ∈ and we have the estimation:




(4.23) ku2 kFm,ξ ≤ C 1l − Πξ v − P − λ u1 F ≤ C kvkF0,ξ + ku1 kFm,ξ ≤ C kvkF0,ξ + kvkCN 0,ξ for any (ξ, λ) ∈ T∗ × I. It is very easy to see that equation (4.22) always has a unique solution u ∈ CN given explicitely by:

uj := v − P − λ (u1 + u2 ) , ψj (., ξ) F . 0,ξ From this explicit expression we easily get the following estimation:  

≤ C kvkF0,ξ + ku1 kFm,ξ + ku2 kFm,ξ ≤ kukCN ≤ C kvkF0,ξ + P − λ (u1 + u2 ) F 0,ξ
≤ C kvkF0,ξ + kvkCN , (4.24) ∀(ξ, λ) ∈ Td∗ × I. In conclusion we have proved the surjectivity of the operator Q(ξ, λ) for any values of (ξ, λ) ∈ T∗ × I and we finish by noticing that the inequalities (4.19), (4.23) and (4.24) imply the boundedness of the operator Q(ξ, λ)−1 uniformly with respect to (ξ, λ) ∈ T∗ × I. We define now the following family of N functions: φj (x, ξ) := e−i<ξ,x> ψj (x, ξ), ∀(x, ξ) ∈ Ξ, 1 ≤ j ≤ N, (4.25) with the family {ψj }1≤j≤N defined in Lemma 4.3. Lemma 4.7. The functions {φj }1≤j≤N defined in (4.25) have the following properties: a) φj ∈ C ∞ (Ξ); b) φj (x + γ, ξ) = φj (x, ξ), c) φj (x, ξ + γ ∗ ) = e−i<γ ∗ ,x> ∀(x, ξ) ∈ Ξ, ∀γ ∈ Γ; ∀(x, ξ) ∈ Ξ, ∀γ ∗ ∈ Γ∗ ; φj (x, ξ), d) For any α ∈ Nd and any s ∈ R there exists a strictly positive constant Cα,s such that:
≤ Cα,s , ∀ξ ∈ X ∗ . ∂ξα φj (., ξ) K s,ξ (4.26) Proof. The properties (a), (b) and (c) follow easily from Lemma 4.3 and the definition (4.25). In order to prove property (d) we consider (9.23) and write:
∂ξα φj (., ξ) 2 Ks,ξ ≤ 1 X < ξ + γ ∗ >2s |E| ∗ ∗ γ ∈Γ 25 2
∗ α ∂d ξ φj (γ , ξ) , (4.27) where we have used the notation:
∗ α ∂d ξ φj (γ , ξ) := 1 |E| Z e−i<γ ∗ ,y> E We recall once again the identity
∂ξα φj (y, ξ) dy. (4.28)
l ∗ = < γ ∗ >−2l 1l − ∆y e−i<γ ,y> , ∀l ∈ N,
and taking into account the Γ-periodicity of the function ∂ξα φj (y, ξ) with respect to the variable y ∈ X we integrate by parts in (4.28) and deduce that ∀α ∈ Nd and ∀l ∈ N there exists a constant Cα,l > 0 such that we have the estimation: Z  
l
∗ 1 ∗ −2l α φ (γ ∗ , ξ) ≤ C e−i<γ ,y> 1l − ∆y ∂ξα φj (y, ξ) dy , ∀ξ ∈ X ∗ , ∀γ ∗ ∈ Γ∗ . (4.29) < γ > ∂d α,l ξ j |E| E e−i<γ ∗ ,y> Coming back to (4.27), taking l ≥ s/2 and using the estimation (4.29) and Plancherel identity we obtain the following estimation: Z   2

l 1 2 2 2 ≤ Cα,l ∂ξα φj (., ξ) K , ∀ξ ∈ E ∗ . (4.30) 1l − ∆y ∂ξα φj (y, ξ) dy ≤ Cα,s s,ξ |E| E In order to extend now to an arbitrary ξ ∈ X ∗ we notice that for any ξ ∈ X ∗ there exist η ∈ E ∗ and γ ∗ ∈ Γ∗ such that ξ = η + γ ∗ and using property (c) of the functions {φ1 , . . . , φN } we see that
∂ξα φj (., ξ) = < D + η + γ ∗ >s e−i<γ Ks,ξ ∗ ,.> =
∂ηα φj (., η + γ ∗ )
∂ηα φj (., η) L2 (E) = Ks,η+γ ∗ e−i<γ = ∗ ,.>
∂ηα φj (., η)
< D + η >s ∂ηα φj (., η) L2 (E) = Ks,η+γ ∗
∂ηα φj (., η) We denote by K0 := K0,0 ≡ L2 (T) ≡ L2 (E) and for any ξ ∈ X ∗ we define the linear operators: n
o , ∀u ∈ K0 , R+ (ξ)u := u, φj K0 1≤j≤N ∀u ∈ CN , R+ (ξ)u := X = uj φj (., ξ). Ks,η ≤ Cα,s . (4.31) (4.32) 1≤j≤N

We evidently have that R+ (ξ) ∈ B K0 ; CN , R− (ξ) ∈ B CN ; K0 and (using (4.26)) both are BC ∞ functions of ξ ∈ X ∗ . With these operators we can now define the following Grushin type operator:  
Pξ − λ R− (ξ) P(ξ, λ) := (4.33) ∈ B Km,ξ × CN ; K0 × CN , ∀(ξ, λ) ∈ X ∗ × I. R+ (ξ) 0 Proposition 4.8. With the above notations, the following statements are true:

∞ ∗ N N a) As a function of (ξ, λ) ∈ X ∗ × I, we have that P ∈ C X × I; B K × C ; K × C and for any α ∈ Nd m,0 0

and any k ∈ N we have that ∂ξα ∂λk P (ξ, λ) ∈ B Km,ξ × CN ; K0 × CN uniformly in (ξ, λ) ∈ X ∗ × I. b) If we considere P(ξ, λ) as an unbounded operator in K0 × CN with domain Km,ξ × CN then for any (ξ, λ) ∈ X ∗ × I, it is self-adjoint and unitarily equivalent with the operator Q(ξ, λ). c) The operator P(ξ, λ) has an inverse:  0 E (ξ, λ) E0 (ξ, λ) := 0 E− (ξ, λ) 0 (ξ, λ) E+ 0 E−,+ (ξ, λ) uniformly bounded with respect to (ξ, λ) ∈ X ∗ × I. 
∈ B K0 × CN ; Km,ξ × CN , (4.34)

d) As a function of (ξ, λ) ∈ X ∗ × I, we have that E0 ∈ C ∞ X ∗ × I; B K0 × CN ; Km,0 × CN and for any α ∈ Nd

and any k ∈ N we have that ∂ξα ∂λk P (ξ, λ) ∈ B K0 × CN ; Km,ξ × CN uniformly in (ξ, λ) ∈ X ∗ × I. 26 Proof. Point (a) follows clearly from the smoothness of the maps R− (ξ) and R+ (ξ) (proved above) and the arguments in Exemple 9.21. b) Let us define the operator:   σξ 0 U (ξ) := : Ks,ξ × CN → Fs,ξ × CN (4.35) 0 1l and let us
notice that it is evidently unitary ∀(ξ, s) ∈ X ∗ × R. Using Remark 9.23 we know that Pξ − λ = σ−ξ P − λ σξ on Km,ξ . If we use the relations (4.14), (4.25) and (4.32) we deduce that for any u ∈ CN : e− (ξ)u = σ−ξ R N X j=1 N X
uj σ−ξ ψj (., ξ) = uj φj (., ξ) = R− (ξ)u. j=1 In a similar way, using now (4.13), (4.25) and (4.31) we obtain that ∀u ∈ K0 we have that: o n n
o

e+ (ξ) σξ u = = R+ (ξ)u. = u, φj (., ξ) K0 R σξ u, ψj (., ξ) F 0,ξ 1≤j≤N 1≤j≤N We conclude that we have the following equality on Km,ξ × CN : P(ξ, λ) = U (ξ)−1 Q(ξ, λ)U (ξ), ∀(ξ, λ) ∈ X ∗ × I. (4.36) c) follows easily from point (b).

d) Let us notice that (a) and (c) imply easily that E0 ∈ C ∞ X ∗ × I; B K0 × CN ; Km,0 × CN . The last property of E0 can be proved by recurence, differentiating the equality PE0 = 1l valid on K0 × CN . For example, if |α| + k = 1 we can write that   ∂ξα ∂λk E0 = −E0 ∂ξα ∂λk P E0 and thus we have the estimation ∂ξα ∂λk E0 B(K0 ×CN ;Km,ξ ×CN ) ≤ kE0 kB(K0 ×CN ;Km,ξ ×CN ) ∂ξα ∂λk P B(Km,ξ ×CN ;K0 ×CN ) kE0 kB(K0 ×CN ;Km,ξ ×CN ) , where the three factors of the right hand side are clearly uniformly bounded with respect to (ξ, λ) ∈ X ∗ × I. 5 Construction of the effective Hamiltonian Let us recall our hypothesis (see Section 2): {Bǫ }|ǫ|≤ǫ0 is a family of magnetic fields satisfying Hypothesis H.1, {Aǫ }|ǫ|≤ǫ0 is an associated family of vector potentials and {pǫ }|ǫ|≤ǫ0 is a family of symbols satisfying Hypothesis H.2 - H.6. As we have already noticed in Remark 9.6, the symbol at ǫ = 0, p0 (x, y, η) does not depend on the first variable x ∈ X ; thus if we denote by p0 (y, η) := p0 (0, y, η) and by rǫ (x, y, η) := pǫ (x, y, η) − p0 (y, η), we notice that the symbol p0 verifies the hypothesis of Section 4 and thus p0 ∈ S1m (T) being real and elliptic and can write pǫ = p0 + rǫ , lim rǫ = 0, in S1m (X × T). ǫ→0 (5.1) We apply the constructions from Section 4 to the operator P0 := Op(p0 ). We obtain that for any compact interval I ⊂ R, for any λ ∈ I and any ξ ∈ X ∗ one can construct the operators R± (ξ) as in (4.31) and (4.32) in order to use (4.33) and define the operator  
P0,ξ − λ R− (ξ) P0 (ξ, λ) := (5.2) ∈ B Km,ξ × CN ; K0 × CN , R+ (ξ) 0 that will verify all the properties listed in Proposition 4.8. In particular P0 (., λ) ∈ S00 X ; B Km,ξ × CN ; K0 × CN

, (5.3) uniformly for λ ∈ I. Moreover, it follows that P0 (ξ, λ) is invertible and its inverse denoted by E0 (ξ, λ) is given (as in (4.34)) by  0  0
E (ξ, λ) E+ (ξ, λ) E0 (ξ, λ) := ∈ B K0 × CN ; Km,ξ × CN (5.4) 0 0 E− (ξ, λ) E−,+ (ξ, λ) and has the property that
E0 (., λ) ∈ S00 X ; B K0 × CN ; Km,ξ × CN , 27 (5.5) uniformly for λ ∈ I. Let us consider now the following operator:   qǫ (x, ξ) − λ R− (ξ) Pǫ (x, ξ, λ) := , R+ (ξ) 0 λ ∈ I, ǫ ∈ [−ǫ0 , ǫ0 ], (x, ξ) ∈ Ξ (5.6) where we recall that qǫ (x, ξ) := Op(e pǫ (x, ., ξ, .)), peǫ (x, y, ξ, η) :=

pǫ (x, y, ξ + η). Taking into account the example 9.21 from the Appendix we notice that qǫ ∈ S00 X ; B Km,ξ ; K0 uniformly in ǫ ∈ [−ǫ0 , ǫ0 ]; thus

(5.7) Pǫ (x, ξ, λ) ∈ S00 X ; B Km,ξ × CN ; K0 × CN , uniformly with respect to (λ, ǫ) ∈ I × [−ǫ0 , ǫ0 ].
Lemma 5.1. The operator Pǫ,λ := OpAǫ (Pǫ (., ., λ)) belongs to B Kǫm (X 2 ) × L2 (X ; CN ); K(X 2 ) × L2 (X ; CN ) uniformly with respect to (λ, ǫ) ∈ I × [−ǫ0 , ǫ0 ]. Moreover, considering Pǫ,λ as an unbounded linear operator in the Hilbert space K(X 2 ) × L2 (X ; CN ) it defines a self-adjoint operator on the domain Kǫm (X 2 ) × L2 (X ; CN ). Proof. If we denote by R∓,ǫ := OpAǫ (R∓ (ξ)) we can write  Peǫ − λ Pǫ,λ = R+,ǫ R−,ǫ 0  . (5.8)
2 Taking into account Lemma 3.13 we may conclude that Peǫ ∈ B Kǫm (X 2 ); K(X ) uniformly with respect to ǫ ∈
∗ 0 N [−ǫ0 , ǫ0 ]. Noticing that R− (ξ) = R+ (ξ) and belongs to S0 (X ; B C ; K0 ) , Proposition 9.27 implies that
R−,ǫ = R∗+,ǫ ∈ B L2 (X ; CN ); K(X 2 ) uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ]. This gives us the first part of the statement of the Lemma. The selfadjointness follows from the self-adjointness of Peǫ in K(X 2 ) on the domain Kǫm (X 2 ) and this follows from Proposition 3.14.

Lemma 5.2. The operator E0,ǫ,λ := OpAǫ E0 (., λ) belons to B K(X 2 ) × L2 (X ; CN ); Kǫm (X 2 ) × L2 (X ; CN ) uniformly with respect to (λ, ǫ) ∈ I × [−ǫ0 , ǫ0 ]. Proof. We can write E0,ǫ,λ = with  E0ǫ,λ E0−,ǫ,λ E0+,ǫ,λ E0−+,ǫ,λ  , (5.9)

0 0 (., λ) . (., λ) , E0−+,ǫ,λ := OpAǫ E−+ E0±,ǫ,λ := OpAǫ E±

From (5.5) it follows that E0 (., λ) ∈ S00 X ; B K0 × CN ; K0 × CN uniformly with respect to (λ, ǫ) ∈ I × [−ǫ0 , ǫ0 ]. In order to prove the boundedness result in the Lemma it is enough to show that  
e m,ǫ 0 Q E0,ǫ,λ ∈ B K(X 2 ) × L2 (X ; CN ); K(X 2 ) × L2 (X ; CN ) , (5.10) 0 1l
E0ǫ,λ := OpAǫ E 0 (., λ) , e m,ǫ is defined before Definition 2.4, with some suitable uniformly with respect to (λ, ǫ) ∈ I × [−ǫ0 , ǫ0 ]; here Q e m,ǫ corresponds to the operator Qm,ǫ from identifications. In that Definition we also argued that the operator Q Remark 9.26 transformed by doubling the variables starting from the operator valued symbol qm,ǫ . We may thus
e m,ǫ is obtained by the OpAǫ quantization of a symbol from S00 X ; B(K0 ; Km,ξ ) . Taking into conclude that Q

0 (ξ, λ) ∈ S00 X ; B(CN ; Km,ξ ) , the property (5.10) follows from account that E 0 (ξ, λ) ∈ S00 X ; B(K0 ; Km,ξ ) and E+ the Composition Theorem 9.24 a) and from the Proposition 9.27. Theorem 5.3. For a sufficiently small ǫ0 > 0 we have that for any (λ, ǫ) ∈ I × [−ǫ0 , ǫ0 ] the operator Pǫ,λ from Lemma 5.1 has an inverse denoted by  
E(ǫ, λ) E+ (ǫ, λ) Eǫ,λ := ∈ B K(X 2 ) × L2 (X ; CN ); Kǫm (X 2 ) × L2 (X ; CN ) , (5.11) E− (ǫ, λ) E−+ (ǫ, λ) uniformly with respect to (ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I. Moreover we have that

lim ρǫ,λ = 0 in S00 X ; B(K0 × CN ; Km,ξ × CN ) . Eǫ,λ = E0,ǫ,λ + Rǫ,λ , Rǫ,λ = OpAǫ ρǫ,λ , ǫ→0 In particular we have that
−+ E−+ (ǫ, λ) = OpAǫ Eǫ,λ , uniformly with respect to λ ∈ I.
−+ 0 lim Eǫ,λ = E−+ (., λ) in S00 X ; B(CN ; CN ) , ǫ→0 28 (5.12) Proof. We begin the proof with the following remarks. 1. The symbol E 0 (ξ, λ) appearing in (5.4) does
not depend on x ∈ X and on ǫ ∈ [−ǫ0 , ǫ0 ]. We can thus consider 0 that E 0 (ξ, λ) ∈ S0,ǫ X ; B(K0 × CN ; Km,ξ × CN ) uniformly for λ ∈ I. 2. The symbol P0 (ξ, λ) appearing in (5.2) does
not depend on x ∈ X and on ǫ ∈ [−ǫ0 , ǫ0 ]. We can thus consider 0 X ; B(Km,ξ × CN ; K0 × CN ) uniformly for λ ∈ I. that P0 (ξ, λ) ∈ S0,ǫ 3. From the relations (5.1), (5.2) and (5.6) it follows that  ′  qǫ (x, ξ) 0 Pǫ (x, ξ, λ) − P0 (ξ, λ) = , 0 0

where q′ǫ (x, ξ) := Op reǫ (x, .ξ, .) and reǫ (x, y, ξ, η) := rǫ (x, y, ξ + η). The fact that lim rǫ = 0 in S1m X × T and ǫ→0

that the map S1m X × T ∋ rǫ 7→ q′ǫ ∈ S00 X ; B(Km,ξ ; K0 ) is continuous (by an evident generalization of property (9.32)), we conclude that   (5.13) lim Pǫ (x, ξ, λ) − P0 (ξ, λ) = 0 ǫ→0
in S00 X ; B(Km,ξ × CN ; K0 × CN ) uniformly with respect to λ ∈ I.
0 Let us come back to the proof of the Theorem and denote by Pǫ,λ := OpAǫ P0 (ξ, λ) . We can write that 0 0 E0,ǫ,λ + Pǫ,λ − Pǫ,λ ) E0,ǫ,λ Pǫ,λ E0,ǫ,λ = Pǫ,λ
in B K(X 2 ) × L2 (X : CN ); K(X 2 ) × L2 (X ; CN ) . Using the Composition Theorem 9.24 and the above remarks, we conclude that
Pǫ,λ E0,ǫ,λ = 1l + OpAǫ sǫ,λ
in B K(X 2 ) × L2 (X : CN ); K(X 2 ) × L2 (X ; CN ) , where lim sǫ,λ = 0, ǫ→0 (5.14) (5.15) (5.16)
in S00 X ; B(K0 × CN ; K0 × CN ) uniformly with respect to λ ∈ I. Aǫ It follows from Proposition 9.28 that for
ǫ0 > 0 small enough, the operator 1l + Op (sǫ,λ ) is invertible in 2 2 N 2 2 N B K(X ) × L (X ; C ); K(X ) × L (X ; C ) for any (ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I and it exists a symbol tǫ,λ such that lim tǫ,λ = 0, (5.17) ǫ→0
in S00 X ; B(K0 × CN ; K0 × CN ) uniformly with respect to λ ∈ I and Let us define  −1 1l + OpAǫ (sǫ,λ ) = 1l + OpAǫ (tǫ,λ ). (5.18)   Eǫ,λ := E0,ǫ,λ 1l + OpAǫ (tǫ,λ ) and let us notice that it is a right inverse for Pǫ,λ . As the operator Pǫ,λ is self-adjoint, it follows that Eǫ,λ defined above is also a left inverse for it. The other properties in the statement of the Theorem are evident now. Remark 5.4. The operator E−+ (ǫ, λ) defined in (5.12) will be the effective Hamiltonian associated to the Hamiltonian Pǫ and the interval I. Its importance will partially be explained in the following Corollary. Corollary 5.5. Under the assumptions of Theorem 5.3, for any λ ∈ I and any ǫ ∈ [−ǫ0 , ǫ0 ] the following equivalence is true:

⇐⇒ 0 ∈ σ E−+ (ǫ, λ) . (5.19) λ ∈ σ Peǫ Proof. The equality Pǫ,λ Eǫ,λ =  1lK(X 2 ) 0 0 1lL2 (X ;CN ) is equivalent with the following system of equations: 
Peǫ − λ E(ǫ, λ) + R−,ǫ E− (ǫ, λ)  
 e Pǫ − λ E+ (ǫ, λ) + R−,ǫ E−+ (ǫ, λ)  R+,ǫ E(ǫ, λ)   R+,ǫ E+ (ǫ, λ) 29 = = = =  1lK(X 2 ) , 0, 0, 1lL2 (X ;CN ) . (5.20) 
If 0 ∈ / σ E−+ (ǫ, λ) the second equality in (5.20) implies that
R−,ǫ = − Peǫ − λ E+ (ǫ, λ)E−+ (ǫ, λ)−1 and by substituting this value in the first equality in (5.20) we obtain
  Peǫ − λ E(ǫ, λ) − E+ (ǫ, λ)E−+ (ǫ, λ)−1 E− (ǫ, λ) = 1lK(X 2 ) . It follows that λ ∈ / σ(Peǫ ). Suppose now that λ ∈ / σ(Peǫ ); then the second equality in (5.20) implies that
−1 E+ (ǫ, λ) = − Peǫ − λ R−,ǫ E−+ (ǫ, λ). After substituting this last expression in the last equality in (5.20) we get
−1 R−,ǫ E−+ (ǫ, λ) = 1lL2 (X ;CN ) . −R+,ǫ Peǫ − λ
It follows that 0 ∈ / σ E−+ (ǫ, λ) and we obtain the following identity (valid in this case):
−1 R−,ǫ . E−+ (ǫ, λ)−1 = −R+,ǫ Peǫ − λ ∗ We recall that for any γ ∗ ∈ Γ∗ we denote by σγ ∗ the operator of multiplication with the character ei<γ ,.> on ′ S (X ; CN ) and we define now Υγ ∗ as the operator of multiplication with the function σγ ∗ ⊗ σ−γ ∗ on the space S ′ (X 2 ). We shall need further the following commutation property. Lemma 5.6. For any γ ∗ ∈ Γ∗ the following equality is true ∀(ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I:     0 Υγ ∗ 0 Υγ ∗ (5.21) Pǫ,λ = Pǫ,λ 0 σγ ∗ 0 σγ ∗ as operators on S (X × T) × S (X ; CN ) (identifying the test functions on the torus with the associated periodic distributions). Proof. From equality (3.21) we deduce that for any γ ∗ ∈ Γ∗ we have the following equality on S (X × T): Υγ ∗ Peǫ = Peǫ Υγ ∗ . (5.22) Taking now into account Lemma 4.7 and the definitions (4.31) and (4.32) we obtain that R+ (ξ + γ ∗ ) = R+ (ξ) σγ ∗ , R− (ξ + γ ∗ ) = σ−γ ∗ R− (ξ), ∀ξ ∈ X ∗ , ∀γ ∗ ∈ Γ∗ . (5.23) Repeating the computations done in Exemple 9.4 we obtain for any u ∈ S (X ; CN ): Z h i
∗ ei<ζ,x−z> ω Aǫ (x, z) R− (ζ)ei<γ ,z> u(z) (y)dz dζ R−,ǫ σγ ∗ u (x, y) = ¯ = = ei<γ =e =e Z ,x> i<γ ∗ ,x> i<γ ∗ ,x> Ξ N ∗ Ξ Z Z ei<ζ−γ ∗ ,x−z> ω Aǫ (x, z) [R− (ζ)u(z)] (y)dz dζ ¯ = Ξ ¯ = ei<ζ,x−z> ω Aǫ (x, z) [R− (ζ + γ ∗ )u(z)] (y)dz dζ Ξ e i<ζ,x−z>
¯ = Υγ ∗ R−,ǫ u (x, y), ω Aǫ (x, z) [σ−γ ∗ R− (ζ)u(z)] (y)dz dζ concluding that on S (X ; C ) we have the equality R−,ǫ σγ ∗ = Υγ ∗ R−,ǫ . (5.24) In a similar way we obtain that on S (X × T) we have the equality: R+,ǫ Υγ ∗ = σγ ∗ R+,ǫ . (5.25) Remark 5.7. Of course the inverse of the operator Pǫ,λ verifies a commutation equation similar to (5.21):     0 Υγ ∗ 0 Υγ ∗ , ∀γ ∗ ∈ Γ∗ , (5.26) Eǫ,λ = Eǫ,λ 0 σγ ∗ 0 σγ ∗ on S (X × T) × S (X ; CN ) and for any (ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I. 30 6 The auxiliary Hilbert spaces V0 and L0 The procedure we use after [14] for coming back from the operator Peǫ to the basic Hamiltonian Pǫ supposes to consider the extension of the pseudodifferential operator Peǫ to the tempered distributions and a restriction of this one to some Hilbert spaces of distributions that we introduce and study in this section. Definition 6.1. Let us consider the following complex space (denoting by δγ := τγ δ with δ the Dirac distribution of mass 1 supported in {0} and γ ∈ Γ):     X V0 := w ∈ S ′ (X ) | ∃f ∈ l2 (Γ) such that w = fγ δ−γ ,   γ∈Γ endowed with the quadratic norm: kwkV0 := sX |fγ |2 , γ∈Γ ∀w ∈ V0 . It is evident that V0 is a Hilbert space and is canonically unitarily equivalent with l2 (Γ). The Hilbert space V0 has a ’good comparaison property’ with respect to the scale of magnetic Sobolev spaces introduced in [19]. In order to study this relation let us choose a family of vector potentials {Aǫ }|ǫ|≤ǫ0 having ∞ components of class Cpol (X ) and defining the magnetic fields {Bǫ }|ǫ|≤ǫ0 satisfying Hypothesis H.1. −s Lemma 6.2. For any s > d and for any ǫ ∈ [−ǫ0 , ǫ0 ] we have the algebraic and topologic inclusion V0 ֒→ HA (X ), ǫ uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ]. P fγ δ−γ ∈ V0 . Then we have (in S ′ (X )): Proof. We use the operator Qs,ǫ from Remark 9.26. Let u = γ∈Γ g := Q−s,ǫ u = X fγ Q−s,ǫ δ−γ . γ∈Γ A computation made in S ′ (X ) shows that (for s > d) we have that Q−s,ǫ δ−γ belongs in fact to C(X ) (as Fourier transform of an integrable function) and moreover: Z
x+y
, η δ−γ (y)dy dη = Q−s,ǫ δ−γ (x) = (2π)−n ei<η,x−y> ωAǫ (x, y) q−s,ǫ 2 Ξ Z x−γ
−n = (2π) ei<η,x+γ> ωAǫ (x, −γ) q−s,ǫ , η dη. 2 X∗ From this last formula we may deduce that for any N ∈ N there exists a strictly positive constant CN > 0 such that for any ǫ ∈ [−ǫ0 , ǫ0 ] and for any x ∈ X we have the estimation:
Q−s,ǫ δ−γ (x) ≤ CN < x + γ >−N . Choosing N > d we notice that for any x ∈ X : |g(x)| ≤ CN X γ∈Γ  |fγ | < x + γ >−N ≤ CN  X γ∈Γ 1/2  |fγ |2 < x + γ >−N   X γ∈Γ 1/2 < x + γ >−N  . We may conclude that g ∈ L2 (X) and kgkL2(X ) ≤ CN kuk2V0 . Finally this is equivalent with the fact that Qs,ǫ g ∈ −s (X ) and there exists a strictly positive constant C such that HA ǫ ∀u ∈ V0 , ∀ǫ ∈ [−ǫ0 , ǫ0 ]. kukH−s (X ) ≤ CkukV0 , Aǫ We shall need a property characterizing the elements from V0 (replacing the property proposed in [14] that is not easy to generalise to our situation and moreover its proof given in [14] and [13] has some gaps that make necessary some modifications in the arguments!). Lemma 6.3. For any s > d there exists a strictly positive constant Cs > 0 such that the following inequality is true: X ∀u ∈ S (X ), ∀ǫ ∈ [−ǫ0 , ǫ0 ]. (6.1) |u(γ)|2 ≤ Cs kuk2Hs (X ) , γ∈Γ Aǫ 31 Proof. For any fixed u ∈ S (X ) let us denote by v := Qs,ǫ u ∈ S (X ). Then u = Q−s,ǫ v and thus for any N ∈ N and for any x ∈ X we can write that:   Z
N x+y
i<η,x−y> −2N ¯ , η v(y)dy dη. u(x) = e <x−y > ωAǫ (x, y) 1l − ∆η q−s,ǫ 2 Ξ This equality implies that we can find two strictly positive constants C and C ′ such that for any ǫ ∈ [−ǫ0 , ǫ0 ] and for any x ∈ X one has the estimation: Z |u(x)| ≤ C < x − y >−2N |v(y)|dy, X |u(x)| 2 ≤ C 2 Z X <x−y > −2N dy  Z X <x−y > −2N 2 |v(y)| dy  ≤ C′ Z X < x − y >−2N |v(y)|2 dy. We choose now N ∈ N large enough and notice that   Z X X  |u(γ)|2 ≤ C ′ < γ − y >−2N  |v(v)|2 dy ≤ C ′′ kvk2L2 (X ) ≤ Cs kuk2Hs X γ∈Γ Aǫ (X ) γ∈Γ . For any γ ∗ ∈ Γ∗ we use the notation σγ ∗ also for the operator of multiplication with the character ei<γ the space of tempered distributions (that it evidently leaves invariant). Proposition 6.4. We have the following charaterization of the vectors from V0 : ∗ ,.> on ∞ a) Given any vector u ∈ V0 there exists a vector u0 ∈ HA (X ) such that ǫ X u = σγ ∗ u0 . (6.2) γ ∗ ∈Γ∗ ∞ Moreover the map V0 ∋ u 7→ u0 ∈ HA (X ) is continuous uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ]. ǫ P ∞ σγ ∗ u0 converges in S ′ (X ) and its sum denoted by u belongs in fact b) Given any u0 ∈ HA (X ) the series ǫ γ ∗ ∈Γ∗ to V0 . Moreover the map ∞ HA (X ) ǫ ∋ u0 7→ u ∈ V0 is continuous uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ]. Proof. We shall use the notation uγ ∗ := σγ ∗ u0 , for any γ ∗ ∈ Γ∗ and for any tempered distribution u0 ∈ S ′ (X ). −s a) Lemma 6.2 implies that for any s > d and any ǫ ∈ [−ǫ0 , ǫ0 ] we have that V0 ⊂ HA (X ) and there exists a ǫ strictly positive constant Cs > 0, independent of ǫ, such that ∀ǫ ∈ [−ǫ0 , ǫ0 ], ∀u ∈ V0 . kukH−s (X ) ≤ Cs kukV0 , Aǫ P Let us choose a real function χ ∈ C0∞ (X ∗ ) such that define γ ∗ ∈Γ∗ (6.3) τγ ∗ χ = 1 on X ∗ . For any distribution u ∈ V0 we u0 := OpAǫ (χ)u. ∞ Due to the fact that χ ∈ S1−∞ (X ) it follows by the properties of magnetic Sobolev spaces (see [19]) that u0 ∈ HA (X ) ǫ ∞ and the map V0 ∋ u 7→ u0 ∈ HAǫ (X ) is continuous uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ]. P −s −s (X ) (X ) to an element v ∈ HA uγ ∗ is convergent in HA We shall prove now that for any s > d the series ǫ ǫ γ ∗ ∈Γ∗ and that there exists a constant C > 0 such that kvkH−s (X ) ≤ Cku0 kHsA ǫ Aǫ (X ) uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ]. We denote by
gγ ∗ := Q−s,ǫ σγ ∗ u0 = σγ ∗ OpAǫ (idl ⊗ τ−γ ∗ )q−s,ǫ u0 (6.4) where we have used (9.4) for the last equality. We notice that the family {< γ ∗ >s (idl ⊗ τ−γ ∗ )q−s,ǫ }|ǫ|≤ǫ0 is bounded as subset of S1s (Ξ) and thus there exists a constant C > 0 such that kgγ ∗ kL2 (X ) ≤ C < γ ∗ >−s ku0 kHsA ǫ 32 (X ) , ∀ǫ ∈ [−ǫ0 , ǫ0 ], ∀γ ∗ ∈ Γ∗ . We conclude that it exists an element g ∈ L2 (X ) such that P γ ∗ ∈Γ∗ gγ ∗ = g in L2 (X ) and we have the estimation kgkL2(X ) ≤ C ′ ku0 kHsA (X ) for any ǫ ∈ [−ǫ0 , ǫ0 ]. Due to the properties of the magnetic pseudodifferential calculus ǫ P −s −s (X ) and (6.4) is true. (X ) to an element v ∈ HA uγ ∗ converges in HA (see [19]) it follows that the series ǫ ǫ γ ∗ ∈Γ∗ We still have to show that v = u as tempered distributions. Let us fix a test function ϕ ∈ S (X ) and compute: E X X X D < v, ϕ > = < uγ ∗ , ϕ > = σγ ∗ OpAǫ (χ)u, ϕ = < σγ ∗ u0 , ϕ > = γ ∗ ∈Γ∗ = γ ∗ ∈Γ∗ X D γ ∗ ∈Γ∗ γ ∗ ∈Γ∗ E OpAǫ (τ−γ ∗ χ)σγ ∗ u, ϕ = X D γ ∗ ∈Γ∗ E u, OpAǫ (τ−γ ∗ χ)ϕ where we have used the relation σγ ∗ u = u verified by all the elements from V0 . Let us also notice that for any s > d we have that X τ−γ ∗ χ = 1 in S1s (Ξ) γ ∗ ∈Γ∗ so that we can write that ϕ = X OpAǫ (τ−γ ∗ χ)ϕ, in S (X ). γ ∗ ∈Γ∗ We conclude that < v, ϕ >=< u, ϕ > for any ϕ ∈ S (X ) and thus v = u. b) During the proof of point (a) we have shown that P for any s > d there exists a constant Cs > 0 such that for −s −s ∞ (X ) (X ) to an element u ∈ HA uγ ∗ converges in HA any u0 ∈ HA (X ) and for any ǫ ∈ [−ǫ , ǫ ] the series 0 0 ǫ ǫ ǫ γ ∗ ∈Γ∗ and the following estimation is true: kukH−s(X ) ≤ Cs ku0 kHsA ǫ Aǫ Let us recall the Poisson formula: X ∞ (X ), ∀ǫ ∈ [−ǫ0 , ǫ0 ]. ∀u0 ∈ HA ǫ (X ) , (2π)d X δ−γ , |E ∗ | σγ ∗ = γ ∗ ∈Γ∗ (6.5) in S ′ (X ). (6.6) γ∈Γ Let us first suppose that u0 ∈ S (X ). Multiplying in the equality (6.6) with u0 we obtain the following equality: u = (2π)d X u0 (−γ)δ−γ , |E ∗ | in S ′ (X ). (6.7) γ∈Γ Lemma 6.3 implies that u ∈ V0 and for any s > d we have the estimation: kukV0 ≤ Cs ku0 kHsA ǫ ∀u0 ∈ S (X ), ∀ǫ ∈ [−ǫ0 , ǫ0 ]. (X ) , (6.8) ∞ We come now to the general case u0 ∈ HA (X ). Let us fix some ǫ ∈ [−ǫ0 , ǫ0 ] and some s > d. Using the fact ǫ s s that S (X ) is dense in HAǫ (X ) we can choose a sequence {uk0 }k∈N∗ ⊂ S (X ) such that u0 = lim uk0 in HA (X ). ǫ kր∞ For each element uk0 we can associate, as we proved above, an element uk ∈ V0 such that the following inequalities hold: kuk kV0 ≤ Cs kuk0 kHsA (X ) , ∀k ∈ N∗ , ǫ k l ku − u kV0 ≤ Cs kuk0 − ∀(k, l) ∈ [N∗ ]2 . ul0 kHsA (X ) , ǫ It follows that there exists v ∈ V0 such that v = lim uk in V0 and the following estimation is valid: kր∞ kvkV0 ≤ Cs ku0 kHsA ǫ (X ) . (6.9) P uγ ∗ . But we know We still have to prove that the element v ∈ V0 obtained above is exactly the limit u = γ ∗ ∈Γ∗ P σγ ∗ uk0 so that from (6.5) we deduce that we have the estimations: that uk = γ ∗ ∈Γ∗ kuk − ukH−s (X ) ≤ Cs kuk0 − u0 kHsA ǫ Aǫ (X ) → kր∞ 0. −s In conclusion uk → u in HA (X ) and uk → v in V0 . But Lemma 6.2 implies that V0 is continuously embedded ǫ kր∞ kր∞ −s (X ) and we conclude that v = u. in HA ǫ 33 Lemma 6.5. Let us consider the map ψ defined in (2.1). For any vector v ∈ L2 (X ) the series X
wv := ψ ∗ v ⊗ δ−γ γ∈Γ
converges in S ′ (X 2 ) and satisfies the identity idl ⊗ τα wv = wv , ∀α ∈ Γ. Proof. Let us denote by wγ the general term of the series defining wv ; then for any ϕ ∈ S (X 2 ) we have that Z
v(x)ϕ(x, x + γ) dx. hwγ , ϕi = ψ ∗ v ⊗ δ−γ , ϕ = hv ⊗ δ−γ , ψ ∗ (ϕ)i = X But using the definition of test functions it follows that for any N ∈ N there exists a defining semi-norm νN (ϕ) on S (X 2 ) such that |ϕ(x, x + γ)| ≤ νN (ϕ) < x >−N < γ >−N , ∀x ∈ X , ∀γ ∈ Γ. We deduce that for any N ∈ N there exists a defining semi-norm µN (ϕ) on S (X 2 ) such that: It follows that the series P |hwγ , ϕi| ≤ µN (ϕ)kvkL2 (X ) < γ >−N , ∀ϕ ∈ S (X 2 ). wγ converges in S ′ (X 2 ) and its sum w ∈ S ′ (X 2 ) satisfies the inequality γ∈Γ |hw, ϕi| ≤ ρ(ϕ)kvkL2 (X ) , ∀ϕ ∈ S (X 2 ), (6.10) for some defining semi-norm ρ(ϕ) on S (X 2 ). Finally let us notice that for any α ∈ Γ we can write that

idl ⊗ τα wγ = ψ ∗ v ⊗ δ−(γ+α) = wγ+α
and thus idl ⊗ τα wv = wv . Definition 6.6. Let us consider the map ψ defined in (2.1). We define the following complex space:     X
L0 := w ∈ S ′ (X 2 ) | ∃v ∈ L2 (X ) such that w = ψ ∗ v ⊗ δ−γ   γ∈Γ endowed with the quadratic norm kwkL0 := kvkL2 (X ) . Lemma 6.7. The complex space L0 is a Hilbert space and is embedded continuously into S ′ (X 2 ). Proof. The space L0 is evidently a Hilbert space canonically unitarily equivalent with L2 (X ) and the continuity of the embedding into S ′ (X 2 ) follows easily from (6.10). −s (X ) ⊗ L2 (T) Lemma 6.8. For any s > d and any ǫ ∈ [−ǫ0 , ǫ0 ] we have a continuous embedding L0 ֒→ HA ǫ uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ]. Proof. Let us fix some s > d and some vector v ∈ L2 (X ) and define X
u := ψ ∗ v ⊗ δ−γ ∈ L0 , γ∈Γ Q′−s,ǫ := Q−s,ǫ ⊗ idl, gγ :=
Q′−s,ǫ ◦ ψ ∗ (v ⊗ δ−γ ). A straightforward computation in S ′ (X 2 ) shows that we have the equality: Z x+y−γ
ei<η,x−y+γ> ωAǫ (x, y − γ) q−s,ǫ gγ (x, y) = v(y − γ) , η dη. ¯ 2 X By estimating the integral in the right hand side above we obtain that for any sufficiently large N ∈ N there exists CN > 0 such that |gγ (x, y)| ≤ CN |v(y − γ)| < x − y + γ >−N , 34 ∀(x, y) ∈ X 2 , ∀γ ∈ Γ. ′ ′′ We conclude that for some suitable constants CN , CN , . . . we get   X 2 |gγ (x, y)| γ∈Γ  2  ≤ CN X  |v(y − γ)|2 < x − y + γ >−N   X γ∈Γ γ∈Γ ′ ≤ CN X  < x − y + γ >−N  ≤ |v(y − γ)|2 < x − y + γ >−N , γ∈Γ ′′ and the integral of the last expression over X × E is bounded by CN kvk2L2 (X ) . P ′ gγ = Q−s,ǫ u, we notice that (idl ⊗ τα )g = g for any α ∈ Γ (because by Lemma 6.5 the If we denote by g := γ∈Γ vector u has this property). Thus g ∈ L2 (X ) ⊗ L2 (T), kgkL2 (X )⊗L2 (T) ≤ √ C ′′ kvkL2 (X ) . It follows that −s u = Q′s,ǫ g ∈ HA (X ) ⊗ L2 (T), ǫ ′′′ kukH−s(X )⊗L2 (T) ≤ CN kvkL2 (X ) , Aǫ ∀ǫ ∈ [−ǫ0 , ǫ0 ], ∀v ∈ L2 (X ). We shall obtain a characterization of the space L0 that is similar to our Proposition 6.4. In order to do that we need some technical results contained in the next Lemma. We shall use the notation:
∞ s (X ) ⊗ L2 (T) := ∩ HA HA (X ) ⊗ L2 (T) ǫ ǫ s∈R with the natural projective limit topology. ∞ Lemma 6.9. Suppose given some u0 ∈ HA (X ) ⊗ L2 (T) and for any γ ∗ ∈ Γ∗ let us denote by uγ ∗ := Υγ ∗ u0 . For ǫ P −s (X ) ⊗ L2 (T) and the sum denoted by uγ ∗ converges in HA any s > d there exists Cs > 0 such that the series ǫ γ ∗ ∈Γ∗ −s v ∈ HA (X ) ⊗ L2 (T) satisfies the estimation: ǫ kvkH−s (X )⊗L2 (T) ≤ Cs ku0 kHsA ǫ Aǫ (X )⊗L2 (T) , ∞ (X ) ⊗ L2 (T), ∀ǫ ∈ [−ǫ0 , ǫ0 ]. ∀u0 ∈ HA ǫ (6.11) Proof. From (9.4) it follows that on S (X ) we have the equality Q−s,ǫ σγ ∗ = σγ ∗ OpAǫ (idl ⊗ τ−γ ∗ )q−s,ǫ so that finally

 
Q−s,ǫ ⊗ idl uγ ∗ = Υγ ∗ OpAǫ (idl ⊗ τ−γ ∗ )q−s,ǫ ⊗ idl u0 . (6.12) Taking into account that the family {< γ ∗ >s (idl ⊗ τ−γ ∗ )q−s,ǫ }(ǫ,γ ∗ )∈[−ǫ0 ,ǫ0 ]×Γ∗ is a bounded subset of S s (Ξ), it follows the existence of a constant C > 0 such that for any ǫ ∈ [−ǫ0 , ǫ0 ] one has the estimation:
∞ (X ) ⊗ L2 (T). (6.13) ∀u0 ∈ HA Q−s,ǫ ⊗ idl uγ ∗ L2 (X )⊗L2 (T) ≤ C < γ ∗ >−s ku0 kHsA (X )⊗L2 (T) , ǫ ǫ it follows that the series P γ ∗ ∈Γ∗
Q−s,ǫ ⊗ idl uγ ∗ converges in L2 (X ) ⊗ L2 (T) uniformly for ǫ ∈ [−ǫ0 , ǫ0 ]. The stated inequality follows now by summing up the estimation (6.13) over all Γ∗ . ∞ Proposition 6.10. For any u ∈ L0 there exists a vector u0 ∈ HA (X ) ⊗ L2 (T) such that ǫ u = X Υγ ∗ u0 , in S ′ (X 2 ). (6.14) γ ∗ ∈Γ∗ ∞ Moreover, the application L0 ∋ u 7→ u0 ∈ HA (X ) ⊗ L2 (T) is continuous uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ]. ǫ Proof. We recall the notation uγ ∗ := Υ∗γ u0 and, as in the proof of point (a) of Proposition 6.4 we fix some real function χ ∈ C0∞ (X ) satisfying the following identity on X : X τγ ∗ χ = 1. γ ∗ ∈Γ∗ 35 
For any u ∈ L0 let us denote by u0 := OpAǫ (χ) ⊗ idl u. We notice that χ ∈ S1−∞ (Ξ) and using Lemma 6.8 we ∞ deduce that u0 ∈ HA (X ) ⊗ L2 (T) and that the continuity property in the end of the Proposition is clearly true. ǫ We still have to verify the equality (6.14). P 1. First let us notice that following Lemma 6.9, the series uγ ∗ converges in S ′ (X 2 ). γ ∗ ∈Γ∗ 2. An argument similar to that in the proof of Proposition 6.4 a) proves that on S (X 2 ) we have the equality X
(6.15) OpAǫ τ−γ ∗ χ ⊗ idl = idl. γ ∗ ∈Γ∗
3. From (9.4) we have that OpAǫ τ−γ ∗ χ = σ−γ ∗ OpAǫ (χ)σγ ∗ .
P ∗ ψ v ⊗ δ−γ ; thus for any γ ∗ ∈ Γ∗ we have that: 4. For any u ∈ L0 there exists v ∈ L2 (X ) such that u = γ∈Γ X Υγ ∗ u = Υγ ∗ ψ ∗ v ⊗ δ−γ γ∈Γ
= X X ∗

ψ ∗ (idl ⊗ σγ ∗ )(v ⊗ δ−γ ) = ψ v ⊗ δ−γ = u. γ∈Γ γ∈Γ Using the last two remarks above we deduce that for any u ∈ L0 we have the equalities:


   Aǫ Op τ−γ ∗ χ ⊗ idl u = σ−γ ∗ OpAǫ (χ)σγ ∗ ⊗ idl σ−γ ∗ ⊗ σγ ∗ u =
= Υ−γ ∗ OpAǫ (χ) ⊗ idl u = Υ−γ ∗ u0 = u−γ ∗ . We apply now equality (6.15) to the vector u ∈ L0 in order to obtain that: X X u = u−γ ∗ = uγ ∗ , γ ∗ ∈Γ∗ γ ∗ ∈Γ∗ as tempered distributions. In order to prove the reciprocal statement of Proposition 6.10 we need a technical Lemma similar to Lemma6.3. Lemma 6.11. For any s > d there exists Cs > 0 such that the following estimation holds: sZ X |u(x, x)|2 dx ≤ Cs kukHsA ǫ (X )⊗L2 (T) , ∀u ∈ S (X × T), ∀ǫ ∈ [−ǫ0 , ǫ0 ]. (6.16)
Proof. Let us fix
some u ∈ S (X × T), ǫ ∈ [−ǫ0 , ǫ0 ] and let us define v := Qs,ǫ ⊗ idl u ∈ S (X × T). It follows that u = Q−s,ǫ ⊗ idl v and we deduce that for any N ∈ N (that we shall choose sufficiently large) we have the identity: Z h
x + z
i < x − z >−2N ei<ζ,x−z> ωAǫ (x, z) (idl − ∆ζ )N q−s,ǫ u(x, y) = ¯ ∀(x, y) ∈ X 2 . , ζ v(z, y) dz dζ, 2 Ξ ′ We deduce that there exist the strictly positive constants CN , CN , . . . such that the following estimations hold: Z Z −2N 2 ′ |u(x, y)| ≤ CN < x−z > |v(z, y)| dz, |u(x, y)| ≤ CN < x − z >−2N |v(z, y)|2 dz. X X In conclusion we have that: Z XZ |u(x, x)|2 dx = X γ∈Γ ′ ≤ CN XZ Z γ∈Γ E X τ−γ E |u(x, x)|2 dx = XZ γ∈Γ E |u(x + γ, x + γ)|2 dx = ′′ < x − z + γ >−2N |v(z, x)|2 dz dx ≤ CN ′′′ ≤ CN Z Z E X XZ Z γ∈Γ E X XZ γ∈Γ E |u(x + γ, x)|2 dx ≤ < z − γ >−2N |v(z, x)|2 dz dx ≤ ′′′ |v(z, x)|2 dz dx = CN kvk2L2 (X ×T) ≤ Cs2 kuk2Hs Aǫ (X )⊗L 2 (T) . We come now to the reciprocal statement of Proposition 6.10. ∞ 2 ∗ ∗ Proposition uγ ∗ := Υγ ∗ u0 . Then P 6.12. Suppose given u0 2∈ HAǫ (X ) ⊗ L (T) and for any γ ∈ Γ let us consider ∞ uγ ∗ converges in S (X ) to an element u ∈ L0 . Moreover, the application HA (X ) ⊗ L2 (T) ∋ u0 7→ the series ǫ γ ∗ ∈Γ∗ u ∈ L0 is continuous uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ]. 36 ∞ Proof. For any s > d and any u0 ∈ HA (X ) ⊗ L2 (T), Lemma 6.9 implies that the series ǫ P γ ∗ ∈Γ∗ uγ ∗ converges in −s −s (X ) ⊗ L2 (T) and there exists Cs > 0 such that the following estimation (X ) ⊗ L2 (T) to an element u ∈ HA HA ǫ ǫ holds: s kukH−s(X )⊗L2 (T) ≤ Cs ku0 kHsA (X )⊗L2 (T) , (X ) ⊗ L2 (T), ∀ǫ ∈ [−ǫ0 , ǫ0 ]. (6.17) ∀u0 ∈ HA ǫ ǫ Aǫ We still have to prove that u ∈ L0 and that the continuity property stated above is true.
As in the proof of Proposition 6.4 b) we make use of the Poisson formula (6.6). Once we notice that ψ ∗ idl ⊗ σγ ∗ = Υγ ∗ , we conclude that for any u0 ∈ S (X 2 ) one has the identity:   X
(2π)d X ∗ (6.18) uγ ∗ = ψ idl ⊗ δ−γ  u0 . ∗| |E ∗ γ γ∈Γ
P ψ ∗ idl ⊗ δ−γ belongs to S ′ (X × T) and we deduce that the identity (6.18) also holds for γ∈Γ
u0 ∈ S (X × T). In this case, ψ ∗ idl ⊗ δ−γ · u0 also belongs to S ′ (X 2 ) and for any ϕ ∈ S (X 2 ) we can write that: Z


∗ ∗ ∗ ψ idl ⊗ δ−γ · u0 , ϕ = ψ idl ⊗ δ−γ , u0 ϕ = idl ⊗ δ−γ , ψ u0 ϕ = ϕ(x, x + γ)u0 (x, x) dx. But we notice that X Let us denote by v0 (x) := u0 (x, x) so that we obtain a test function v0 ∈ S (X ) and an equality:

ψ ∗ idl ⊗ δ−γ · u0 = ψ ∗ v0 ⊗ δ−γ .
Let us further denote by v := (2π)d /|E ∗ | v0 ∈ S (X ) ⊂ L2 (X ). If we insert (6.19) into (6.18) we obtain: u := X uγ ∗ = γ ∗ ∈Γ∗ X
ψ ∗ v ⊗ δ−γ ∈ L0 . (6.19) (6.20) γ∈Γ ∞ Let us verify now the continuity property. Suppose given u0 ∈ HA (X ) ⊗ L2 (T) and suppose given a sequence ǫ s {u0,k }k∈N∗ ⊂ S (X × T) and some s > d such that u0 = lim u0,k in HA (X ) ⊗ L2 (T). Let us also introduce the ǫ kր∞ notations: vk (x) := (2π)d u0,k (x, x), ∀x ∈ X ; |E ∗ | uk := X
ψ ∗ vk ⊗ δ−γ ∈ L0 . γ∈Γ From Lemma 6.11 we deduce that there exists a strictly positive constant Cs such that for any ǫ ∈ [−ǫ0 , ǫ0 ] and for any pair of indices (k, l) ∈ [N∗ ]2 the following estimations hold: kuk − ul kL0 := kvk − vl kL2 (X ) ≤ Cs ku0,k − u0,l kHsA ǫ kuk kL0 ≤ Cs ku0,k kHsA ǫ (X )⊗L2 (T) , (6.21) (X )⊗L2 (T) . (6.22) From (6.21) we deduce that there exists v ∈ L2 (X ) limit of the sequence {vk }k∈N∗ in L2 (X ) and moreover, using also (6.22), that it satisfies the estimation: kvkL2 (X ) ≤ Cs ku0 kHsA ǫ Let us denote by u e := P ψ ∗ v ⊗ δ−γ γ∈Γ
(X )⊗L2 (T) , ∀ǫ ∈ [−ǫ0 , ǫ0 ]. (6.23) ∈ L0 . From (6.23) we deduce that ke ukL0 ≤ Cs ku0 kHsA ǫ (X )⊗L2 (T) , ∀ǫ ∈ [−ǫ0 , ǫ0 ]. (6.24) P −s uγ ∗ in HA (X )⊗L2 (T). From (6.20) we know that In order to end the proof we have to show that u e = u := ǫ ∗ ∗ γ ∈Γ P Υγ ∗ u0,k . If we use now the inequality (6.17) with u0 replaced by u0,k − u0 , we obtain the estimation: uk := γ ∗ ∈Γ∗ kuk − ukH−s (X )⊗L2 (T) ≤ Cs ku0,k − u0 kHsA ǫ Aǫ (X )⊗L2 (T) . −s From this we deduce that u = lim uk in HA (X ) ⊗ L2 (T). But from (6.21) we deduce that u e = lim uk in L0 and ǫ kր∞ kր∞ −s (X ) ⊗ L2 (T). In conclusion u e = u and the proof is finished. thus, due to Lemma 6.8 also in HA ǫ 37 Proceeding as in Lemma 6.5 we can show that the following definition is meaningful, the series appearing in the definition of the space Ls (ǫ) being convergent as tempered distribution, and the space with the associated norm s being a Hilbert space canonically isomorphic with HA (X ) and continuously embedded into S ′ (X 2 ). ǫ Definition 6.13. For any s ∈ R and any ǫ ∈ [−ǫ0 , ǫ0 ], we define the following subspace of tempered distributions:     X
∗ s Ls (ǫ) := (X ), w ≡ w = , w ∈ S ′ (X 2 ) | ∃v ∈ HA ψ v ⊗ δ v −γ ǫ   γ∈Γ endowed with the quadratic norm: kwv kLs (ǫ) := kvkHsA ǫ (X ) . (6.25) Remark 6.14. For any w ∈ Ls (ǫ) we have the identity: (idl ⊗ τα )w = w, ∀α ∈ Γ. e s,ǫ introduced in Definition 2.4 b). Lemma 6.15. We recall the notation Q 1. We have the equality: Ls (ǫ) = n e s,ǫ w ∈ L0 w ∈ S ′ (X 2 ) | Q 2. On Ls (ǫ) the definition norm is equivalent with the following norm: kwk′Ls (ǫ) := e s,ǫ w Q L0 o . (6.26) . (6.27) 3. If s ≥ 0, then Ls (ǫ) is continuously embedded into L0 , uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ].
P ∗ s ψ v ⊗ δ−γ . But, we know Proof. 1. For any w ∈ Ls (ǫ), there exists a vector v ∈ HA (X ) such that w ≡ wv = ǫ γ∈Γ that by definition we have that Qs,ǫ v ∈ L2 (X ), so that we deduce that X

e s,ǫ wv = ψ ∗ Qs,ǫ ⊗ idl ψ ∗ wv = Q ψ ∗ (Qs,ǫ v) ⊗ δ−γ ∈ L0 . γ∈Γ e s,ǫ w belongs to L0 . By the definition of this last space it follows Reciprocally let w ∈ S ′ (X 2 ) be such that Q that there exists f ∈ L2 (X ) such that X
e s,ǫ w = Q ψ ∗ f ⊗ δ−γ . γ∈Γ It follows that w = X
ψ ∗ (Q−s,ǫ f ) ⊗ δ−γ . γ∈Γ s HA (X ) ǫ But then we have that v = Q−s,ǫ f ∈ and in conclusion w belongs to Ls (ǫ). 2. This result follows from the Closed Graph Theorem. s 3. This result follows from the continuous embedding of HA (X ) into L2 (X ) for any s ≥ 0 uniformly for ǫ ǫ ∈ [−ǫ0 , ǫ0 ]. Lemma 6.16. For any m ∈ R+ and for any ǫ ∈ [−ǫ0 , ǫ0 ] we have the following topological embedding:
Lm (ǫ) ֒→ S ′ X ; Hm (T) , (6.28) uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ]. Proof. From
the Lemmas 6.15 c) and 6.8 it follows that we have the following topological embedding: Lm (ǫ) ֒→ S ′ X ; L2 (T) uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ]. From here on we proceed as in the proof of the secong inclusion in Lemma 3.17. the sesquilinear form (3.45) to the canonical sesquilinear form
Extending by continuity
(., .)m on S ′ X ; Hm (T) × S X ; Hm (T) we can write it in the following way

∀(u, v) ∈ S ′ X ; Hm (T) × S (X × T) (6.29) (u, v)m = u, (idl⊗ < DΓ >2m )v 0 ,
and may extend it to S ′ X ; L2 (T) × S (X × T). e m,ǫ u = Q e −m,ǫ f , e m,ǫ u ∈ L0 . Because u = Q e −m,ǫ Q Let us choose now some u ∈ Lm (ǫ) and let us denote by f := Q for any v ∈ S (X × T) we shall have the equality
e −m,ǫ (idl⊗ < DΓ >2m )v . (u, v)m = f, Q (6.30) 0 38 
From the fact that L0 is
continuously embedded into S ′ X ; L2 (T) , it follows the existence of a defining semi-norm |.|l on S X ; L2 (T) such that e −m,ǫ (idl⊗ < DΓ >2m )v , |(u, v)m | ≤ kf kL0 Q ∀v ∈ S (X × T). l (6.31) Noticing that kf kL
0 = kukLm (ǫ) and applying Lemma 3.16, we deduce the existence of a defining semi-norm |.|−m,k on S X ; H−m (T) such that |(u, v)m | ≤ CkukLm (ǫ) (idl⊗ < DΓ >2m )v But (idl⊗ < DΓ >2m )v 7 −m,k −m,k , ∀v ∈ S (X × T). (6.32) = kvkm,k so that the statement of the Lemma follows from (6.32). The proof of Theorem 1.3 The main technical results discussed in this section concern some continuity properties of the operators Pǫ,λ and Eǫ,λ defined in Section 5 extended to some spaces of tempered distributions of the type considered in Section 6. We shall suppose that the Hypothesis H.1 - H.6 are satisfied and we shall use the notations introduced in Sections 1 and 2. Let us just recall that: ◦ ◦ • The operator Pǫ := OpAǫ (pǫ ) with pǫ (y, η) := p(y, y, η), defines a self-adjoint operator in L2 (X ) on the domain m HA (X ). ǫ e m (X 2 ) and • The operator Peǫ defines a self-adjoint operator Peǫ′ in the Hilbert space L2 (X 2 ) with domain H Aǫ another self-adjoint operator Peǫ′′ in the Hilbert space L2 (X × T) with the domain Kǫm (X 2 ). Lemma 7.1. For any ǫ ∈ [−ǫ0 , ǫ0 ] we have that:
1. Peǫ ∈ B Lm (ǫ); L0 uniformly in ǫ ∈ [−ǫ0 , ǫ0 ]. 2. The operator Peǫ considered as an unbounded operator in the Hilbert space L0 defines a self-adjoint operator Peǫ′′′ having domain Lm (ǫ) and this self-adjoint operator is unitarily equivalent with Pǫ . Proof. 1. Let us choose two test functions v and ϕ from S (X ). Using formula (2.4) we obtain that h
i  ∗
 ∀(x, y) ∈ X 2 . ψ Peǫ ψ ∗ (v ⊗ ϕ) (x, y) = OpAǫ [(idl ⊗ τy ⊗ idl)pǫ ]◦ v (x)ϕ(y),
∗ In this equality we insert ϕ(y) ≡ ϕλ (y) := λ−d θ y+γ for some (λ, γ) λ R ∈ R+ × Γ and for any y ∈ X , where we ∞ denoted by θ a test function of class C0 (X ) that satisfies the condition X θ(y)dy = 1. With this choice we consider the limit for λ ց 0 as tempered distribution on X 2 . Taking into account that for λ ց 0 we have that ϕλ converges in S ′ (X ) to δ−γ and using Hypothesis H.6, we conclude that
ψ ∗ Peǫ ψ ∗ (v ⊗ δ−γ ) =
Pǫ v ⊗ δ−γ , Extending by continuity we can write the equality

ψ ∗ Peǫ ψ ∗ (v ⊗ δ−γ ) = Pǫ v ⊗ δ−γ , We conclude that for any u ∈ Lm (ǫ) of the form u ≡ uv := m for some v ∈ HA (X ) we can write: ǫ  Peǫ u = ψ ∗  X γ∈Γ X ∀v ∈ S (X ), ∀γ ∈ Γ. ∀v ∈ S ′ (X ), ∀γ ∈ Γ. ψ ∗ v ⊗ δ−γ γ∈Γ (7.1)
 X

ψ ∗ Peǫ ψ ∗ (v ⊗ δ−γ ) = ψ ∗ (Pǫ v) ⊗ δ−γ . γ∈Γ
m The first statement of the Lemma follows now from the fact that Pǫ ∈ B HA (X ); L2 (X ) uniformly with respect ǫ to ǫ ∈ [−ǫ0 , ǫ0 ]. 39 2. Let us notice that the linear operator defined by Uǫs v := s (X ) → Ls (ǫ), Uǫs : HA ǫ X ψ ∗ v ⊗ δ−γ γ∈Γ
is in fact a unitary operator for any pair (s, ǫ) ∈ R × [−ǫ0 , ǫ0 ]. Following the arguments from the proof of the first m point of the Lemma we have the following equality Peǫ Uǫs = Uǫs Pǫ valid on HA (X ) (the domain of self-adjointness ǫ of Pǫ ). We shall study now the effective Hamiltonian E−+ (ǫ, λ) defined in Theorem 5.3. The following two technical N results will be used in proving the boundedness and self-adjointness of E−+ (ǫ,
λ) in V0 . N 0 Lemma 7.2. Suppose given an operator-valued symbol q ∈ S0 X ; B(C ) that is hermitian (i.e. q(x, ξ)∗ = q(x, ξ), ∀(x, ξ) ∈ Ξ) and verifies the following invariance property: (idl ⊗ τγ ∗ )q = q, ∀γ ∗ ∈ Γ∗ . Then, for any
ǫ ∈ [−ǫ0 , ǫ0 ] the operator OpAǫ (q) belongs to B VN uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ] and is self-adjoint. 0

Moreover, the application S00 X ; B(CN ) ∋ q 7→ OpAǫ (q) ∈ B VN is continuous uniformly with respect to 0 ǫ ∈ [−ǫ0 , ǫ0 ]. Proof. The invariance with respect to translations from Γ∗ assumed in the statement implies that the operatorvalued symbol q is in fact a Γ∗ -periodic function with respect to the second variable ξ ∈ X ∗ and thus can be
′ N decomposed in a Fourier series (as tempered distributions in S Ξ; B(C ) ): Z X i<ξ,α> ∗ −1 q̂α (x)e , q̂α (x) := |E | q(x, ξ) = e−i<ξ,α> q(x, ξ) dξ. (7.2) E∗ α∈Γ Due to the regularity of the symbol functions we deduce that for any β ∈ Nd and for any k ∈ N there exists a strictly positive constant Cβ,k such that
∀x ∈ X , ∀α ∈ Γ (7.3) ∂xβ q̂α (x) ≤ Cβ,k < α >−k ,

and we conclude that the series in (7.2) converges in fact in BC ∞ Ξ; B(CN ) ≡ S00 X ; B(CN ) . From (7.2) we deduce that X

OpAǫ (q)u (x) = Qα u (x), ∀x ∈ X , ∀u ∈ S (X : CN ), (7.4) α∈Γ where Qα is the linear operator defined on S (X ; CN ) by the following oscillating integral: Z
x + y
u(y) dy dη ¯ = ωAǫ (x, x + α) q̂α (x + α/2)(τ−α u)(x). Qα u (x) := ei<η,x−y+α> ωAǫ (x, y) q̂α 2 Ξ (7.5) Both equalities (7.4) and (7.5) may be extended by continuity to any u ∈ S ′ (X ; CN ). Let us consider then  2 N P f γ δ−γ ∈ VN . Then we can write: u ≡ uf = 0 for some f ∈ l (Γ) γ∈Γ Qα u = X ωAǫ (−γ − α, −γ) q̂α (−γ − α/2) f γ δ−α−γ = γ∈Γ X ωAǫ (−γ, α − γ) q̂α (−γ + α/2) f γ−α δ−γ . (7.6) γ∈Γ If we use now the formula (7.6) in (7.4) we get OpAǫ (q)u = X feγ δ−γ , (7.7) γ∈Γ feγ := X ωAǫ (−γ, α − γ) q̂α (−γ + α/2) f γ−α = α∈Γ X ωAǫ (−γ, −α) q̂γ−α − α∈Γ γ + α
f α. 2 (7.8)  N Let us verify that fe ∈ l2 (Γ) . In fact from (7.3) and (7.8) it follows that for any k ∈ N (sufficiently large) there exists Ck > 0 such that sX sX X |feγ | ≤ Ck < γ − α >−k |f α | ≤ Ck < γ − α >−k < γ − α >−k |f α |2 , α∈Γ α∈Γ α∈Γ so that we have the estimation: kfek2[l2 (Γ)]N = X |feγ |2 ≤ C ′ γ∈Γ 40 X |f α |2 = C ′ kf k2[l2 (Γ)]N . α∈Γ (7.9) 
From (7.7) and (7.9) we clearly deduce the fact that OpAǫ (q) ∈ B VN uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ] and 0

uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ] the continuity of the application S00 X ; B(CN ) ∋ q 7→ OpAǫ (q) ∈ B VN 0 clearly follows from (7.8) and (7.2). P g γ δ−γ In order to prove the self-adjointness of OpAǫ (q) we fix a second element v ∈ VN 0 of the form v ≡ vg =  N for some g ∈ l2 (Γ) . Then we notice that  Aǫ  Op (q)v  gγ e γ∈Γ P = = = X VN 0  X feγ , gγ = γ∈Γ P ωAǫ (−γ, −α) q̂γ−α − α∈Γ Let us point out the following evident equalities:  q̂(x)]∗ = q̂−α (x); in order to deduce that   OpAǫ (q)u , v gγ δ−γ e γ∈Γ CN = γ+α 2
(7.10) gα . ωAǫ (−γ, −α) = ωAǫ (−α, −γ) X (α,γ)∈Γ2  ωAǫ (−γ, −α) q̂γ−α γ + α
− f α , gγ 2 (7.11)  = CN      X γ + α
gγ f α , ωAǫ (−α, −γ) q̂α−γ − = = u , OpAǫ (q)v N . gα fα , e 2 CN V0 CN 2 α∈Γ (α,γ)∈Γ Remark 7.3. Let us point out that a shorter proof of the boundedness of OpAǫ (q) on VN 0 may be obtained by using the Proposition 6.4 characterizing the distributions from V0 . The proof we have given has the advantage  2 N of giving the explicit form of the operator OpAǫ (q) when we identify VN (see (7.7) and (7.8)). 0 with l (Γ) Moreover, the self-adjointness is a very easy consequence of these formulae. In order to prove that the effective Hamiltonian E−+ (ǫ, λ) satisfies the hypothesis of the Lemma 7.2 we shall need the commutation properties we have proved at the end of Section 5, that we now recall in the following Lemma. Lemma 7.4. With the notations introduced in Lemma 5.6 and Remark 5.7, for any γ ∗ ∈ Γ∗ and for any (ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I the following equalities are true:  R−,ǫ σγ ∗ = Υγ ∗ R−,ǫ (7.12) R+,ǫ Υγ ∗ = σγ ∗ R+,ǫ ,  E(ǫ, λ) Υγ ∗ = Υγ ∗ E(ǫ, λ)    E+ (ǫ, λ) σγ ∗ = Υγ ∗ E+ (ǫ, λ) (7.13) E− (ǫ, λ) Υγ ∗ = σγ ∗ E− (ǫ, λ)    E−+ (ǫ, λ) σγ ∗ = σγ ∗ E−+ (ǫ, λ). Proof. The equalities (7.12) are exactly the equalities (5.24) and (5.25) that we have proved in Section 5. The equalities (7.13) follow from (5.26) and (5.11).
uniformly with respect to Lemma 7.5. Under the Hypothesis of Theorem 5.3, we have that E−+ (ǫ, λ) ∈ B VN 0 (ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I and is self-adjoint on the Hilbert space VN . 0
−,+
−,+ Aǫ Proof. We recall that E−+ (ǫ, λ) := Op Eǫ,λ where Eǫ,λ ∈ S00 X ; B(CN ) . This Lemma will thus follow −,+ directly from Lemma 7.2 once we have shown that Eǫ,λ is hermitian and Γ∗ -periodic in the second variable ∗ ξ∈X . In order to prove the symmetry we use the fact that the operator Eǫ,λ is self-adjoint on K(X 2 ) × L2 (X ; CN ) and  ∗ deduce that E−+ (ǫ, λ) is self-adjoint on the Hilbert space L2 (X ; CN ). Thus we have the equality E−+ (ǫ, λ) = E−+ (ǫ, λ) from which we deduce that  −,+ ∗ −,+
= 0. − Eǫ,λ OpAǫ Eǫ,λ
As the application OpAǫ : S ′ (Ξ) → B S (X ); S ′ (X ) is an isomorphism (see [25]) it follows the symmetry relation  −,+ ∗ −,+ Eǫ,λ = Eǫ,λ . 41 For the Γ∗ -periodicity we use the last equality in (7.13) that can also be written as σ−γ ∗ E−+ (ǫ, λ)σγ ∗ = E−+ (ǫ, λ). Considering now the equality (9.4), that evidently remains true also for the OpAǫ quantization, we can write
−+ . σ−γ ∗ E−+ (ǫ, λ)σγ ∗ = OpAǫ (idl ⊗ τ−γ ∗ )Eǫ,λ Repeating the above argument based on the injectivity of the quantization map ([25]) we conclude that (idl ⊗ −+ −+ = Eǫ,λ for any γ ∗ ∈ Γ∗ . τ−γ ∗ )Eǫ,λ We shall now study the continuity properties of the operators R±,ǫ , E± (ǫ, λ) and E(ǫ, λ) acting on the distribution spaces V0 or L0 by using the characterizations of these spaces obtained in Section 6 (Propositions 6.4, 6.10 and 6.12) as well as the commutation properties recalled in Lemma 7.4. We shall suppose the hypothesis of Theorem 5.3 are satisfied and (ǫ,
λ) ∈ [−ǫ0 , ǫ0 ] × I. uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ]. Lemma 7.6. R+,ǫ ∈ B L0 ; VN 0

Proof. Let us recall that
R+,ǫ = OpAǫ R+ with R+ ∈ S00 X ; B(K0 ; CN ) so that finally we deduce that R+,ǫ ∈ B S ′ (X ; K0 ); S ′ (X ; CN ) . Moreover, from Proposition 9.27 we deduce that for any s ∈ R we have that R+,ǫ ∈  s N
s B HA (X ) ⊗ K0 ; HA (X ) uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ]. ǫ ǫ ∞ Suppose fixed some u ∈ L0 ; P from Proposition 6.10 we deduce the existence of a unique u0 ∈ HA (X ) ⊗ K0 ≡ ǫ ∞ 2 ′ 2 ∗ HAǫ (X ) ⊗ L (T) such that u = Υγ u0 with convergence in S (X ). In fact Lemma 6.9 implies the convergence γ∗

′ of the above series in S X ; K0 . Using now also the second equation in (7.12) we can write that in S ′ X ; CN we have the equalities X X R+,ǫ u = R+,ǫ Υγ ∗ u0 = σγ ∗ R+,ǫ u0 . γ∗ γ∗  ∞ N But we have seen that R+,ǫ u0 ∈ HA (X ) and thus Proposition 6.4 b) implies that R+,ǫ u ∈ VN 0 . The fact ǫ
N that R+,ǫ ∈ B L0 ; V0 uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ] follows now from this result and the following three remarks: 1. The above mentioned continuity property of R+,ǫ that follows from Proposition 9.27. ∞ 2. The uniform continuity of the application L0 ∋ u 7→ u0 ∈ HA (X ) ⊗ K0 with respect to ǫ ∈ [−ǫ0 , ǫ0 ], that ǫ follows from Proposition 6.10. ∞ 3. The uniform continuity of the application HA (X ) ∋ R+,ǫ u0 7→ R+,ǫ u ∈ VN 0 with respect to ǫ ∈ [−ǫ0 , ǫ0 ], ǫ that follows from Proposition 6.4 b).
uniformly with respect to (ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I. Lemma 7.7. E− (ǫ, λ) ∈ B L0 ; VN 0

− − Proof. Let us recall that E− (ǫ, λ) = OpAǫ Eǫ,λ with Eǫ,λ ∈ S00 X ; B(K0 ; CN ) uniformly with respect to (ǫ, λ) ∈ [−ǫ0 , ǫ0 ]×I. We continue as in the above proof of Lemma 7.6. Considering E− (ǫ, λ)
as a magnetic pseudodifferential operator it can be extended to an operator E− (ǫ, λ) ∈ B S ′ (X ; K0 ); S ′ (X ; CN ) ; moreover we notice that L0
is continuously embedded into S ′ (X ; K0 ) and we conclude that we have indeed that E− (ǫ, λ) ∈ B L0 ; S ′ (X ; CN ) . ∞ From for any u ∈ L0 there exists some u0 ∈ HA (X ) ⊗ K0 such that u = ǫ P Proposition 6.10 we conclude that ′ Υγ ∗ u0 , the series converging in S (X ; K0 ). Using this identity and the third equality in (7.13) we obtain that γ ∗ ∈Γ∗ E− (ǫ, λ)u = X E− (ǫ, λ)Υγ ∗ u0 = γ∗ X σγ ∗ E− (ǫ, λ)u0 . γ∗  s N
s Taking into account that for any s ∈ R we have that E− (ǫ, λ) ∈ B HA (X ) ⊗ K0 ; HA (X ) uniformly with ǫ ǫ respect to (ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I, the proof of the Lemma ends similarly to the proof of Lemma 7.6 above.
Lemma 7.8. E+ (ǫ, λ) ∈ B VN 0 ; Lm (ǫ) uniformly with respect to (ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I. 42 

+ + Proof. Let us recall that E+ (ǫ, λ) = OpAǫ Eǫ,λ with Eǫ,λ ∈ S00 X ; B(CN ; Km,ξ ) uniformly with respect to
(ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I. We conclude that E+ (ǫ, λ) ∈ B S ′ (X ; CN ); S ′ (X ; Km,0 ) . Noticing that by
Lemma 6.2 ′ N N ′ the space VN 0 embeds continuously into S (X ; C ) we conclude that E+ (ǫ, λ) ∈ B V0 ; S (X ; Km,0 ) uniformly with respect to (ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I.  ∞ N Suppose now fixed some u ∈ VN ; we know from Proposition 6.4 that there exists an element u0 ∈ HA (X ) 0 ǫ P σγ ∗ u0 converging as tempered distribution and such that the application VN such that u = 0 ∋ u 7→ u0 ∈ ∗ γ ∈Γ∗  ∞ N HAǫ (X ) is continuous uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ]. Using this result and the second equation in (7.13) we obtain that X X
E+ (ǫ, λ)u = E+ (ǫ, λ)σγ ∗ u0 = Υγ ∗ E+ (ǫ, λ)u0 . γ ∗ ∈Γ∗ γ ∗ ∈Γ∗ e m,ǫ E+ (ǫ, λ)u ∈ L0 . Using now Lemma 6.15, in order to prove that E+ (ǫ, λ)u ∈ Lm (ǫ) all we have to prove is that Q e In order to do that we shall need two of the properties of the operator Qm,ǫ that we have proved in the previous sections. First we know from (3.22) that e m,ǫ Υγ ∗ = Υγ ∗ Q e m,ǫ , Q ∀γ ∗ ∈ Γ∗ .
e m,ǫ = OpAǫ e Secondly, at the end of the proof of Lemma 5.2 we have shown that Q qm,ǫ with e qm,ǫ ∈
to ǫ ∈ [−ǫ , ǫ ]. If we use The Composition Theorem 9.24 we notice S00 (X ; B(Km,ξ ; K0 ) uniformly with respect 0 0
+ ∈ S00 X ; B(CN ; K0 ) uniformly with respect to (ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I. Applying then Proposition that e qm,ǫ ♯Bǫ Eǫ,λ 
 e m,ǫ E+ (ǫ, λ) ∈ B H∞ (X ) N ; H∞ (X ) ⊗ K0 uniformly with respect to (ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I. We 9.27 gives that Q Aǫ Aǫ conclude that X
e m,ǫ E+ (ǫ, λ)u = e m,ǫ E+ (ǫ, λ)u0 , Q Υγ ∗ Q γ ∗ ∈Γ∗ and this last element belongs to L0 as implied by Proposition 6.12. The conclusion of the Lemma follows now from the following remarks:  ∞ N 1. The application VN is continuous uniformly with respect to (ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I, as 0 ∋ u 7→ u0 ∈ HAǫ (X ) proved in Proposition 6.4 a). ∞ e m,ǫ E+ (ǫ, λ)u0 7→ Q e m,ǫ E+ (ǫ, λ)u ∈ L0 is continuous uniformly with respect 2. The application HA (X ) ⊗ K0 ∋ Q ǫ to (ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I, as proved in Proposition 6.12.
Lemma 7.9. R−,ǫ ∈ B VN 0 ; Lm (ǫ) uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ].

Proof. Let us recall that R−,ǫ = OpAǫ R− with R− ∈ S00 X ; B(CN ; Km,ξ ) as implied by (4.22) and (4.26). Using now the first equality in (7.12) we notice that R−ǫ σγ ∗ = Υγ ∗ R−,ǫ , ∀γ ∗ ∈ Γ∗ and the arguments from the proof of Lemma 7.8 may be repeated and one obtains the desired conclusion of the Lemma.
Lemma 7.10. E(ǫ, λ) ∈ B L0 ; Lm (ǫ) uniformly with respect to (ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I.

Proof. Let us recall that E(ǫ, λ) = OpAǫ Eǫ,λ with Eǫ,λ ∈ S00 X ; B(K0 ; Km,ξ ) uniformly with respect to (ǫ, λ) ∈
[−ǫ0 , ǫ0 ]×I. As magnetic pseudodifferential operator we can then extend it to E(ǫ, λ) ∈ B S ′ (X ; K0 ); S ′ (X ; K
m,0 ) . Recalling that we have a continuous embedding L0 ֒→ S ′ (X ; K0 ) we deduce that E(ǫ, λ) ∈ B L0 ; S ′ (X ; Km,0 ) . We ∞ use now Proposition 6.10 and the first equality in (7.13) and write that for any u ∈ L0 there exists u0 ∈ HA (X )⊗K0 ǫ such that: X X
E(ǫ, λ)u = E(ǫ, λ)Υγ ∗ u0 = Υγ ∗ E(ǫ, λ)u0 , γ ∗ ∈Γ∗ γ ∗ ∈Γ∗ with convergence in the sense of tempered distributions on X 2 . From Proposition 6.10 we deduce that the applica∞ tion L0 ∋ u 7→ u0 ∈ HA (X ) ⊗ K0 is continuous uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ] and from The Composition ǫ
Theorem 9.24 we deduce that e qm,ǫ ♯Bǫ Eǫ,λ ∈ S00 X ; B(K0 ) and the proof of the Lemma ends exactly as the proof of Lemma 7.8. 43 Now we shall prove a variant of Theorem 5.3 in the frame of the Hilbert spaces V0 and L0 . Theorem 7.11. We suppose verified the hypothesis of Theorem 5.3 and use the same notations; then we have that

N N (7.14) Eǫ,λ ∈ B L0 × VN Pǫ,λ ∈ B Lm (ǫ) × VN 0 ; Lm (ǫ) × V0 , 0 ; L0 × V0 , uniformly with respect to (ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I. Moreover, for any (ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I the operator Pǫ,λ is invertible and its inverse is Eǫ,λ . Proof. The boundedness properties in (7.14) follow from Lemmas 7.1 (a), 7.5, 7.6, 7.7, 7.8, 7.9 and 7.10. Concerning the invertibility of Pǫ,λ let us recall that in Theorem 5.3
we have proved that the operator Pǫ,λ considered as operator in B Kǫm (X 2 ) × L2 (X
; CN ); K(X 2 ) × L2 (X ; CN ) is invertible and its inverse is Eǫ,λ ∈ B K(X 2 ) × L2 (X ; CN ); Kǫm (X 2 ) × L2 (X ; CN ) . From (5.7)
we recall that Pǫ,λ is a magnetic pseudodifferential operator with symbol Pǫ of class S00 X ; B(Km,ξ × CN ; K0 × CN ) uniformly with respect to (ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I. Applying Proposition 9.7 we deduce that
Pǫ,λ ∈ B S (X ; Km,0 ) × S (X ; CN ); S (X ; K0 ) × S (X ; CN ) , (7.15) and extending by continuity we also have that
Pǫ,λ ∈ B S ′ (X ; Km,0 ) × S ′ (X ; CN ); S ′ (X ; K0 ) × S ′ (X ; CN ) . Similarly, the operator Eǫ,λ appearing in Theorem 5.3 has a symbol of class S00 X ; B(K0 × CN ; Km,ξ and thus defines first an operator of the form
Eǫ,λ ∈ B S (X ; K0 ) × S (X ; CN ); S (X ; Km,0 ) × S (X ; CN ) , (7.16)
× CN ) (7.17) and extending by continuity we also have that
Eǫ,λ ∈ B S ′ (X ; K0 ) × S ′ (X ; CN ); S ′ (X ; Km,0 ) × S ′ (X ; CN ) . (7.18) From the first inclusion in Lemma 3.17 it follows that S (X ; Km,0 ) ֒→ Kǫm (X 2 ), so that from the invertibility implied by Theorem 5.3 (see above in this proof), it also follows that the operator Pǫ,λ appearing in (7.15) is invertible and its inverse is the operator Eǫ,λ appearing in (7.17). As both operators Pǫ,λ and Eǫ,λ are symmetric, by duality we deduce that also the operators appearing in (7.16) and (7.18) are the inverse of one another. This property, together with the embeddings Lm (ǫ) ֒→ S ′ (X ; Km,0 ) given by Lemma 6.16, L0 ֒→ S ′ (X ; Ko ) given by Lemma 6.8 and V0 ֒→ S ′ (X ) given by Lemma 6.2 allow us to end the proof of the Theorem. We come now to the proof of the main result of this paper. Proof of the Theorem 1.3 We proceed exactly as in the proof of Corollary 5.5. We start from the equality   0 idlL0 Pǫ,λ Eǫ,λ = 0 idlVN 0 and use the fact that Peǫ′′′ is a self-adjoint operator in L0 that is unitarily equivalent with Pǫ (by Lemma 7.1)

so that we deduce that σ Peǫ′′′ = σ Pǫ . Then we can write that

−1
= E(ǫ, λ) − E+,ǫ (ǫ, λ)E−+ (ǫ, λ)−1 E−,ǫ (ǫ, λ) (7.19) 0∈ / σ E−+ (ǫ, λ) =⇒ λ ∈ / σ Peǫ′′′ , and Peǫ′′′ − λ
−1
R−,ǫ . (7.20) =⇒ 0 ∈ / σ E−+ (ǫ, λ) , and E−+ (ǫ, λ)−1 = −R+,ǫ Peǫ′′′ − λ


In conclusion we
have obtained that λ ∈ σ Peǫ′′′ ⇔ 0 ∈ σ E−+ (ǫ, λ) and this implies that λ ∈ σ Pǫ ⇔ 0 ∈ σ E−+ (ǫ, λ) .  λ∈ / σ Peǫ′′′
An imediate consequence of Theorem 1.3 is the following result concerning the stability of spectral gaps for the operator Pǫ : Proof of Corollary C.0.2 We apply Theorem 1.3 and

the arguments from its proof above, taking I = K

and ǫ0 > 0 sufficiently small. Knowing that dist K, σ P0 > 0, we deduce that we also have dist K, σ Pe0′′′ > 0 and thus we have the estimation:
−1 sup Pe0′′′ − λ < ∞. (7.21) B(L0 ) λ∈K 44 From (7.20) we deduce that and thus

−1 λ ∈ K =⇒ 0 ∈ / σ E−+ (0, λ) and E−+ (0, λ)−1 = −R+,0 Pe0′′′ − λ R−,0 , sup E−+ (0, λ)−1 λ∈K B(VN 0 ) < ∞. (7.22) From Theorem 5.3 it follows that for any (ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × K:
−+ S−+ (ǫ, λ) := OpAǫ Sǫ,λ , E−+ (ǫ, λ) = E−+ (0, λ) + S−+ (ǫ, λ),
−+ lim Sǫ,λ = 0 in S 0 X ; B(C N ) , ǫ→0 uniformly with respect to λ ∈ K. (7.23) (7.24) −+ We notice that the symbol Sǫ,λ (x, ξ) is Γ∗ -periodic in the second variable ξ ∈ X ∗ , so that from Lemma (7.2) we deduce that (7.25) lim kS−+ (ǫ, λ)kB(VN ) = 0, ǫ→0 0 uniformly with respect to λ ∈ K. From (7.22), (7.23) and (7.25) imply that for ǫ0 > 0 sufficiently small, the magnetic pseudodifferential
operator E−+ (ǫ, / σ E−+ (ǫ, λ) and
λ) is invertible in B(V0 ) for any (ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × K; in conclusion 0 ∈ thus λ ∈ / σ Pǫ for any (ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × K. The arguments elaborated in the proof of Corollary 1.4 allow to obtain an interesting relation between the spectra of the operators Pǫ and P0 , under some stronger hypothesis. Hypothesis I.1 Under the conditions of Hypothesis H.1 we suppose further that for any pair (j, k) of indices between 1 and d the family {ǫ−1 Bǫ,jk }0<|ǫ|≤ǫ0 are bounded subsets of BC ∞ (X ). Hypothesis I.2 We suppose that pǫ (x, y, η) = p0 (y, η) + rǫ (x, y, η) where p0 is a real valued symbol from S1m (T) with m > 0 and the family {ǫ−1 rǫ }0<|ǫ|≤ǫ0 is a bounded subset of S1m (X × T), each symbol rǫ being real valued. Hypothesis I.3 The symbol p0 is elliptic; i.e. there exist C > 0, R > 0 such that p0 (y, η) ≥ C|η|m for any (y, η) ∈ Ξ with |η| ≥ R. Remark 7.12. If we come back to the proofs of Theorem 5.3, Theorem 9.24 (of Composition) and Proposition 9.28 and suppose the Hypothesis I.1 - I.3 to be true, we can prove the following fact that extends our property (5.12): ∀I  ⊂ R compact interval, ∃ǫ0 > 0, ∃N ∈ N, such that:  ∀(ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I, E−+ (ǫ, λ) = E−+ (0, λ) + S−+ (ǫ, λ), n o  the familly −+ ǫ−1 Sǫ,λ (|ǫ|,λ)∈(0,ǫ0 ]×I
−+ , S−+ (ǫ, λ) := OpAǫ Sǫ,λ
N 0 is a bounded subset of S X ; B(C ) . (7.26) −+ (x, ξ) with respect to the variable ξ ∈ X ∗ and from Once again we notice the Γ∗ -periodicity of the symbol Sǫ,λ Lemma 7.2 we deduce that there exists a strictly positive constant C1 such that the following estimation is true: kS−+ (ǫ, λ)kB(VN ) ≤ C1 ǫ, 0 ∀(ǫ, λ) ∈ [−ǫ0 , ǫ0 ] × I. (7.27) Using Lemma 7.6 and 7.9 we conclude that there exists a strictly positive constant C2 such that the following estimation is true: kR+,ǫ kB(L0 ;VN ) + kR−,ǫ kB(VN ;Lm (ǫ)) ≤ C2 ∀ǫ ∈ [−ǫ0 , ǫ0 ]. (7.28) 0 0 Proof of Proposition 1.5 For M ⊂ R and δ > 0 we use the notation Mδ := {t ∈ R | dist(t, M ) ≤ δ}. Then we have to prove the following inclusions:

∀ǫ ∈ [0, ǫ0 ]. (7.29) σ Pǫ ∩ I ⊂ σ P0 Cǫ ∩ I,

∀ǫ ∈ [0, ǫ0 ]. (7.30) σ P0 ∩ I ⊂ σ Pǫ Cǫ ∩ I,



Suppose there exists λ ∈ I such that dist λ, σ P0 > Cǫ. From Lemma 7.1 we know that σ P0 = σ Pe0′′′

so that we deduce that dist λ, σ Pe0′′′ > Cǫ and conclude that:
−1 Pe0′′′ − λ 45 B(L0 ) ≤ (Cǫ)−1 . (7.31) 
−1
R−,0 . Using these From (7.20) we also deduce that 0 ∈ / σ E−+ (0, λ) and E−+ (0, λ)−1 = −R+,0 Pe0′′′ − λ facts together with (7.28) and (7.31) we obtain the estimation: E−+ (0, λ)−1 B(VN 0 ) ≤ C22 (Cǫ)−1 . (7.32) Using (7.27) and (7.32) we also obtain the following estimation: E−+ (0, λ)−1 B(VN 0 ) · kS−+ (ǫ, λ)kB(VN ) ≤ C1 C22 C −1 , 0 ∀ǫ ∈ [−ǫ0 , ǫ0 ]. (7.33) If we choose now C > 0 such that C > C1 C22 , we notice that the operator E−+ (ǫ, λ) = E−+ (0, λ) + S
−+ (ǫ, λ)
is invertible in B(VN ) and thus we deduce that 0 ∈ / σ E (ǫ, λ) . It follows then that λ ∈ / σ P −+ ǫ for any 0 ǫ ∈ [−ǫ0 , ǫ0 ] and the inclusion (7.29) follows. Let us prove now (7.30). Let us suppose that for some ǫ with |ǫ| ∈ (0, ǫ0 ] there exists some λ ∈ I such that





dist λ, σ Pǫ > Cǫ. Recalling that σ Pǫ = σ Peǫ′′′ we deduce that we also have dist λ, σ Peǫ′′′ > Cǫ and thus
−1 ≤ (Cǫ)−1 . (7.34) Peǫ′′′ − λ B(L0 )
−1
R−,ǫ . Using these facts together We also deduce that 0 ∈ / σ E−+ (ǫ, λ) and E−+ (ǫ, λ)−1 = −R+,ǫ Peǫ′′′ − λ with (7.28) and (7.31) we obtain the estimation: E−+ (ǫ, λ)−1 B(VN 0 ) ≤ C22 (Cǫ)−1 . (7.35) It follows like above that the
operator E−+ (0, λ) = E−+ (ǫ, λ)
− S−+ (ǫ, λ) is invertible in B(VN 0 ) and thus we deduce that 0 ∈ / σ E−+ (0, λ) . It follows then that λ ∈ / σ P0 and the inclusion (7.30) follows. Remark 7.13. The relations (7.29) and (7.30) clearly imply that the boundaries of the spectral gaps of the operator Pǫ are Lipshitz functions of ǫ in ǫ = 0. 8 Some particular situations 8.1 The simple spectral band In this subsection we shall find some explicit forms for the principal part of the effective Hamiltonian E−+ (ǫ, λ). We shall suppose the Hypothesis H.1 - H.6 to be satisfied. For the beginning we concentrate on the operator P0 = Op(p0 ) with p0 ∈ S1m (T) a real elliptic symbol. We know tht P0 has a self-adjoint realization as operator acting in L2 (X ) with the domain Hm (X ) (the usual Sobolev space of order m). From Lemma 9.19 we obtain that τγ P0 = P0 τγ , ∀γ ∈ Γ and thus we can use the Floquet theory in order to study the spectrum of the operator P0 . We shall consider the following spaces similar to the one used in Section 3 but with one variable less:  SΓ′ (Ξ) := v ∈ S ′ (Ξ) | v(y + γ, η) = ei<η,γ> v(y, η) ∀γ ∈ Γ, v(y, η + γ ∗ ) = v(y, η) ∀γ ∈ Γ∗ (8.1) endowed with the topology induced from S ′ (Ξ).  F0 (Ξ) := v ∈ L2loc (Ξ) ∩ SΓ′ (Ξ) | v ∈ L2 (E × E ∗ )
1/2 R R . that is a Hilbert space for the norm kvkF0 (Ξ) := |E ∗ |−1 E E ∗ |v(x, ξ)|2 dx dξ 
∀s ∈ R, Fs (Ξ) := v ∈ SΓ′ | < D >s ⊗idl v ∈ F0 (Ξ) ,
that is a Hilbert space with the norm kvkFs (Ξ) := k < D >s ⊗idl vkF0 (Ξ) . The following Lemma and its prof are completely similar with the case discussed in Section 3. Lemma 8.1. With the above notations the following statements are true: 1. The operator U0 : L2 (X ) → F0 (Ξ) defined by X
U0 u (x, ξ) := ei<ξ,γ> u(x − γ), γ∈Γ ∀(x, ξ) ∈ Ξ, is a unitary operator. The inverse of the operator U0 is the operator W0 : F0 (Ξ) → L2 (X ) defined as Z
W0 v (x) := |E ∗ |−1 v(x, ξ) dξ, ∀x ∈ X . E∗ 46 (8.2) (8.3) (8.4) (8.5) 2. For any s ∈ R the restriction of the operator U0 to Hs (X ) defines a unitary operator U0 : Hs (X ) → Fs (Ξ). Lemma 8.2. With the above notations the following statements are true: 1. The operator Pb0 := P0 ⊗ idl leaves invariant the subspace SΓ′ (Ξ). 2. Considered as an unbounded operator in the Hilbert space F0 (Ξ), the operator Pb0 is self-adjoint and lower semi-bounded on the domain Fm (Ξ) and is unitarily equivalent to the operator P0 . Proof. The first conclusion follows easily from Lemma 9.19. For the second conclusion let us notice that (8.4) implies the following equality: U0 P0 u = Pb0 U0 u, ∀u ∈ Hm (X ), (8.6) that together withLemma 8.1 implies that Pb0 and P0 are unitarily equivalent. Let us notice that for any v ∈ F0 (Ξ) and for any ξ ∈ X ∗ , the restriction v(·, ξ) defines a function on X that is of class F0,ξ and by Remark 9.17 this last space is canonically unitarily equivalent with L2 (E); moreover we have the equality Z 1/2 kvkF0 (Ξ) = |E ∗ |−1/2 kv(·, ξ)k2F0,ξ dξ . E∗ The following periodicity is evidently true: F0,ξ = F0,ξ+γ ∗ for any γ ∗ ∈ Γ∗ . Moreover, it is easy to see that we can make the following identification: Z ⊕ F0,ξ dξ. F0 (Ξ) ≡ T∗,d Similarly, we also have the following relations: Fm,ξ+γ ∗ = Fm,ξ , ∀γ ∗ ∈ Γ∗ ; Fm (Ξ) ≡ Z ⊕ Fm,ξ dξ. T∗,d Taking into account the Remark 9.23 we notice that for any ξ ∈ X ∗ the operator P0 induces on the Hilbert space F0,ξ a self-adjoint operator with domain Fm,ξ that we shalll denote by Pb0 (ξ); we evidently have the periodicity Pb0 (ξ + γ ∗ ) = Pb0 (ξ) for any γ ∗ ∈ Γ∗ . b If we idntify K0 with L2loc ∩ SΓ′ (Ξ) ≡ L2 (E), the same Remark 9.23
implies that the operator P0 (ξ) is unitarily equivalent with the operator P̌0 (ξ) that is induced by Op (idl ⊗ τ−ξ )p on the space K0 ; this is a self-adjoint lower m semi-bounded operator on the domain Km,ξ (identified with Hloc (X )∩SΓ′ (X ), with the norm k < D+ξ >m ·kL2 (E) ). More precisely, we can write that ∀ξ ∈ X ∗ . (8.7) P̌0 (ξ) = σ−ξ Pb0 (ξ)σξ ,
Lemma 8.3. For any z ∈ C \ ∪ ∗ σ P̌0 (ξ) the application ξ∈X X ∗ ∋ ξ 7→ P̌0 (ξ) − z is of class C ∞ (X ∗ ).
−1 ∈ B(K0 ) (8.8) Proof. From the Example 9.21 follows that the application X ∗ ∋ ξ 7→ P̌0 (ξ) − z
−1 ∈ B(Km,0 ; K0 ) is of class C ∞ (X ∗ ). But from the second rezolvent equality: P̌0 (ξ) − z
−1 − P̌0 (ξ0 ) − z
−1 = P̌0 (ξ) − z
−1

−1 P̌0 (ξ0 ) − P̌0 (ξ) P̌0 (ξ0 ) − z follows the continuity of the application (8.8) and the existence of the derivatives follows by usual arguments. Lemma 8.4. The following equality is true: Pb0 = Z ⊕ T∗ 47 Pb0 (ξ) dξ. (8.9) Proof. First let us notice the equality
Pb0 u (·, ξ) = Pb0 (ξ)u(·, ξ), ∀ξ ∈ T∗ , ∀u ∈ F0 (Ξ). (8.10) Then, from (8.7) we deduce the equality: Pb0 (ξ) + i
−1 = σξ P̌0 (ξ) + i
−1 σ−ξ , ∀ξ ∈ X ∗ and from Lemma 8.3 we obtain the continuity of the application T∗ ∋ ξ 7→ Pb0 (ξ) + i and that together with (8.10) imply equality (8.9).
−1
∈ B L2 (E) , Remark 8.5. Let us notice that Km,ξ is compactly embedded into K0 and thus, the operator P̌0 (ξ) has compact rezolvent for any ξ ∈ X ∗ ; it is clearly lower semi-bounded uniformly with respect to ξ ∈ X ∗ (taking into account

that P̌0 (ξ + γ ∗ ) = σ−γ ∗ P̌0 (ξ)σγ ∗ , ∀γ ∗ ∈ Γ∗ ). We deduce that σ Pb0 (ξ) = σ P̌0 (ξ) = {λj (ξ)}j≥1 , where for any ξ ∈ X ∗ and any j ≥ 1, λj (ξ) is a real finitely degenerated eigenvalue and lim λj (ξ) = ∞ ∀ξ ∈ X ∗ ; we can j→∞ always renumber the eigenvalues and suppose that λj (ξ) ≤ λj+1 (ξ) for any j ≥ 1 and for any ξ ∈ X ∗ . Due to the Γ∗ -periodicity of Pb(ξ) we have that λj (ξ + γ ∗ ) = λj (ξ) for any j ≥ 1, for any ξ ∈ X ∗ and for any γ ∗ ∈ Γ∗ . These are the Floquet eigenvalues of the operator Pb0 . Lemma 8.6. For each j ≥ 1 the function T∗,d ∋ ξ 7→ λj (ξ) ∈ R is continuous on T∗,d uniformly in j ≥ 1. Proof. From the uniform lower semi-boundedness it follows the existence of some real number c ∈ R such that
−1 λj (ξ) ≥ c + 1 for any j ≥ 1 and for any ξ ∈ X ∗ . We can thus define the operators R(ξ) :=
P̌0 (ξ) − c , for ∗ ∞ ∗ any ξ ∈ X and due to the result in Lemma 8.3 we obtain a function of class C X ; B(K0 ) ; for any ξ ∈ X ∗

−1 the operator R(ξ) is a bounded self-adjoint operator on K0 and σ R(ξ) = { λj (ξ) − c }j≥1 . Applying now the Min-Max Principle (see [32]) we obtain easily that: λj (ξ) − c
−1 − λj (ξ0 ) − c
−1 ≤ kR(ξ) − R(ξ0 )kB(K0 ) , ∀(ξ, ξ0 ) ∈ [X ∗ ]2 , ∀j ≥ 1.


∞ Lemma 8.7. We have the following spectral decomposition: σ P0 = σ Pb0 = ∪ Jk with Jk := λk T∗,d is a k=1 compact interval in R. Proof. Lemma 8.2 states that the operators P0 and Pb0 are unitarily equivalent and thus they have the same spectrum. From Theorem XIII.85 (d) in [32] it follows that: λ ∈ σ Pb0 ∞
⇐⇒ ∀ǫ > 0, 
ξ ∈ T∗,d | σ P̌0 (ξ) ∩ (λ − ǫ, λ + ǫ) 6= ∅ > 0. Let us denote by M := ∪ Jk . If λ ∈ M , it exist ξ0 ∈ T∗,d and k ≥ 1 such that λ = λk (ξ0 ). Due to the continuity k=1 of λk (ξ) we know that for any ǫ > 0 there exists a neighborhood V of ξ0 in T∗,d such that |λk (ξ) − λ| ≤ ǫ for any

ξ ∈ V . It follows that λ ∈ σ Pb0 and thus M ⊂ σ Pb0 .
On the other hand it is evident by definitions that σ Pb0 ⊂ M and thus we need to prove that M is a closed set. Let us fix some λ ∈ M ; it follows that there exists a sequence {µl }l≥1 ⊂ M such that µl → λ. We deduce that l→∞ for any l ≥ 1 there exists a point ξ l ∈ T∗,d and an index kl ≥ 1 such that µl = λkl (ξ l ). The manifold T∗,d being compact it follows that we may suppose that it exists ξ ∈ T∗,d such that ξ l → ξ. Due to the uniform continuity l→∞ of the functions λj with respect to j ≥ 1 we deduce that λkl (ξ) → λ. Taking into account that λj (ξ) → ∞ for l→∞ j→∞ any ξ ∈ T∗,d we deduce that the sequence {jl }l≥1 becomes constant above some rank. We conclude that λ = λjl (ξ) for l sufficiently large and that means that λ ∈ M . If we suppose that Hypothesis H.7 is satisfied, i.e. there exists k ≥ 1 such that Jk is a simple spectral band for P0 , then we have some more regularity for the Floquet eigenvalue λk (ξ). Lemma 8.8. Under Hypothesis H.7, if Jk is a simple spectral band for P0 , then the function λk (ξ) is of class C ∞ (T∗,d ). 48 Proof. Let us fix a circle C in the complex plane having its center on the real axis and such that: Jk is contained in the open interior domain delimited by C and all the other spectral bands Jl with l 6= k are contained in the exterior open
domain delimited by C (that is unbounded). In particular, for such a choice we get that the distance d C , σ(P̌0 ) > 0. With the above noations let us define the following operator: I
−1 i P̌0 (ξ) − z dz, ∀ξ ∈ X ∗ . (8.11) Πk (ξ) := − 2π C
−1 One verifies easily that it is an orthogonal projection on the subspace Nk (ξ) := ker P̌0 (ξ) − z , that is a complex vector space of dimension 1. Moreover,
it is easy to verify using Lemma 8.3 that the application T∗,d ∋ ξ 7→ Πk (ξ) ∈ B(K0 ) is of class C ∞ T∗,d ; B(K0 ) . Let us now fix some point ξ0 ∈ T∗,d and some vector φ(ξ0 ) ∈ Nk (ξ0 ) having norm kφ(ξ0 )kK0 = 1. We can find a sufficiently small open neighborhood V0 of ξ0 in T∗,d such that kΠk (ξ)φ(ξ0 )kK0 ≥ We denote by 1 , 2 ∀ξ ∈ V0 . −1 φ(ξ) := kΠk (ξ)φ(ξ0 )kK0 Πk (ξ)φ(ξ0 ), ∀ξ ∈ V0 .
For any ξ ∈ V0 the vector φ(ξ) is a norm one vector that generates the subspace Nk (ξ) and φ ∈ C ∞ V0 ; K0 . We choose now c ∈ R as in the proof of Lemma 8.6. Then, for any ξ ∈ V0 we have that
−1
−1 λk (ξ) − c φ(ξ) = P̌0 (ξ) − cidl φ(ξ) and we conclude that λk (ξ) − c
−1 =  P̌0 (ξ) − cidl Using Lemma 8.3 we conclude finally that λk ∈ C ∞ (V0 ).
−1  φ(ξ) , φ(ξ) K0 . Lemma 8.9. With the above definitions and notations the following statements are true: 1. For any (s, ξ) ∈ R × X ∗ the Hilbert spaces Ks,ξ and Fs,ξ are stable under complex conjugation. 2. ∀ξ ∈ X ∗ and ∀γ ∗ ∈ Γ∗ we have that P̌0 (ξ + γ ∗ ) = σ−γ ∗ P̌0 (ξ)σγ ∗ and λj (ξ + γ ∗ ) = λj (ξ) for any j ≥ 1. 3. If the symbol p0 verifies the property p0 (x, −ξ) = p0 (x, ξ) (8.12) then the following relations hold: P̌0 (ξ)u = P̌0 (−ξ)u, ∀u ∈ Km,ξ , ∀ξ ∈ X ∗ . λj (−ξ) = λj (ξ), Πk (ξ)u = Πk (−ξ)u, ∀j ≥ 1. (8.13) (8.14) ∗ ∀u ∈ K0 , ∀ξ ∈ X , (8.15) for any simple spectral band Jk of P0 . Proof. The first statement follows directly from the definitions (9.22) and (9.24), while the second statement follows from Remark 8.5.
As we know that P̌0 (ξ) is induced by P0,ξ := Op (idl ⊗ τ−ξ )p on the Hilbert space K0 , it is enough to prove that P0,ξ u = P0,−ξ u for any u ∈ S (X ). In fact, for any x ∈ X we have that: Z

x+y ¯ = P0,ξ u (x) = ei<η,y−x> p0 , ξ + η u(y) dy dη 2 Ξ Z Z


x+y x+y = ei<η,x−y> p0 ¯ = ei<η,x−y> p0 ¯ = P0,−ξ u (x). , ξ − η u(y) dy dη , −ξ + η u(y) dy dη 2 2 Ξ Ξ Let us fix now some point ξ ∈ X ∗ and some vector u ∈ Km,ξ ; following statement (1) of this Lemma and (8.13), the vector u is an eigenvector of P̌0 (ξ) for the eigenvalue λj (ξ) if and only if u is eigenvector of P̌0 (−ξ) for the eigenvalue λj (ξ). We deduce that {λj (−ξ)}j≥1 = {λj (ξ)}j≥1 ; as both sequences are monotonuous we conclude that λj (−ξ) = λj (ξ) for any j ≥ 1 so that we obtain (8.14). Finally (8.15) follows from (8.13) and (8.11). 49 The next Lemma is very important for the construction in the Grushin problem under Hypothesis H.7; we shall prove it following the ideas in [17]. Lemma 8.10. Supposing that Hypothesis H.7 is also satisfied and supposing that p(y, −η) = p(y, η) for any (y, η) ∈ Ξ, we can construct a function φ having the following properties: 1. φ ∈ C ∞ (Ξ; Klm,0 ) for any l ∈ N. 2. φ(y + γ, η) = φ(y, η), ∀(y, η) ∈ Ξ, ∀γ ∈ Γ. 3. φ(y, η + γ ∗ ) = e−i<γ ∗ 4. kφ(·, η)kK0 = 1, ∀η ∈ X ∗ . 5. φ(y, η) = φ(y, −η) ∀(y, η) ∈ Ξ. ,y> φ(y, η), ∀(y, η) ∈ Ξ, ∀γ ∗ ∈ Γ∗ .
6. φ(·, η) ∈ Nk (η) = ker P̌0 (η) − λk (η) , ∀η ∈ X ∗ .
Proof. First we shall construct a function φ ∈ C X
∗ ; K0 that satisfies properties (2)-(6). We begin by recalling that Πk ∈ C ∞ X ; B(K0 ) and deducing that there exists some δ > 0 such that for any pair of points (ξ ′ , ξ ′′ ) ∈ [E ∗ ]2 with |ξ ′ − ξ ′′ | < δ the following estimation is true: kΠk (ξ ′ ) − Πk (ξ ′′ )kB(K0 ) < 1 . 2 (8.16) We decompose the vectors of X ∗ with respect to the dual basis {e∗j }1≤j≤d associated to Γ, and write ξ = t1 e∗1 + · · · + td e∗d or ξ = (t1 , . . . , td ) for any ξ ∈ X ∗ . In particular we have that ξ ∈ E ∗ if and only if −(1/2) ≤ tj < (1/2) for any j = 1, . . . , d. The construction of the function φ will be done by induction on the number d of variables (t1 , . . . , td ). We start with the case d = 1, or equivalently, we consider only momenta of the type (t, 0) with t ∈ R and 0 ∈ Rd−1 . We choose some n ∈ N∗ such that n−1 < δ and consider a division of the interval [0, 1/2] ⊂ R: 0 = τ0 < τ1 < . . . < τn = 1/2, τj = j , ∀0 ≤ j ≤ n. 2n Let us fix some vector ψ(0) ∈ Nk ((0, 0) such that kψ(0)kK0 = 1 and ψ(0) = ψ(0); this last property may be realized noticing that Lemma 8.9 implies that if v ∈ Nk ((0, 0)) then also v ∈ Nk ((0, 0)) and if kψ(0)kK0 = 1 then it exists f ∈ R such that ψ(0) = eif ψ(0) and we can replace ψ(0) by e−if /2 ψ(0). We use now (8.16) and deduce that kΠk ((t, 0))ψ(0)kK0 ≥ Thus we can define: 1 , 2 ∀t ∈ [0, τ1 ]. −1 ∀t ∈ [0, τ1 ]. ψ(t) := kΠk ((t, 0))ψ(0)kK0 Πk ((t, 0))ψ(0), We notice that this function verifies the following properties:
ψ ∈ C [0, τ1 ]; K0 ; ψ(0) = ψ(0); kψ(t)kK0 = 1, ∀t ∈ [0, τ1 ]; ψ(t) ∈ Nk ((t, 0)), ∀t ∈ [0, τ1 ]. (8.17) (8.18) But let us notice that we can use (8.16) once more starting with ψ(τ1 ), noticing that we also have kΠk ((t, 0))ψ(τ1 )kK0 ≥ and defining 1 , 2 ∀t ∈ [τ1 , τ2 ]. −1 ψ(t) := kΠk ((t, 0))ψ(τ1 )kK0 Πk ((t, 0))ψ(τ1 ), ∀t ∈ [τ1 , τ2 ]. (8.19) We notice that lim ψ(t) = ψ(τ1 ) and conclude that the function ψ : [0, τ2 ] → K0 , defined by (8.17) and (8.19) is tցτ1 continuous in t = τ1 and also verifies
ψ ∈ C [0, τ2 ]; K0 ; ψ(0) = ψ(0); kψ(t)kK0 = 1, ∀t ∈ [0, τ2 ]; ψ(t) ∈ Nk ((t, 0)), ∀t ∈ [0, τ2 ]. (8.20) Continuing in this way, after a finite number of steps (n steps) we obtain a function ψ : [0, (1/2)] → K0 verifying the properties:
ψ ∈ C [0, 1/2]; K0 ; ψ(0) = ψ(0); kψ(t)kK0 = 1, ∀t ∈ [0, 1/2]; ψ(t) ∈ Nk ((t, 0)), ∀t ∈ [0, 1/2]. (8.21) 50 We extend now this function to the interval [−(1/2), (1/2)] by defining ψ(−t) := ψ(t) for any t ∈ [0, 1/2]. It verifies the properties:
ψ ∈ C [−1/2, 1/2]; K0 ; ψ(t) ∈ Nk ((t, 0)), ∀t ∈ [−1/2, 1/2]. (8.22) The second property above follows easily from (8.13). We take now into account the second statement of the Lemma 8.9 and notice that it implies the equality P̌0 ((1/2, 0)) = σ−e∗1 P̌0 ((−1/2, 0))σe∗1 and from that we deduce that σ−e∗1 ψ(−1/2) is an eigenvector of P̌0 ((1/2, 0)) for the eigenvalue λk ((−1/2, 0)) = λk ((1/2, 0)) (following (8.14)). We deduce that it exists κ ∈ R such that σ−e∗1 ψ(−1/2) = eiκ ψ(1/2). Let us define now φ(t) := eitκ ψ(t) for any t ∈ [−1/2, 1/2]. It will evidently have the following properties:
(8.23) φ ∈ C [−1/2, 1/2]; K0 ; φ(t) ∈ Nk ((t, 0)), ∀t ∈ [−1/2, 1/2]; φ(1/2) = σ−e∗1 φ(−1/2). We extend this function to R by the following reccurence relation: φ(t) := σ−e∗1 φ(t − 1), ∀t ∈ [j − 1/2, j + 1/2], j ∈ Z. We obtain a function φ : R → K0 verifying the following properties:
 φ ∈ C R; K0 .     ∀t ∈ R, ∀l ∈ Z.  φ(t + l) = σ−le∗1 φ(t), ∀t ∈ R. kφ(t)kK0 = 1,    φ(t) = φ(−t), ∀t ∈ R.   φ(t) ∈ Nk ((t, 0)), ∀t ∈ R. (8.24) (8.25) Let us denote now by t′ := (t1 , . . . , td−1 ) ∈ Rd−1 and suppose by hypothesis that we have constructed a function ψ : Rd−1 → K0 satisfying the following properties:
 ψ ∈ C Rd−1 ; K0 .     ∀t′ ∈ Rd−1 , ∀l′ ∈ Zd−1 .  ψ(t′ + l′ ) = σ−<e∗′,l′ > ψ(t′ ), ′ d−1 ′ ∀t ∈ R . kψ(t )kK0 = 1, (8.26)  ′ d−1  ′ ) = ψ(−t′ ),  ψ(t ∀t ∈ R .   ψ(t′ ) ∈ Nk ((t′ , 0)), ∀t′ ∈ Rd−1 . We want to construct now a function φ : Rd → K0 satisfying the same properties as ψ above (with d − 1 replaced by d). For doing that let us introduce the following notations t = (t′ , td ) with t′ = (t1 , . . . , td−1 ) ∈ Rd−1 for any t = (t1 , . . . , td ) ∈ Rd . e ′ , 0) := ψ(t′ ) for any t′ ∈ Rd−1 . Repeating the argumet at the beginning of this We start by defining first ψ(t proof, we extend step by step our function ψe on subsets of the form Rd−1 × [τj , τj+1 ] with 0 ≤ j ≤ n − 1. Finally e ′ , −td ) := ψ(−t e ′ , td ) for any t′ ∈ Rd−1 and any we extend it to the subset Rd−1 × [−1/2, 1/2] by the definition ψ(t d−1 e td ∈ [0, 1/2]. We obtain in this way a function ψ : R × [−1/2, 1/2] → K0 verifying the properties: 
ψe ∈ C Rd−1 × [−1/2, 1/2]; K0 .     e ′ , td ), e ′ + l′ , td ) = σ−<e∗′ ,l′ > ψ(t  ∀t′ ∈ Rd−1 , ∀l′ ∈ Zd−1 , ∀td ∈ [−1/2, 1/2].  ψ(t ′ d−1 ′ e ∀t ∈ R , ∀td ∈ [−1/2, 1/2]. kψ(t , td )kK0 = 1,   ′ ′ e e  ψ(t , t ) = ψ(−t , −td ), ∀t′ ∈ Rd−1 , ∀td ∈ [−1/2, 1/2].    e ′ d ′ ′ ψ(t , td ) ∈ Nk ((t , 0)), ∀t ∈ Rd−1 , ∀td ∈ [−1/2, 1/2]. (8.27) Now we come once more to the second statement of the Lemma 8.9 and notice that: P̌0 ((t′ , 1/2)) = σ−e∗d P̌0 ((t′ , −1/2))σe∗d , ∀t′ ∈ Rd−1 . e ′ , −1/2) is a normed eigenvector of P̌0 ((t′ , 1/2)) for the eigenvalue λk ((t′ , −1/2)) = We deduce that σ−e∗d ψ(t ′ λk ((t , 1/2)) (we made use of point (3) in Lemma 8.9). We conclude that it exists a function κ′ : Rd−1 → R such that e ′ , 1/2), e ′ , −1/2) = eiκ′ (t′ ) ψ(t ∀t′ ∈ Rd−1 . (8.28) σ−e∗d ψ(t 51 
From the continuity of the function ψe ∈ C Rd−1 × [−1/2, 1/2]; K0 we deduce the continuity of the function ′ eiκ : Rd−1 → U(1) and also of the function κ′ : Rd−1 → R. We take into account now (8.28) and the second equality in (8.27) with t′ ∈ Rd−1 and l′ ∈ Zd−1 and get ′ ′ ′ e ′ , −1/2) = e ′ + l′ , −1/2) = σ−e∗ σ−<e∗′ ,l′ > ψ(t e ′ + l′ , 1/2) = σ−e∗ ψ(t eiκ (t +l ) ψ(t d d ′ ′ ′ ′ ′ ′ ′ ′ ′ e ′ + l′ , 1/2). e ′ , 1/2) = eiκ (t ) ψ(t = σ−<e∗′ ,l′ > eiκ (t ) ψ(t It follows that ei[κ (t +l )−κ (t )] = 1 and the function κ′ : Rd−1 → R being continuous we deduce that for any l′ ∈ Zd−1 there exists n(l′ ) ∈ Z such that κ′ (t′ + l′ ) − κ′ (t′ ) = 2πn(l′ ), ∀t′ ∈ Rd−1 . (8.29) But from (8.28) we deduce that ′ ′ e ′ , 1/2), e ′ , −1/2) = eiκ (−t ) ψ(−t σ−e∗d ψ(−t ∀t′ ∈ Rd−1 . After complex conjugation and making use of (8.27) we get ′ ′ e ′ , −1/2), e ′ , 1/2) = e−iκ (−t ) ψ(t σe∗d ψ(t or equivalently ′ ∀t′ ∈ Rd−1 , ′ e ′ , 1/2), e ′ , −1/2) = eiκ (−t ) ψ(t σ−e∗d ψ(t ′ ′ ′ ∀t′ ∈ Rd−1 . ′ We use once again (8.28) in order to obtain the equality eiκ (−t ) = eiκ (t ) , or equivalently the relation κ′ (−t′ ) − κ′ (t′ ) ∈ 2πZ. As the function κ′ is continuous, we conclude that the difference κ′ (−t′ ) − κ′ (t′ ) must be constant; as in t′ = 0 this difference is by definition 0 we conclude that κ′ (−t′ ) = κ′ (t′ ), ∀t′ ∈ Rd−1 . (8.30) We consider now (8.29) and notice that for any l′ ∈ Zd−1 we can choose the point t′ := −(1/2)l′ ∈ Rd−1 verifying the relation t′ + l′ = −t′ and thus we conclude that n(l′ ) = 0 for any l′ ∈ Zd−1 obtaining the equalities κ′ (t′ + l′ ) = κ′ (t′ ), ∀t′ ∈ Rd−1 , ∀l′ ∈ Zd−1 . (8.31) We can now define φe : Rd−1 × [−1/2.1/2] → K0 by the following equality: e ′ , td ), e ′ , td ) := eiκ′ (t′ )td ψ(t φ(t ∀(t′ , td ) ∈ Rd−1 × [−1/2, 1/2]. From (8.30) and (8.31) the function φe has all the properties (8.27) and it also has the following property: e ′ , −1/2), e ′ + l′ , 1/2) = σ<e∗′ ,l′ > σ−e∗ φ(t φ(t d In fact we see that ∀t′ ∈ Rd−1 , ∀l′ ∈ Zd−1 . (8.32) (8.33) e ′ , −1/2) = e ′ , 1/2) = e−iκ′ (t′ )/2 σ−<e∗′ ,l′ > σ−e∗ ψ(t e ′ + l′ , 1/2) = eiκ′ (t′ )/2 σ−<e∗′ ,l′ > ψ(t e ′ + l′ , 1/2) = eiκ′ (t′ )/2 ψ(t φ(t d e ′ , −1/2). = σ−<e∗′ ,l′ > σ−e∗d φ(t As in the case d = 1 we extend the function φe to the entire Rd by the following relation: e ′ , td − 1), e ′ , td ) := σ−e∗ φ(t φ(t d ∀t′ ∈ Rd−1 ∀td ∈ [j − 1/2, j + 1/2], j ∈ Z. (8.34)
We evidently obtain a function of class C X ∗ ; K0 satisfying the properties (2)-(6) in our Lemma. We end up our construction by a regularization procedure. Let us choose a real, non-negative, even function R χ ∈ C0∞ (X ∗ ) and such that X ∗ χ(t)dt = 1. For any δ > 0 we denote by χδ (ξ) := δ −d χ(ξ/δ) and define: Z e e dη, φδ (ξ) := χδ (ξ − η)φ(η) ∀ξ ∈ X ∗ . (8.35) X∗
Evidently φeδ ∈ C ∞ X ∗ ; K0 for any δ > 0. Moreover we also have that: Z Z Z ∗ ∗ ∗ e e e χδ (ξ − η)φ(η + γ ) dη = χδ (ξ + γ − η)φ(η) dη = φδ (ξ + γ ) = X∗ X∗ X∗ 52 e dη = (8.36) χδ (ξ − η)σ−γ ∗ φ(η) = φeδ (ξ) = Z Z = σ−γ ∗ φeδ (ξ), X∗ ∀ξ ∈ X ∗ , ∀γ ∗ ∈ Γ∗ ; Z e dη = e χδ (ξ − η)φ(η) χδ (ξ − η)φ(−η) dη = e dη = χδ (ξ + η)φ(η) X∗ Z X∗ (8.37) X∗ e dη = φeδ (−ξ), χδ (−ξ − η)φ(η) ∀ξ ∈ X ∗ . We shall prove now the following continuity relation: e ∀θ > 0, ∃δ0 > 0 : kφeδ (ξ) − φ(ξ)k K0 ≤ θ, ∀δ ∈ [0, δ0 ], ∀ξ ∈ X ∗ . (8.38) In fact, let us notice that: e kφeδ (ξ) − φ(ξ)k K0 ≤ Z X∗ e − δη) − φ(ξ)k e kφ(ξ K0 |χ(η)| dη ≤ C Z suppχ e − δη) − φ(ξ)k e kφ(ξ K0 dη. e for all γ ∗ ∈ Γ∗ and for all ξ ∈ X ∗ e + γ ∗ ) = σ−γ ∗ φ(ξ) The function φe being continuous and satisfying the relation φ(ξ we conclude that it is uniformly continuous on X ∗ and the support of χ being compact, the above estimation implies the stated continuity in δ > 0. Thus we conclude that for δ ∈ [0, δ0 ] with δ0 > 0 small enough we have Πk (ξ)φeδ (ξ) and we can define: K0 ≥ 1 , 2 ∀ξ ∈ X ∗ e ∀ξ ∈ X ∗ . φ(ξ) := kΠk (ξ)φeδ0 (ξ)k−1 K0 Πk (ξ)φδ0 (ξ),
It remains to prove that φ ∈ C ∞ X ∗ ; Klm,0 for any l ∈ N because all the other properties from the statement of our Lemma are clearly satisfied considering the
definition of φ. The case l = 0 is also clear from the definition. Let us notice that P̌0 ∈ C ∞ X ∗ ; B(Km,0 ; K0 ) and we know that for any ξ0 ∈ X ∗ there exists a constant C0 > 0 such that   ∀u ∈ Km,0 kukKm,0 ≤ C0 P̌0 (ξ0 )u K0 + kukK0 , and evidently kukKm,0 ≤ C0  P̌0 (ξ)u K0 +
P̌0 (ξ0 ) − P̌0 (ξ) u K0  + kukK0 , ∀ξ ∈ X ∗ , ∀u ∈ Km,0 . Choosing a sufficiently small neighborhood V0 of ξ0 ∈ X ∗ we deduce that it exists C0′ > 0 such that   ∀u ∈ Km,0 . kukKm,0 ≤ C0′ P̌0 (ξ)u K + kukK0 , ∀ξ ∈ V0 , 0 Thus, with c ∈ R the constant introduced in the proof of Lemma 8.6, if we take u := P̌0 (ξ) − c v ∈ K0 we obtain that
−1 P̌0 (ξ) − c v ∀v ∈ K0 . ≤ C0′′ kvkK0 , ∀ξ ∈ V0 ,
−1 v for some Km,0
−1 We conclude that P̌0 (ξ) − c ∈ B(K0 ; Km,0 ) uniformly for ξ in any compact subset of X ∗ .
Considering now the proof of Lemma 8.3 once again we obtain that in fact φ ∈ C ∞ X ∗ ; Km,0 and we get the case l = 1.
In order to obtain the general situation l ∈ N let us notice that for any ξ ∈ X ∗ we know that φ(ξ) ∈ D P̌0 (ξ)l
and P̌0 (ξ)l φ(ξ) = λk (ξ)l φ(ξ) for any l ∈ N. We also have that P̌0 (·)l ∈ C ∞ X ∗ ; B(K0 ; Klm,0 ) . The fact that l P̌0 (ξ)l is induced by the operator P0,ξ on the Hilbert space of tempered distributions K0 and we know that P0l is
elliptic of order lm, allows us to conclude by an argument similar to the one above, that φ ∈ C ∞ X ∗ ; Klm,0 for any l ∈ N. Remark 8.11. The arguments used in the proof of Lemma 4.7 allow to deduce from properties (1) - (3) from Lemma 8.10 that for any α ∈ Nd and for any s ∈ R there exists a constant Cα,s > 0 such that:
∂ξα φ (·, ξ) K ≤ Cα,s , ∀ξ ∈ X ∗ . (8.39) s,ξ 53 Proof of Proposition 1.6 We shall repeat the construction of the Grushin operator (4.33) from Section 4 under the Hypothesis of Proposition 1.6. We shall prove that in this case we can take N = 1 and φ1 (x, ξ) = φ(x, ξ) the function obtained in Lemma 8.10. Due to Lemma 8.10 and Remark 8.11 this function has all the properties needed in Lemma 4.7. It is thus possible to obtain the operator P0 (ξ, λ) and the essential problem is to prove its invertibility in order to obtain a result similar to Proposition 4.8. Frm that point the proof of Proposition 1.6 just repeats the arguments of Section 5. As in Section 4, for any ξ ∈ X ∗ we construct the operators: R+ (ξ) ∈ B(K0 , C), R+ (ξ)u := (u, φ(·, ξ))K0 , ∀u ∈ K0 , (8.40) R− (ξ) ∈ B(C, K0 ), R− (ξ)c := cφ(·, ξ), ∀c ∈ C. (8.41)

0 X ; B(Km,ξ ; C) and R− ∈ S0 X ; B(C; K0 ) and from Example 9.21 we deduce It is evident that R+ ∈
that P̌0 (·) − λidl ∈ S00 X ; B(Km,ξ ; K0 ) uniformly for λ ∈ I. S00 For any (ξ, λ) ∈ X ∗ × I we define P0 (ξ, λ) :=  P̌0 (ξ) R+ (ξ) R− (ξ) 0 
∈ B Km,ξ × C; K0 × C . We denote by Aξ := Km,ξ × C and by Bξ := K0 × C and we notice that
P0 (·, λ) ∈ S00 X ; B(A• ; B• uniformly in λ ∈ I. (8.42) (8.43) In order to construct an inverse for P0 (ξ, λ) we make the following choices: 0 E+ (ξ, λ) ∈ B(C; Km,ξ ), 0 E+ (ξ, λ)c := cφ(·, ξ), 0 E− (ξ, λ) ∈ B(K0 ; C), 0 E− (ξ, λ)u := (u, φ(·, ξ))K0 ,
0 0 E−+ (ξ, λ) ∈ B(C), E−+ (ξ, λ)c := λ − λk (ξ) c,  E 0 (ξ, λ) ∈ B(K0 ; Km,ξ ), E 0 (ξ, λ) := [idl − Πk (ξ)] P̌0 (ξ) − ∀c ∈ C, ∀u ∈ K0 , ∀c ∈ C,  λ [idl − Πk (ξ)] , (8.44) (8.45) (8.46) (8.47) where λk (ξ) is the Floquet eigenvalue generating the simple spectral band Jk and Πk (ξ) is the orthogonal projection on Nk (ξ) as defined in (8.11), with the circle C chosen to contain the interval I ⊂ R into its interior domain. Let us notice that the operator E 0 (ξ, λ) is well defined; in fact ℜange [idl − Πk (ξ)] is the orthogonal complement in K0 of Nk (ξ) := ker P̌0 (ξ) − λ (linearly generated by φ(·, ξ)) and is a reducing subspace for P̌0 (ξ). Thus P̌0 (ξ) induces on the space ℜange [idl − Πk (ξ)] a self-adjoint operator having the spectrum {λj (ξ)}j6=k . If λ ∈ I and I is as in the hypothesis (i.e. I ∩ Jl = ∅ for any l 6= k), the distance from λ to the spectrum of this induced operator is bounded below by a strictly positive constant C > 0 that does not depend on (ξ, λ) ∈ X ∗ × I; thus the norm of the rezolvent of this induced operator, in point λ ∈ I ⊂ R is bounded above by C −1 . This
rezolvent is exactly our operator E 0 (ξ, λ) defined in (8.47). Recalling that to λ ∈ I and the formula (8.11) for the operator P̌0 (ξ) − λ ∈ S00 X ; B(Km,ξ ; K0 ) uniformly with respect
Πk (ξ) one proves that E 0 (·, ξ) ∈ S00 X ; B(K0 ; Km,ξ ) uniformly with respect to λ ∈ I.

0
0 0 (ξ, λ) ∈ S00 X ; B(C) (ξ, λ) ∈ S00 X ; B(K0 ; C) , E−+ By definition we have that E+ (ξ, λ) ∈ S00 X ; B(C; Km,ξ ) , E− uniformly for λ ∈ I and thus if we define for any (ξ, λ) ∈ X ∗ × I:  0  0 E (ξ, λ) E+ (ξ, λ) E0 (ξ, λ) := ∈ B(Bξ ; Aξ ) (8.48) 0 0 E− (ξ, λ) E−+ (ξ, λ) we obtain an operator-valued symbol:
E0 (· · · , λ) ∈ S00 X ; B(B• ; A• ) , uniformly in λ ∈ I. (8.49) We shall verify now that for each (ξ, λ) ∈ X ∗ × I this defines an inverse for the operator P0 (ξ, λ): P0 (ξ, λ) E0 (ξ, λ) = idl, on K0 × C. (8.50) We can write: P0 (ξ, λ) E0 (ξ, λ) = 
0 P̌0 (ξ) − λ E 0 (ξ, λ) + R− (ξ)E− (ξ, λ) 0 R+ (ξ)E (ξ, λ) 54 
0 0 P̌0 (ξ) − λ E+ (ξ, λ) + R− (ξ)E−+ (ξ, λ) . 0 R+ (ξ)E+ (ξ, λ) For any u ∈ K0 we can write:
   −1 P̌0 (ξ) − λ E 0 (ξ, λ)u = P̌0 (ξ) − λ [idl − Πk (ξ)] P̌0 (ξ) − λ [idl − Πk (ξ)] u = [idl − Πk (ξ)] u, 0 R− (ξ)E− (ξ, λ)u = (u, φ(·, ξ))K0 φ(·, ξ) = Πk (ξ)u    −1 R+ (ξ)E 0 (ξ, λ)u = [idl − Πk (ξ)] P̌0 (ξ) − λ [idl − Πk (ξ)] u , φ(·, ξ) K0 = 0. For c ∈ C we can write that:
0

P̌0 (ξ) − λ E+ (ξ, λ)c = c P̌0 (ξ) − λ φ(·, ξ) = c λk (ξ) − λ φ(·, ξ), 0 R− (ξ)E−+ (ξ, λ)c =
λ − λk (ξ) cφ(·, ξ), 0 R+ (ξ)E+ (ξ, λ)c = c (φ(·, ξ), φ(·, ξ)) K0 = c. These identities imply (8.50). The property of being a left inverse is verified by very similar computations or by taking into account the self-adjointness of both P0 (ξ, λ) and E0 (ξ, λ). From this point one can repeat identically the arguments in the proof of Theorem 1.3 noticing that (5.12) −+ and (8.46) imply that we can take Eλ,ǫ (x, ξ) as in (1.38). 8.2 The constant magnetic field In this subsection we prove Proposition 1.7. Thus we suppose that the symbols pǫ do not depend on the first argument and the magnetic field has constant components:  P Bjk (ǫ) dxj ∧ dxk , Bjk (ǫ) = −Bkj (ǫ) ∈ R,  Bǫ = 12 1≤j,k≤d (8.51)  lim Bjk (ǫ) = 0. ǫ→0 Using the transversal gauge (1.28) we associate to these magnetic fields some vector potentials Aǫ := (Aǫ,1 , . . . , Aǫ,d ) satisfying: 1 X Bjk (ǫ)xk . (8.52) Aǫ,j (x) = 2 1≤k≤d Proof of Proposition 1.7 (1) We use formula (2.5) from Lemma 2.2 noticing that the linearity of the functions Aǫ,j and the definition of ωA imply that ωτ−x Aǫ (y, ỹ) = ωAǫ (y, ỹ) ei<Aǫ (x),y−ỹ> = ωAǫ +Aǫ (x) (y, ỹ). We deduce that for any u ∈ S (X 2 ) and for any (x, y) ∈ X 2 we have that:   

 χ∗ Peǫ (χ∗ )−1 u (x, y) = idl ⊗ σA(x) (idl ⊗ Pǫ ) idl ⊗ σ−A(x) u (x, y). (8.53) (8.54) It follows that the operator Peǫ , that is an unbounded self-adjoint operator in L2 (X 2 ) denoted in Proposition 3.15 by Peǫ′ , is unitarily equivalent with the oprtator idl ⊗ Pǫ with Pǫ self-adjoint unbounded operator in



L2 (X ). It follows that σ Peǫ′ = σ Pǫ . From Proposition 3.15 we deduce that σ Peǫ′ = σ Peǫ′′ where Peǫ′′ is
the self-adjoint realization of Peǫ in the space L2 X × T . Finally, from Corollary 5.5 we deduce that for any (λ, ǫ) ∈ I × [−ǫ0 , ǫ0 ] we have the equivalence relation: λ ∈ σ Peǫ′′
⇐⇒
0 ∈ σ E−+ (ǫ, λ) ,  N where E−+ (ǫ, λ) is considered as a bounded self-adjoint operator on L2 (X ) . In order to prove the second point of Proposition 1.7 we shall use the magnetic translations Tǫ,a := σAǫ (a) τa for any a ∈ X , that define a family of unitary operators in L2 (X ). Lemma 8.12. For any two families of Hilbert
spaces with temperate variation {Aξ }ξ∈X ∗ and {Bξ }ξ∈X ∗ and for any operator-valued symbol q ∈ S00 X ; B(A• ; B• ) the following equality holds:
Tǫ,a OpAǫ (q) = OpAǫ (τa ⊗ idl)q Tǫ,a , 55 ∀a ∈ X . (8.55) Proof. From Lemma 9.19 it follows that
τa OpAǫ (q) = Opτa Aǫ (τa ⊗ idl)q τa . (8.56) We notice that τa Aǫ = Aǫ − Aǫ (a), so that the equality (8.56) becomes
τa OpAǫ (q) = Op(Aǫ − Aǫ (a)) (τa ⊗ idl)q τa . Then, for any u ∈ S (X ; A0 ) and for any x ∈ X we can write that Z   x+y
, η u(y) dy dη. ¯ ei<η,x−y> e−i<Aǫ (a),x−y> ωAǫ (x, y) q σ−Aǫ (a) OpAǫ (q)σAǫ (a) u (x) = 2 Ξ R Noticing that < Aǫ (a), x − y >= − [x,y] Aǫ (a), the above formula implies that OpAǫ (q)σAǫ (a) = σAǫ (a) Op(Aǫ −Aǫ (a)) (q). (8.57) (8.58) From (8.57) and (8.58) we deduce that

Tǫ,a OpAǫ (q) = σAǫ (a) τa OpAǫ (q) = σAǫ (a) Op(Aǫ − Aǫ (a)) (τa ⊗ idl)q τa = OpAǫ (τa ⊗ idl)q σAǫ (a) τa =
= OpAǫ (τa ⊗ idl)q Tǫ,a .
Proof of Proposition 1.7 (2) The operator Pǫ,λ := Op Pǫ (·, ·, λ) from Theorem 5.3 has its symbol defined in (5.6):   qǫ (x, ξ) − λ R− (ξ) Pǫ (x, ξ, λ) := , ∀(x, ξ) ∈ Ξ, ∀(λ, ǫ) ∈ I × [−ǫ0 , ǫ0 ] R+ (ξ) 0
where qǫ (x, ξ) := Op peǫ (x, ·, ξ, ·) with peǫ (x.y.ξ, η) := pǫ (x, y, ξ + η). As we have noticed from the beginning, we suppose that our symbol pǫ does not depend on the first variable so that neither the operator-valued symbol Pǫ will not depend on the first variable. From Lemma 8.12 the operator

Pǫ,λ : S X ; Km,0 × CN → S X ; K0 × CN commutes with the family {Tǫ,a ⊗ idlK0 ×CN }a∈X . Then its inverse appearing in Theorem 5.3,

Eǫ,λ : S X ; K0 × CN → S X ; Km,0 × CN also commutes with the family {Tǫ,a ⊗ idlK0 ×CN }a∈X . From this property we deduce that also the operator E−+ (ǫ, λ) : L2 (X ; CN ) → L2 (X ; CN ) commutes with the family {Tǫ,a ⊗ idlCN }a∈X . Using Lemma 8.12 once again we deduce that :

−1 −+ −+ OpAǫ Eǫ,λ = E−+ (ǫ, λ) = [Tǫ,a ⊗ idlCN ] E−+ (ǫ, λ) [Tǫ,a ⊗ idlCN ] = OpAǫ (τa ⊗ idl)Eǫ,λ , ∀a ∈ X . −+ −+ We conclude that Eǫ,λ (x, ξ) = Eǫ,λ (x − a, ξ) for any (x, ξ) ∈ Ξ and for any a ∈ X . It follows that −+ −+ Eǫ,λ (x, ξ) = Eǫ,λ (0, ξ) for any (x, ξ) ∈ Ξ. The Γ∗ -periodicity follows as in the general case (see the proof of Lemma 7.5). 9 Appendices 9.1 Magnetic pseudodifferential operators with operator-valued symbols Definition 9.1. A family of Hilbert spaces {Aξ }ξ∈X ∗ (indexed by the points in the momentum space) is said to have temperate variation when it verifies the following two conditions: 1. Aξ = Aη as complex vector spaces ∀(ξ, η) ∈ [X ∗ ]2 . 2. There exist C > 0 and M ≥ 0 such that ∀u ∈ A0 we have the estimation: kukAξ ≤ C < ξ − η >M kukAη , 56 ∀(ξ, η) ∈ [X ∗ ]2 . (9.1) Example 9.2. We can take Aξ = Hs (X ), with any s ∈ R endowed with the ξ-dependent norm: Z kukAξ := <ξ+η > 2s X 2 |û(η)| dη 1/2 ∀u ∈ Hs (X ), ∀ξ ∈ X ∗ . , The inequality (9.1) clearly follows from the well known inequality: < ξ + η >2s ≤ Cs < ζ + η >2s < ξ − ζ >2|s| , ∀(ξ, η, ζ) ∈ [X ∗ ]3 , (9.2) where the constant Cs only depends on s ∈ R. For this specific family we shall use the shorter notation Aξ ≡ Hξs (X ). Definition 9.3. Suppose given two families of Hilbert spaces with tempered variation {Aξ }ξ∈X ∗ and {Bξ }ξ∈X ∗ ; suppose also
given m ∈ R, ρ ∈ [0, 1] and Y a finite dimensional real vector space. A function p ∈ C ∞ Y ×
∗ m X ; B(A0 ; B0 ) is called an operator-valued symbol of class Sρ Y; B(A• ; B• ) when it verifies the following property: ∀α ∈ Ndim Y , ∀β ∈ Nd , ∃Cα,β > 0 :
∂yα ∂ξβ p (y, ξ) ≤ Cα,β < ξ >m−ρ|β| , ∀(y, ξ) ∈ Y × X ∗ . B(Aξ ;Bξ ) (9.3)
The space Sρm Y; B(A• ; B• ) endowed with the family of seminorms να,β defined as being the smallest constants Cα,β that satisfy the defining property (9.3) is a metrizable locally convex linear topological space. In case we have for any ξ ∈
X ∗ that Aξ = A0 and Bξ = B0 as algebraic and topological structures, then we use the notation Sρm Y; B(A0 ; B0 ) . If moreover we have that A0 = B0 = C, then we use the simple notation Sρm (Y).
Example 9.4. If p ∈ S1m (X ) and if for any ξ ∈ X ∗ we denote by pξ := (idl ⊗ τ−ξ )p, by Pξ := Op pξ and by p the application Ξ ∋ (x, ξ) 7→ Pξ ∈ B(Hξs+m (X ); Hξs (X )), for some s ∈ R, we can prove that p is an operator valued

symbol of class S00 X ; B(H•s+m (X ); H•s (X )) . Moreover the map S1m (X ) ∋ p 7→ p ∈ S00 X ; B(H•s+m (X ); H•s (X )) is continuous. In fact, let us recall that for any ξ ∈ X ∗ we have denoted by σξ the multiplication operator with the function e on the space S ′ (X ). Then, for any u ∈ S (X ) and for any ξ ∈ X ∗ we have that u ∈ Hξs+m (X ) and we can write: Z Z


x+y x+y
, η u(y) dy dη ¯ = ei<η,x−y> p , η + ξ u(y) dy dη ¯ = Pξ u (x) σ−ξ P0 σξ u (x) = ei<η−ξ,x−y> p 2 2 Ξ Ξ i<ξ,·> and we conclude that Pξ = σ−ξ P0 σξ , ∀ξ ∈ X ∗ . (9.4) m On the other side, for any ξ ∈ X ∗ we notice that
pξ is a symbol of class S1 (X ) and thus, the usual Weyl s+m s calculus implies that Pξ ∈ B H (X ); H (X ) for any s ∈ R. We notice easily that for any multi-index β ∈ Nd we can write:
(9.5) ∂ξβ Pξ = Op ∂ξβ pξ
so that we conclude that Pξ ∈ C ∞ Ξ; B(Hs+m (X ); Hs (X )) (constant with respect to the variable x ∈ X ) for any s ∈ R. Let us further notice that for any u ∈ S (X ) and any ξ ∈ X ∗ we have the equalities: 2 kσ−ξ ukHs (X ) = ξ Z σd ξ u = τξ û, X∗ (9.6) 2 < ξ + η >2s |û(ξ + η)| dη = kuk2Hs(X ) . (9.7) Coming back to (9.4) we deduce that for any u ∈ S (X ) and any ξ ∈ X ∗ we have the estimation: 2 2 2 kPξ ukHs (X ) = kP0 σξ ukHs (X ) ≤ Cs kσξ ukHs+m (X ) = Cs kuk2Hs+m(X ) . ξ ξ Using also the equality (9.5)

we obtain similar estimations for the derivatives of Pξ and conclude that p ∈ S00 X ; B(H•s+m (X ); H•s (X )) and we have the continuity of the map S1m (X ) ∋ p 7→ p ∈ S00 X ; B(H•s+m (X ); H•s (X )) .
m Definition 9.5. We denote by Sρ,ǫ X 2 ; B(A• ; B• ) the linear space of families {pǫ }|ǫ|≤ǫ0 satisfying the following three conditions:
1. ∀ǫ ∈ [−ǫ0 , ǫ0 ], pǫ ∈ Sρm X 2 ; B(A• ; B• ) uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ], 57 
2. lim pǫ = p0 in Sρm X 2 ; B(A• ; B• ) , ǫ→0 3. Denoting the variable in X 2 by (x, y), for any multi-index α ∈ Nd with |α| ≥ 1 we have that lim ∂xα pǫ = 0 ǫ→0
in Sρm X 2 ; B(A• ; B• ) , endowed with the natural locally convex topology of symbols of Hörmander type. As in the case of Definition 9.3, in case we have for any ξ ∈ X
∗ that Aξ = A0 and Bξ = B0 as algebraic and m X 2 ; B(A0 ; B0 ) . If moreover we have that A0 = B0 = C, then topological structures, then we use the notation Sρ,ǫ m 2 we use the simple notation Sρ,ǫ (X ).
m X 2 ; B(A• ; B• ) that do not depend on the first variable x in X 2 we For the families of symbols of type Sρ,ǫ
m X ; B(A• ; B• ) . shall use the notation Sρ,ǫ Let us also notice the following canonical injection:

m m X 2 ; B(A• ; B• ) . X ; B(A• ; B• ) ∋ pǫ 7→ idl ⊗ pǫ ∈ Sρ,ǫ Sρ,ǫ
m X 2 ; B(A• ; B• ) : Remark 9.6. We can obtain the following description of the space Sρ,ǫ
m X 2 ; B(A• ; B• ) then we can write the first order Taylor expansion: If {pǫ }|ǫ|≤ǫ0 ∈ Sρ,ǫ  Z pǫ (x, y, η) = pǫ (0, y, η) + x , 0 1 
∇x pǫ (tx, y, η) dt . Using conditions (2) and (3) from Definition 9.5 we deduce that p0 (x, y, η) = p0 (0, y, η).
We denote by: p0 (y, η) := p0 (0, y, η) as element in Sρm X ; B(A• ; B• ) and by: rǫ (x, y, η) := pǫ (x, y, η) − p0 (y, η).
This last symbol is of class Sρm X 2 ; B(A• ; B• ) uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ].
We have: r0 = 0 and lim rǫ = 0 in Sρm X 2 ; B(A• ; B• ) and evidently: ǫ→0 pǫ (x, y, η) = p0 (y, η) + rǫ (x, y, η). (9.8) Reciprocally, if p0 and
rǫ from the equality (9.8) have the properties stated above, then {pǫ }|ǫ|≤ǫ0 belongs to m X 2 ; B(A• ; B• ) . Sρ,ǫ We shall consider now magnetic pseudodifferential operators associated to operator-valued symbols. Let us first
consider p ∈ Sρm X ; A• ; B• ) and a magnetic field B with components of class BC ∞ (X ); this magnetic field can ∞ (X ) (as for example in the transversal always be associated with a vector potential A with components of class Cpol R gauge). Let us recall the notation ωA (x, y) := exp{−i [x,y] A}. For any u ∈ S (X ; A0 ) we can define the oscillating integral (its existence following from the next Proposition): Z  A  x+y
, η u(y) dy dη, ¯ ∀x ∈ X . (9.9) ei<η,x−y> ωA (x, y) p Op (p)u (x) := 2 Ξ Proposition 9.7. Under the hypothesis described in the paragraph above (9.9) the following facts are true: 1. The integral in (9.9) exists for any x ∈ X as oscillating Bochner integral and defines a function OpA (p)u ∈ S (X ; B0 ). 2. The map OpA (p) : S (X ; A0 ) → S (X ; B0 ) defined by (9.9) and point (1) above is linear and continuous.
 ∗ 3. The formal adjoint OpA (p) : S (X ; B0 → S (X ; A0 ) of the linear continuous operator defined at point
′ (2) above is equal to OpA (p∗ ) where p∗ ∈ Sρm X ; B(B• ; A• ) where m′ = m + 2(MA + MB ) and p∗ (x, ξ) :=  ∗ p(x, ξ) (the adjoint in B(A0 ; B0 ). 4. The operator OpA (p) extends in a natural way to a linear continuous operator S ′ (X ; A0 ) → S ′ (X ; B0 ) that we denote in the same way. 58 Proof. Let us first prove the first two points of the Proposition. Fix some u ∈ S (X ; A0 ) and for the begining let us suppose that p(y, η) = 0 for |η| ≥ R, with some R > 0. Then, for any x ∈ X , the integral in (9.9) exists as a Bochner integral of a B0 -valued function. Let us notice that in this case we can integrate by parts in (9.9) and use the identities:     ei<η,x−y> =< x−y >−2N1 (idl−∆η )N1 ei<η,x−y> , ∀N1 ∈ N; e−i<η,y> =< η >−2N2 (idl−∆y )N2 e−i<η,y> , ∀N2 ∈ N. In this way we obtain the equality: = Z Ξ ei<η,x−y> < x − y >−2N1   OpA (p)u (x) = (9.10)     x+y
 (idl − ∆η )N1 < η >−2N2 (idl − ∆y )N2 ωA (x, y) p dy dη. ¯ , η u(y) 2 From this one easily obtains the following estimation: ∃C(N1 , N2 ) > 0, ∃k(N2 ) ∈ N such that for any l ∈ N we have  A  Op (p)u (x) ≤ ≤ C Z Ξ B0 <x−y > −2N1 <η> −2N2 <x>+<y> × sup sup < y >l |α|≤2N2 y∈X
k(N2 )
∂ α u (y) <η> A0 m+MB +MA <y> −l  dy dη ¯ × , where MA and MB are the constants appearing in condition (9.1) with respect to each of the two families {Aξ }ξ∈X ∗ and {Bξ }ξ∈X ∗ . We make the following choices: 2N2 ≥ m + MA + MB + d, l = 2N1 + k(N2 ) + d + 1, 2N1 ≥ k(N2 ) and we obtain that  A  Op (p)u (x) B0 ≤ C(N1 ) < x >−2N1 +k(N2 ) sup sup < y >l |α|≤2N2 y∈X
∂ α u (y) A0 , ∀x ∈ X . (9.11) Similar estimations may be obtained for the derivatives ∂xβ OpA (p)u and this finishes the proof of the first two points of the Proposition for the ”compact support” situation we have considered first. For the general case we consider a cut-off function ϕ ∈ C0∞ (X ∗ ) that is equal to 1 for |η| ≤ 1. For any R ≥ 1 we define then the ’approximating’
symbols pR (y, η) := ϕ(η/R)p(y, η) that has compact support in the η-variable and belongs to Sρm X ; B(A• ; B• ) uniformly with respect to R ∈ [1, ∞). We write the operator OpA (pR ) under a form similar to (9.10) with the choice of the parameters N1 and N2 as above; then the Dominated Convergence Theorem allows us to take the limit R ր ∞ obtaining now for OpA (p) the integral expression (9.10) that is well defined and verifies the estimation (9.11). The last two points of the statement of the Proposition follow in a standard way from the equality: Z  Z    A     Op (p)u (x) , v(x) dx = u(y) , OpA (p∗ )v (y) dy, ∀(u, v) ∈ S (X ; A0 ) × S (X ; B0 ). (9.12) X B0 A0 X m (X 2 ) and let us define, as in Subsection 1.3: Example 9.8. Let us consider a family {pǫ }|ǫ|≤ǫ0 of class S1,ǫ
peǫ (x, y, ξ, η) := pǫ (x, y, ξ + η); qǫ (x, ξ) := Op peǫ (x, ·, ξ, ·) . The following two statements are true:
0 1. {qǫ }|ǫ|≤ǫ0 ∈ S0,ǫ X ; B(H•s+m (X ); H•s (X ) for any s ∈ R. 2. If the family of magnetic fields {Bǫ }|ǫ|≤ǫ0 satisfies Hypothesis H.1 and if the associated vector potentials are choosen as in (1.28), then we have that:

OpAǫ (qǫ ) ∈ B S (X ; Hs+m (X )); S (X ; Hs (X )) ∩ B S ′ (X ; Hs+m (X )); S ′ (X ; Hs (X )) , (9.13) for any s ∈ R. Moreover,

OpAǫ (qǫ ) ∈ B S (X 2 ); S (X 2 ) ∩ B S ′ (X 2 ); S ′ (X 2 ) , and all the continuities are uniform with respect to ǫ ∈ [−ǫ0 , ǫ0 ]. 59 (9.14) Proof. We begin by verifying point (1). By similar arguments as in Example 9.4 we prove that for any ǫ ∈ [−ǫ0 , ǫ0 ]
and s ∈ R we have that qǫ ∈ S00 X ; B(H•s+m (X ); H•s (X )) uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ] and the following application is continuous:
S1m (X 2 ) ∋ pǫ 7→ qǫ ∈ S00 X ; B(H•s+m (X ); H•s (X ) , ∀s ∈ R, uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ]. Point (1) follows then clearly. Concerning the second point of the Proposition, let us notice that (9.13) and the uniformity with respect to ǫ ∈ [−ǫ0 , ǫ0 ] follow easily from Proposition 9.7 and its proof. In order to prove (9.14) let us notice that pe′ǫ (x, ·, ξ, ·) :=< ξ >−|m| peǫ (x, ·, ξ, ·) defines a symbol of class S0m (X
) uniformly with respect to ((x, ξ), ǫ) ∈ Ξ×[−ǫ0 , ǫ0 ] and we can view the element pe′ǫ as a function in BC ∞ Ξ; S0m (X ) . Then, the operator-valued symbol q′ǫ (x, ξ) :=< ξ >−|m| qǫ (x, ξ) has the following property:

∀(α, β) ∈ [Nd ]2 , ∂xα ∂ξβ q′ǫ (x, ξ) ∈ B S (X ) , uniformly with respect to ((x, ξ), ǫ) ∈ Ξ × [−ǫ0 , ǫ0 ]. Denoting ss (x, ξ) :=< ξ >s for any s ∈ R and writing OpAǫ (qǫ ) = OpAǫ (s|m| q′ ), the proof of Proposition 9.7 implies (9.14) uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ]. 9.2 Some spaces of periodic distributions We shall use the following notations: • SΓ′ (X ) := {u ∈ S ′ (X ) | τγ u = u, ∀γ ∈ Γ}, the space of Γ-periodic distributions on X . • S (T) := C ∞ (T) with the usual Fréchet topology, We have the evident identification: S (T) ∼ = {ϕ ∈ E (X ) | τγ ϕ = ϕ, ∀γ ∈ Γ} = SΓ′ (X ) ∩ E (X ). • S ′ (T) is the dual of S (T). • We shall denote by < ·, · >T the natural bilinear map defined by the duality relation on S ′ (T) × S (T) and by (·, ·)T the natural sesquilinear map on S ′ (T) × S (T) obtaind by extending the scalar product from L2 (T). Remark 9.9. It is well known that the spaces SΓ′ (X ) and S ′ (T) can be identified through the following topological isomorphism: X ∼ < i(u), ϕ >:=< u, i : S ′ (T) → SΓ′ (X ); τγ ϕ >T , ∀(u, ϕ) ∈ S ′ (T) × S (X ). (9.15) γ∈Γ ∞ In order P to give an explicit form to the inverse of the isomorphism i let us fix some test function φ ∈ C0 (X ) such τγ φ = 1 (it is easy to see that there exist enough such functions). Then we can easily verify that: that γ∈Γ < i−1 (v), θ >T =< v, φθ >, ∀(v, θ) ∈ SΓ′ (X ) × S (T). Remark 9.10. For any distribution u ∈ SΓ′ (X ) ∼ = S ′ (T) we have the Fourier series decomposition: X u = û(γ ∗ ) := |E|−1 < u, σ−γ ∗ >T , û(γ ∗ )σγ ∗ , (9.16) (9.17) γ ∗ ∈Γ∗ ∗ where σγ ∗ (y) := ei<γ ,y> , ∀y ∈ T, ∀γ ∗ ∈ Γ∗ and the series converges as tempered distribution. In particular, if u ∈ L2 (T) we also have the equality: X kuk2L2 (T) = |E| |û(γ ∗ )|2 . (9.18) γ ∗ ∈Γ∗ Remark 9.11. A simple computation allows to prove that for any s ∈ R and any γ ∗ ∈ Γ∗ the following equality is true in S ′ (X ): (9.19) < D >s σγ ∗ = < γ ∗ >s σγ ∗ . s ′ ′ ∼ Using now (9.17) we deduce that < D > induces on S (T) = SΓ (X ) a well-defined operator, denoted by < DΓ >s , explicitely given by the following formula: X < DΓ >s u := ∀u ∈ S ′ (T). (9.20) < γ ∗ >s û(γ ∗ )σγ ∗ , γ ∗ ∈Γ∗ 60 Definition 9.12. Given any s ∈ R we define the complex linear space:  Hs (T) := u ∈ S ′ (T) | < DΓ >s u ∈ L2 (T) endowed with the quadratic norm kukHs (T) := k< DΓ >s ukL2 (T) for which it becomes a Hilbert space. Let us notice that for any u ∈ S ′ (T) we have the following equivalence relation: X u ∈ Hs (T) ⇐⇒ |E|−1 < γ ∗ >2s |û(γ ∗ )|2 < ∞, γ ∗ ∈Γ∗ and the formula in the right hand side of the above equivalence relation is equal to kuk2Hs(T) . From these facts it follows easily that S (T) is dense in Hs (T). s+m Lemma 9.13. Let p ∈ S1m (X ) and let us denote by P := Op(p). Then for any s ∈ R and for any u ∈ Hloc (X ) ∩ ′ s ′ S (X ) we have that P u ∈ Hloc (X ) ∩ S (X ). Proof. It is clear from the definitions that we have P u ∈ S ′ (X ). For any relatively compact open subset Ω ⊂ X we can choose two positive test functions ψ and χ of class C0∞ (X ) such that: ψ = 1 on Ω and χ = 1 on a neighborhood Vψ of the support of ψ. Then χu ∈ Hs+m (X ) and we know that P χu ∈ Hs (X ). Thus the Lemma will follow if we prove that the restriction of P (1 − χ)u to Ω is of class C ∞ (Ω). For that, let us choose φ ∈ C0∞ (Ω) and notice that  Z  t hP (1 − χ)u, φi = u, (1 − χ)P (ψφ) = u, K(x, y)φ(y)dy , X where for any N ∈ N we have the following equality (obtained by the usual integration by parts method): Z
 
x+y , −η dη. ¯ ei<η,x−y> |x − y|−2N ∆N K(x, y) := 1 − χ(x) ψ(y) η p 2 X It is evident that K ∈ S (X 2 ), so that v(y) :=< u(·), K(·, y) > is a function of class S (X ) and we have the equality: hP (1 − χ)u, φi = < v, φ >, from which we conclude that P (1 − χ)u ∈ C ∞ (Ω). Corollary 9.14. The space Hs (T) can be identified with the usual Sobolev space of order s on the torus that is s defined as Hloc (X ) ∩ SΓ′ (X ). Proof. The Corollary follows from Lemma 9.13 using the fact that we have the equality < DΓ >s =< D >s on SΓ′ (X ) and thus for any u ∈ SΓ′ (X ) the following relations hold: s u ∈ Hs (T) ⇔ < DΓ >s u ∈ L2 (T) ⇔ < D >s u ∈ L2loc (X ) ∩ SΓ′ (X ) ⇔ u ∈ Hloc (X ) ∩ SΓ′ (X ). Remark 9.15. Let us notice the following facts to be used in the paper. 1. For any pair of real numbers (s, t) the application < DΓ >s : Ht+s (T) → Hs (T) is an isometric isomorphism. 2. For any pair (u, v) of test functions from S (T) the following equalities are true: (u, v)Hs (T) = (< DΓ >s u, < DΓ >s v)T = u, < DΓ >2s v
T . 3. The dual of Hs (T) can be canonically identified with the space H−s (T). In fact, using also the above remark, we can identify the dual of Hs (T) with itself via the operator < DΓ >−2s . Example 9.16. For any s ∈ R and for any ξ ∈ X ∗ we define the following operator: X < DΓ + ξ >s : S ′ (T) → S ′ (T), < DΓ + ξ >s u := < γ ∗ + ξ >s û(γ ∗ )σγ ∗ . (9.21) γ ∗ ∈Γ∗ Considering u ∈ SΓ′ (X ) we evidently have that < DΓ + ξ >s u =< D + ξ >s u. We define the following complex linear space:  Ks,ξ := u ∈ S ′ (T) | < DΓ + ξ >s u ∈ L2 (T) , 61 (9.22) endowed with the quadratic norm: 2 kuk2Ks,ξ := k< DΓ + ξ >s ukL2 (T) = |E|−1 X γ ∗ ∈Γ∗ < γ ∗ + ξ >2s |û(γ ∗ )|2 (9.23) that defines a structure of Hilbert space on it. It is clear that Ks,ξ = Hs (T) as complex vector spaces and for ξ = 0 even as Hilbert spaces (having the same scalar product). Similar arguments to those in Example 9.2 show that the family {Ks,ξ }ξ∈X ∗ has temperate variation. s Coming back to Corollary 9.14 we can consider the elements of Ks,ξ as distributions from Hloc (X ) ∩ SΓ′ (X ) and we can define the spaces: Fs,ξ := {u ∈ S ′ (X ) | σ−ξ u ∈ Ks,ξ } . (9.24) It will become a Hilbert space isometrically isomorphic with Ks,ξ (through the operator σ−ξ ) once we endow it with the norm: kukFs,ξ := kσ−ξ ukKs,ξ , ∀u ∈ Fs,ξ . (9.25) Remark 9.17. Let us fix some ξ ∈ X ∗ . 1. Let us denote by: Sξ′ (X ) := Then σ−ξ : Sξ′ (X ) → SΓ′ (X )  u ∈ S ′ (X ) | τ−γ u = ei<ξ,γ> u, ∀γ ∈ Γ . (9.26) is an isomorphism having the inverse σξ . 2. Let us notice that we can write:  F0,ξ = u ∈ S ′ (X ) | σ−ξ u ∈ L2loc (X ) ∩ SΓ′ (X ) = Sξ′ (X ) ∩ L2loc (X ) and conclude that we can identify F0,ξ with L2 (E) and notice that we have the equality of the norms ∼ kukF0,ξ = kukL2 (E) . In fact the isomorphism jξ : F0,ξ → L2 (E) is defined by taking the restriction to E ⊂ X , ∼ i.e. jξ u := u|E , ∀u ∈ F0,ξ . We can obtain an explicit formula for its inverse j−1 : L2 (E) → F0,ξ ; for any ξ v ∈ L2 (E) we define a distribution ṽξ that is equal to σ−ξ v on E and is extended to X by Γ-periodicity. This clearly gives us a distribution from L2loc (X ) ∩ SΓ′ (X ). One can easily see that we have j−1 ξ v = σξ ṽξ . 3. From (9.4) it follows that < D + ξ >s = σ−ξ < D >s σξ . Thus: s s Fs,ξ = {u ∈ S ′ (X ) | σ−ξ u ∈ Hloc (X ) ∩ SΓ′ (X )} = Sξ′ (X ) ∩ Hloc (X ), and for any u ∈ Fs,ξ we have that kukFs,ξ = kσ−ξ ukKs,ξ = (9.27) = k< DΓ + ξ >s σ−ξ ukL2 (T) = kσ−ξ < D >s ukL2 (E) = k< D >s ukL2 (E) . In particular we obtain that  Fs,ξ = u ∈ Sξ′ (X ) | < D >s u ∈ F0,ξ , kukFs,ξ = k < D >s ukF0,ξ . (9.28) Moreover, if s ≥ 0 we have that Fs,ξ = {u ∈ F0,ξ | < D >s u ∈ F0,ξ }.

Definition 9.18. We shall denote by Sρm X × T; B(A• ; B• ) the space of symbols p ∈ Sρm X 2 ; B(A• ; B• ) that are Γ-periodic with respect to the second variable (i.e. p(x, y
+ γ, ξ) = p(x, y, ξ), ∀(x,
y) ∈ X 2 , ∀ξ ∈ X ∗
and ∀γ ∈ Γ).
In a similar way we define the spaces Sρm T; B(A• ; B• ) , Sρm X × T; B(A0 ; B0 ) , Sρm T; B(A0 ; B0 ) , Sρm X × T ,

m
m m m X × T . Let us notice that we have an evident X × T; B(A0 ; B0 ) , Sρ,ǫ X × T; B(A• ; B• ) , Sρ,ǫ Sρ (T), Sρ,ǫ

m m identification of Sρ,ǫ X ; B(A• ; B• ) with a subspace of Sρ,ǫ X × T; B(A• ; B• ) . Lemma 9.19. Under the hypothesis of Proposition 9.7, for any a ∈ X we have the equality:
τa OpA (p) = Opτa A (τa ⊗ idl)p τa . (9.29)
Proof. We start from equality (9.9) with u ∈ S X ; A0 and we get: Z h i x−a+y
ei<η,x−a−y> ωA (x − a, y) p , η u(y) dy dη ¯ = τa OpA (p)u (x) = 2 Ξ Z
x+y ei<η,x−y> ωA (x − a, y − a) p = − a, η u(y − a) dy dη, ¯ ∀x ∈ X . 2 Ξ 62 First let us recall that: ωA (x, y) = e−i Let us notice that Z R [x,y] A [x−a,y−a] A   Z = exp i (x − y), 0  Z = − (x − y), 0 and thus ωA (x − a, y − a) = ωτa A (x, y). 1 1
A (1 − s)x + sy ds
A (1 − s)x + sy − a ds  . (9.30)  Lemma 9.20. For any symbol p ∈ S1m (T) the pseudodifferential operator P := Op(p) induces on T an operator PΓ ∈ B(Ks+m,0 ; Ks,0 ) for any s ∈ R and the application S1m (T) ∋ p 7→ PΓ ∈ B(Ks+m,0 ; Ks,0 ) is continuous. Proof. From equality (9.29) with A = 0 and from the fact that (τγ ⊗idl)p = p, ∀γ ∈ Γ we deduce that P leaves SΓ′ (X ) invariant and thus induces a linear and continuous operator PΓ : S ′ (T) → S ′ (T). If u ∈ Ks+m,0 = Hs+m (T) we can write kPΓ ukKs,0 = k< DΓ >s PΓ ukL2 (T) = k< D >s P ukL2 (E) = < D >s P < D >−s−m < D >s+m u L2 (E) . From the Weyl calculus we know that < D >s P < D >−s−m = Op(q) for a well defined symbol q ∈ S10 (X ) and the map S1m (T) ∋ p 7→ q ∈ S10 (X ) is continuous; by Lemma 9.13 we can find a strictly positive constant C0′ (p) (it is one of the defining seminorms for the topology of S1m (T) and we can find a number N ∈ N (that does not depend on p as seen in the proof of Lemma 9.13) such that < D >s P < D >−s−m v where F := L2 (E) ≤ C0′ (p)kvkL2 (F ) , ∀v ∈ L2loc (X ) ∩ S ′ (X ), ∪ τγ E, and ΓN := {γ ∈ Γ | |γ| ≤ N }. Let us consider now v =< D >s+m u ∈ L2loc (X ) ∩ SΓ′ (X ). γ∈ΓN We deduce that kvk2L2 (F ) = X Z |γ|≤N τγ E 2 2 |v(x)|2 dx ≤ CN kvk2L2 (E) = CN < D >s+m u L2 (E) 2 kuk2Ks+m,0 . = CN We conclude that kPΓ ukKs,0 ≤ CN C0′ (p)kukKs+m,0 . ∗ m Example 9.21. For any symbol p ∈ S1m (T) and for
any point ξ ∈ X we know that (idl ⊗ τ−ξ )p ∈ S1 (T) and
due to Lemma 9.20 the operator Pξ := Op (idl ⊗ τ−ξ )p induces on T a well defined operator PΓ,ξ ∈ B(K
s+m,0 ; Ks,0 for any s ∈ R. From the same Lemma we deduce that the application X ∗ ∋ ξ 7→ PΓ,ξ ∈ B(Ks+m,0 ; Ks,0 is continuous;
α α taking into account that ∂ξ Pξ = Op (idl ⊗ τ−γ )(idl ⊗ ∂ )p we deduce that the previous application is in fact of class C ∞ . Let us prove now that for any s ∈ R we have that
(9.31) PΓ,ξ ∈ S00 T; B(Ks+m,ξ ; Ks,ξ ) and the application
S1m (T) ∋ p 7→ PΓ,ξ ∈ S00 T; B(Ks+m,ξ ; Ks,ξ ) (9.32) is continuous. These two last statements will follow once we have proved that for any α ∈ Nd there exists cα (p) defining seminorm of the topology of S1m (T), such that ∂ξα PΓ,ξ B(Ks+m,ξ ;Ks,ξ ) ≤ cα (p), ∀ξ ∈ X ∗ . (9.33) It is clearly enough to prove the case α = 0. Then, using (9.4) we deduce that for any u ∈ Ks+m,ξ we have that: kPΓ,ξ ukKs,ξ = k< DΓ + ξ >s PΓ,ξ ukL2 (T) = k< D + ξ >s Pξ ukL2 (E) = k< D + ξ >s σ−ξ P σξ ukL2 (E) = = kσ−ξ < D >s P σξ ukL2 (E) = = < D >s P < D >−s−m < D >s+m σξ u < D >s P < D >−s−m σξ < D + ξ >s+m u L2 (E) L2 (E) = . As in the proof of Lemma 9.20 we deduce that < D >s P < D >−s−m v L2 (E) ≤ C0′ (p)kvkL2 (F ) , 63 ∀v ∈ L2loc (X ) ∩ S ′ (X ). We consider a vector w :=< D + ξ >s+m u ∈ L2loc (X ) ∩ SΓ′ (X ) and v := σξ w. Then 2 2 kwk2L2 (E) = CN < D + ξ >s+m u kvk2L2 (F ) = kwk2L2 (F ) ≤ CN 2 L2 (E) 2 = CN kuk2Ks+m,ξ . Thus (9.33) follows for α = 0 with C0 (p) = CN C0′ (p). Lemma 9.22. Let p ∈ S1m (T) be a real elliptic symbol (i.e. ∃C > 0, ∃R > 0 such that p(y, η) ≥ C|η|m for any (y, η) ∈ Ξ with |η| ≥ R), with m > 0. Then the operator PΓ defined in Lemma 9.20 is self-adjoint on the domain Km,0 . Moreover, PΓ is lower semi-bounded and its graph-norm on Km,0 gives a norm equivalent to the defining norm of Km,0 . Proof. Let us first verify the symmetry of PΓ on Km,0 . Due to the density of S (T) in Km,0 and to the fact that
PΓ ∈ B Km,0 ; L2 (T) , it is enough to verify the symmetry of PΓ on S (T). Let us choose two vectors u and v from S (T) so that we have to prove the equality (PΓ u, v)L2 (T) = (u, PΓ v))L2 (T) or equivalently (P u, v)L2 (E) = (u, P v))L2 (E) . (9.34) Identifying S (T) with E (X ) ∩ SΓ′ (X ) and using the definition of the operator P on the space S ′ (X ) one easily verifies that P u also belongs to E (X )∩SΓ′ (X ) and is explicitely given by the following oscillating integral (∀x ∈ X ): Z
x+y
, η u(y) dy dη ¯ = (9.35) P u (x) = ei<η,x−y> p 2 Ξ Z XZ XZ Z x+y
x+y−γ
= , η u(y) dy dη ¯ = , η u(y) dy dη, ¯ ei<η,x−y> p ei<η,x−y+γ> p 2 2 τγ E X ∗ E X∗ γ∈Γ γ∈Γ the series converging in E (X ). Using the Γ-periodicity of p we obtain that: Z XZ Z Z
x+y−γ
, η u(y)v(x) dx dy dη P u (x)v(x) dx = (P u, v)L2 (E) = ¯ = ei<η,x−y+γ> p 2 E E E X∗ γ∈Γ = Z E  u(y)  Z

x + y + γ , η v(x)dx dη ¯ dy = u(y) P v (y) dy = (u, P v)L2 (E) , ei<η,y−x−γ>p 2 E X∗ XZ Z γ∈Γ E and thus we proved the equality (9.34).
In order to prove the self-adjointness of PΓ let us choose some vector u ∈ D PΓ∗ ; thus it exists f ∈ L2 (T) such that we have the equality (PΓ ϕ, u)L2 (T) = (ϕ, f )L2 (T) , ∀ϕ ∈ S (T). Using now the facts that S (T) is dense in S ′ (T) and PΓ is symmetric on S (T), we deduce that (ϕ, f )T = (PΓ ϕ, u)T = (ϕ, PΓ u)T , ∀ϕ ∈ S (T) and thus we obtain the equality PΓ u = f in S ′ (T). By hypothesis PΓ is an elliptic pseudifferential operator of strictly positive order m, on the compact manifold T, so that the usual regularity results imply that u ∈ Km,0 = D(PΓ ). In conclusion PΓ is self-adjoint on the domain Km,0 . The lower semiboundedness property follows from the Gårding inequality: (PΓ u, u)L2 (T) ≥ C −1 kuk2Km/2,0 − Ckuk2L2 (T) , ∀u ∈ Km,0 . (9.36) The equivalence of the norms stated as the last point of the Lemma follows from the Closed Graph Theorem. Remark 9.23. Under the Hypothesis of Lemma 9.22, the same proof also shows that for any ξ ∈ X ∗ , the operator PΓ,ξ from Example 9.21 is self-adjoint and lower semibounded on L2 (T) on the domain Km,ξ . As in Remark 9.17 we m can identify Km,ξ with Hloc (X ) ∩ SΓ′ (X ) (endowed with the norm k< D + ξ >m ukL2 (E) ) and thus we can deduce that the operator Pξ is a self-adjoint operator in the space L2loc (X ) ∩ SΓ′ (X ) on the domain Km,ξ . From (9.4) we know that P = σξ Pξ σ−ξ and we also know that σξ : Ks,ξ → Fs,ξ is a unitary operator for any s ∈ R and for any ξ ∈ X ∗ and we conclude that the operator induced by P in F0,ξ is unitarily equivalent with the operator induced by Pξ in K0,ξ ∼ = L2loc (X ) ∩ SΓ′ (X ). It follows that the operator P acting in F0,ξ with domain Fm,ξ is self-adjoint and lower semibounded. 64 9.3 Properties of magnetic pseudodifferential operators with operator-valued symbols Theorem 9.24. Let us first consider the composition operation. Suppose chosen three families of Hilbert spaces
with m temperate variation {Aξ }ξ∈X ∗ , {Bξ }ξ∈X ∗ and {Cξ }ξ∈X ∗ , two families of symbols {pǫ }|ǫ|≤ǫ0 ∈ Sρ,ǫ X ; B(B• ; C• ) and
′ m {qǫ }|ǫ|≤ǫ0 ∈ Sρ,ǫ X ; B(A• ; B• ) and a family of magnetic fields {Bǫ }|ǫ|≤ǫ0 satisfying Hypothesis H.1 from Section 2 with an associated family of vector potentials {Aǫ }|ǫ|≤ǫ0 given by (1.28). Then 1. There exist a family of symbols
m+m′ X ; B(A• ; C• ) , {pǫ ♯Bǫ qǫ }|ǫ|≤ǫ0 ∈ Sρ,ǫ such that OpAǫ (pǫ )OpAǫ (qǫ ) = OpAǫ (pǫ ♯Bǫ qǫ ). 2. The application


m+m′ m′ X ; B(A• ; C• ) X ; B(A• ; B• ) ∋ (pǫ , qǫ ) 7→ pǫ ♯Bǫ qǫ ∈ Sρ, Sρm X ; B(B• ; C• ) × Sρ, (9.37) is continuous uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ].
m+m′ −ρ 3. There exists a family of symbols {rǫ }|ǫ|≤ǫ0 ∈ Sρ,ǫ X ; B(A• ; C• ) having the following properties: lim rǫ = 0 ǫ→0
′ in Sρm+m −ρ X ; B(A• ; C• ) pǫ ♯Bǫ qǫ = pǫ · qǫ + rǫ , ∀ǫ ∈ [−ǫ0 , ǫ0 ]. (9.38) (9.39) Proof. By a standard cut-off procedure, as in the proof of Proposition 9.7 we may reduce the problem to the case of symbols with compact support in both arguments (x, ξ) ∈ Ξ. A direct computation using Stokes formula and the fact that dBǫ = 0 for any ǫ ∈ [−ǫ0 , ǫ0 ] shows that for point (1) of the Theorem we may take the definition of the composition operation to be the following well defined integral formula: Z Z
Bǫ ¯ dZ, ¯ (9.40) e−2i[[Y,Z]] ω Bǫ (x, y, z) pǫ (X − Y )qǫ (X − Z) dY pǫ ♯ qǫ (X) = Ξ Ξ where we used the notation X := (x, ξ), Y := (y, η), Z := (z, ζ), [[Y, Z]] :=< η, z > − < ζ, y > and Z Fǫ (x, y, z) := Bǫ ω Bǫ (x, y, z) := e−iFǫ (x,y,z) , <x−y+z,x−y−z,x+y−z> with < a, b, c > the triangle with vertices a ∈ X , b ∈ X and c ∈ X . A direct computation (see for example Lemma 1.1 from [19]) shows that all the vectors ∇x
Fǫ , ∇y Fǫ and ∇z Fǫ have the form Cǫ (x, y, z)y + Dǫ (x, y, z)z with
Cǫ and Dǫ functions of class BC ∞ X 3 ; B(X ) satisfying the conditions lim Cǫ = lim Dǫ = 0 in BC ∞ X 3 ; B(X ) . It follows easily then that the derivatives of ω Bǫ (x, y, z) of ǫ→0 ǫ→0 order at least 1 are finite linear combinations of terms of the form C(α,β);ǫ y α z β ω Bǫ (x, y, z) with C(α,β);ǫ ∈ BC ∞ (X 3 ) satisfying the property lim C(α,β);ǫ = 0 in BC ∞ (X 3 ). Applying now the usual integration by parts with respect ǫ→0 to the variables {y, z, η, ζ} we obtain that for any Nj ∈ N (1 ≤ j ≤ 4) and for any X ∈ Ξ the following equality is true: Z Z
1
N1 1
N2 e−2i[[Y,Z]] < η >−2N1 < ζ >−2N2 idl − ∆z pǫ ♯Bǫ qǫ (X) = idl − ∆y × (9.41) 4 4 Ξ Ξ     1
N4 1
N3 −2N4 Bǫ −2N3 ω (x, y, z) idl − ∆η <z> × <y> pǫ (X − Y ) qǫ (X − Z) dY ¯ dZ. ¯ idl − ∆ζ 4 4 First we apply the differentiation operators on the functions they act on and then we eliminate all the monomials of the form y α z β that appear from the differentiation of ω Bǫ by integrating by parts using the formulas: yj e−2i[[Y,Z]] = 1 ∂ζ e−2i[[Y,Z]], 2i j zj e−2i[[Y,Z]] = − 1 ∂η e−2i[[Y,Z]]. 2i j These computations allow us to obtain the following estimation (for some C > 0 and N ∈ N):
pǫ ♯Bǫ qǫ (X) B(A ;C ) ≤ ξ ≤ C max |α|,|β|,|γ|,|δ|≤N Z Z Ξ <η> −2N1 <ζ> −2N2 <y> −2N3 <z> −2N4 Ξ 65 (9.42) ξ α γ β ∂x ∂ξ pǫ (X − Y ) B(Bξ ;Cξ ) δ ∂x ∂ξ qǫ (X − Z) B(Aξ ;Bξ ) dY ¯ dZ, ¯ for any ǫ ∈ [−ǫ0 , ǫ0 ]. We use now (9.1) and (9.3) and obtain the following estimations valid for any ǫ ∈ [−ǫ0 , ǫ0 ]: ∂xα ∂ξβ pǫ (X − Y ) ≤ C<η> 2M B(Bξ ;Cξ ) <ξ−η > ≤ C < η >2M ∂xα ∂ξβ pǫ (X − Y ) m−ρ|β|  sup < ζ > −m+ρ|β| Z∈Ξ B(Bξ−η ;Cξ−η )
∂zα ∂ζβ pǫ (Z) B(Bζ ;Cζ ) ≤  (9.43) . Repeating the same computations for the derivatives of qǫ and choosing suitable large exponents Nj (1 ≤ j ≤ 4) in (9.42) we deduce the existence of two defining seminorms | · |n1 and respectively | · |n2 on the Fréchet space

′ Sρm X ; B(B• ; C• ) and respectively on Sρm X ; A• ; B• ) such that we have the estimation: ′ sup < ξ >−(m+m ) X∈Ξ
pǫ ♯Bǫ qǫ (X) B(Aξ ;Cξ ) ≤ |pǫ |n1 |qǫ |n2 , ∀ǫ ∈ [−ǫ0 , ǫ0 ]. (9.44)
′ The derivatives of pǫ ♯Bǫ qǫ can be estimated in a similar way in order to conclude that pǫ ♯Bǫ qǫ ∈ Sρm+m X ; A• ; C• ) uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ] and that property (2) is valid. Considering now the family of symbols {pǫ ♯Bǫ qǫ }|ǫ|≤ǫ0 , the hypothesis (2) and (3) from the Definition 9.5 follow easily from (9.38) and (9.39). In conclusion there is only point (3) that remains to be proved. By the same arguments as above we can once again assume that the symbols pǫ and qǫ have compact support. We begin by using (9.40) in the equality: = pǫ (X)qǫ (X) − Z 0 pǫ (X − Y )qǫ (X − Z) = (9.45) 1 [hY, ∇X pǫ (X − tY )i qǫ (X − tZ) + pǫ (X − tY ) hZ, ∇X qǫ (X − tZ)i] dt. The first term on the right side of the equality (9.45) will produce the term pǫ qǫ in the equality (9.39) (see also the Lemma 2.1 from [19]). Let us study now the term obtained by replacing (9.40) into (9.45). We eliminate Y and Z by integration by parts as in the beginning of this proof taking also into account the following identities: ηj e−2i[[Y,Z]] = − 1 ∂z e−2i[[Y,Z]], 2i j ζj e−2i[[Y,Z]] = 1 ∂y e−2i[[Y,Z]] . 2i j These operations will produce derivatives of pǫ and qǫ with respect to x ∈ X , that go to 0 for ǫ → 0 in their symbol spaces topology and derivatives of Fǫ with respect to y and z; but these derivatives may be once again transformed by integrations by parts into factors of the form Cǫ ∈ BC ∞ (X 3 ) having limit 0 for ǫ → 0 as elements from BC ∞ (X 3 ). Thus, the estimations proved in the first part of the proof imply that the equality (9.39) holds
R1 ′ with rǫ = 0 sǫ (t)dt with sǫ (t) ∈ Sρm+m −ρ X ; B(A• ; C• ) uniformly with respect to (ǫ, t) ∈ [−ǫ0 , ǫ0 ] × [0, 1] and
′ lim sǫ (t) = 0 in Sρm+m −ρ X ; B(A• ; C• ) uniformly with respect to t ∈ [0, 1]. We conclude that rǫ has the properties ǫ→0 stated in the Theorem. Remark 9.25. The proof of Theorem 9.24 also implies the following fact (that we shall paper): the

use in the
′ ′ operation ♯Bǫ is well defined also as operation: Sρm X ; B(B• ; C• ) × Sρm X ; B(A• ; B• ) → Sρm+m X ; B(A• ; C• ) being bilinear and continuous uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ]. Remark 9.26. As in [20] one can define a family of symbols {qs,ǫ }(s,ǫ)∈R×[−ǫ0 ,ǫ0 ] having the following properties: 1. qs,ǫ ∈ S1s (X ) uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ], 2. qs,ǫ ♯Bǫ q−s,ǫ = 1, 3. ∀s > 0 we have that qs,ǫ (x, ξ) =< ξ >s +µ with some sufficiently large µ > 0 and q0,ǫ = 1. Evidently that for any Hilbert space A we can
identify the symbol qs,ǫ with the operator-valued symbol qs,ǫ idlA and thus we may consider that qs,ǫ ∈ S1s X ; B(A) uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ]. We shall use the notation Qs,ǫ := OpAǫ (qs,ǫ ). Proposition 9.27. Suppose given two Hilbert spaces A and B and for any ǫ ∈ [−ǫ0 , ǫ0 ] a symbol pǫ ∈
S0m X ; B(A; B) , uniformly in ǫ ∈ [−ǫ0 , ǫ0 ]. Then for any s ∈ R the operator OpAǫ (pǫ ) belongs to the space
s+m s (X ) ⊗ A; HA B HA (X ) ⊗ B uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ]. Moreover, the norm of OpAǫ (pǫ ) in the ǫ ǫ
above Banach space is bounded from above by a seminorm of pǫ in S0m X ; B(A; B) , uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ]. 66 Proof. For m = s = 0 the proposition may be proved by the same arguments as in the scalar case: A = B = C (see for example [19]). Also using
the results from [19] we can see that for any t ∈ R the operator Qs,ǫ belongs to t+s t the space B HA (X ); H (X ) uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ]. The proof of the general case folows now Aǫ ǫ from the following identity: OpAǫ (pǫ ) = Q−s,ǫ Qs,ǫ OpAǫ (pǫ ) Q−(s+m),ǫ Qs+m,ǫ
and the fact that qs,ǫ ♯Bǫ pǫ ♯Bǫ q−(s+m),ǫ is a symbol of class S00 X ; B(A; B) uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ] (as implied by the Remark 9.25).
Proposition 9.28. Suppose given a Hilbert space A and a bounded subset {pǫ }|ǫ|≤ǫ0 ⊂ Sρ0 X ; B(A) such that lim pǫ = 0 in this space of symbols. Then, for sufficiently small ǫ0 > 0 the following statements are true: ǫ→0
1. idl + OpAǫ (pǫ ) is invertible in B L2 (X ) ⊗ A for any ǫ ∈ [−ǫ0 , ǫ0 ].

2. It exists a bounded subset of symbols {qǫ }|ǫ|≤ǫ0 from Sρ0 X ; B(A) such that lim qǫ = 0 in Sρ0 X ; B(A) and ǫ→0 the following equality holds:  −1 idl + OpAǫ (pǫ ) = idl + OpAǫ (qǫ ). (9.46) Proof. The first statement above is quite evident once we notice that following Proposition 9.27 we can choose some small enough ǫ0 > 0 such that OpAǫ (pǫ ) ≤ (1/2) for any ǫ ∈ [−ǫ0 , ǫ0 ]. By a straightforward B(L2 (X )⊗A) modification of the arguments given in §6.1 from [20] in order
to deal with operator-valued symbols, we deduce that there exists a bounded subset {rǫ }|ǫ|≤ǫ0 in Sρ0 X ; B(A) such that: −1  idl + OpAǫ (pǫ ) = OpAǫ (rǫ ). (9.47) The equality (9.46) follows if we notice that  −1  −1 idl + OpAǫ (pǫ ) = idl − OpAǫ (pǫ ) idl + OpAǫ (pǫ ) = idl − OpAǫ (pǫ )OpAǫ (rǫ ) and also that Remark 9.25 implies that qǫ := −pǫ ♯Bǫ rǫ has all the stated properties. 9.4 Relativistic Hamiltonians We shall close this subsection with the study of a property that connects the two relativistic Schrödinger Hamiltop  1/2 nians OpAǫ (hR ) and OpAǫ (hN R ) with hR (x, ξ) :=< ξ >≡ 1 + |ξ|2 and hN R (x, ξ) := 1 + |ξ|2 ≡< ξ >2 . We shall use some arguments presented in §6.3 of [20]. Proposition 9.29. There exists a bounded subset {qǫ }|ǫ|≤ǫ0 of symbols from S10 (X ) such that lim qǫ = 0 in S10 (X ) ǫ→0 and  1/2 OpAǫ (hN R ) = OpAǫ (hR ) + OpAǫ (qǫ ). (9.48) Proof. Following [20], if we denote by p− the inverse of the symbol p with respect to the composition ♯Bǫ ,   Z i∞  Aǫ 1/2
− 1 Aǫ Aǫ −1/2 2 Op (hN R ) = Op (hN R )Op − z < ξ > −z dz . (9.49) 2πi −i∞ Recalling the proof of point (3) in Theorem 9.24 we can easily prove that:
< ξ >2 −z ♯Bǫ < ξ >2 −z
−1 = 1 + rǫ,z (9.50) where < z > rǫ,z ∈ S10 (X ) uniformly for (ǫ, z) ∈ [−ǫ0 , ǫ0 ] × iR and lim < z > rǫ,z = 0 in S10 (X ) uniformly with ǫ→0 respect to z ∈ iR. Following the proof of Proposition 9.28, for ǫ0 > 0 sufficiently small there exists a symbol fǫ,z such that < z > fǫ,z ∈ S10 (X ) uniformly with respect to (ǫ, z) ∈ [−ǫ0 , ǫ0 ] × iR, lim < z > fǫ,z = 0 in S10 (X ) uniformly with respect to z ∈ iR and we also have 1 + rǫ,z ǫ→0
− = 1 + fǫ,z . (9.51) From (9.49) and from the properties of the symbol rǫ,z it follows that we can define: < ξ >2 −z
− := < ξ >2 −z
−1 ♯Bǫ 1 + fǫ,z
= 67 < ξ >2 −z
−1 + < ξ >2 −z
−1 ♯Bǫ fǫ,z . (9.52) 
−1 Using (9.52) in (9.49) we notice that the term < ξ >2 −z produces by magnetic quantization a term of the form Aǫ Aǫ −1 Op (hN R )Op (hR ) and using Theorem 9.24 this operator may be put in the form OpAǫ (hR ) + OpAǫ (qǫ′ ) where
−1 qǫ′ ∈ S10 (X ) uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ] with lim qǫ′ = 0 in S10 (X ). If we notice that hN R ♯Bǫ hN R −z ∈ ǫ→0 S10 (X ) uniformly with respect to (ǫ, z) ∈ [−ǫ0 , ǫ0 ] × iR, then we can see that the second term of (9.52) gives in (9.49) by magnetic quantization an expression of the form OpAǫ (qǫ′′ ) with qǫ′′ ∈ S10 (X ) uniformly with respect to ǫ ∈ [−ǫ0 , ǫ0 ] and such that lim qǫ′′ = 0 in S10 (X ). ǫ→0 Acknowledgements: R.Purice aknowledges the CNCSIS support under the Ideas Programme, PCCE project no. 55/2008 Sisteme diferenţiale ı̂n analiza neliniară şi aplicaţii. 68 References [1] W.O. Amrein, A. Boutet de Monvel and V. Georgescu, C0 -Groups, Commutator Methods and Spectral Theory of N-Body Hamiltonians, Birkhäuser, Basel, 1996. [2] N. Athmouni, M. Măntoiu, R. Purice: On the continuity of spectra for families of magnetic pseudodifferential operators, J. Math. Phys. 51, 15 p. (2010). [3] Y. Avron, B. 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