Absolute continuity in periodically bent and twisted tubes

Absolute continuity and band gaps of the spectrum of the Dirichlet Laplacian in periodic waveguides Carlos R. Mamani and Alessandra A. Verri arXiv:1508.02574v4 [math-ph] 7 Jul 2017 March 5, 2022 Abstract Consider the Dirichlet Laplacian operator −∆D in a periodic waveguide Ω. Under the condition that Ω is sufficiently thin, we show that its spectrum σ(−∆D ) is absolutely continuous (in each finite region). In addition, we ensure the existence of at least one gap in σ(−∆D ) and locate it. 1 Introduction and results During the last years the Dirichlet Laplacian operator −∆D restricted to strips (in R2 ) or tubes (in R3 ) has been studied under various aspects. We highlight the particular case where the geometry of these regions are periodic [2, 4, 13, 15, 20, 21]. In this situation, an interesting point is to know under what conditions the spectrum σ(−∆D ) is purely absolutely continuous. On the other hand, since σ(−∆D ) is a union of bands, another question is about the existence of gaps in its structure. In the case of planar periodically curved strips, the absolutely continuity was proved by Sobolev [20] and the existence and location of band gaps was studied by Yoshitomi [21]. The goal of this paper is to prove similar results to those in the three dimensional case. In the following paragraphs, we explain the details. Let r : R → R3 be a simple C 3 curve in R3 parametrized by its arc-length parameter s which possesses an appropriate Frenet frame; see Section 2. Suppose that r is periodic, i.e., there exists L > 0 and a nonzero vector u so that r(s + L) = u + r(s), for all s ∈ R. Denote by k(s) and τ (s) the curvature and torsion of r at the position s, respectively. Pick S 6= ∅; an open, bounded, smooth and connected subset of R2 . Build a tube (waveguide) in R3 by properly moving the region S along r(s); at each point r(s) the cross-section region S may present a (continuously differentiable) rotation angle α(s). Suppose that α(s) is L-periodic. For ε > 0 small enough, one can realize this same construction with the region εS and so obtaining a thin waveguide which is denoted by Ωε . D Let −∆D Ωε be the Dirichlet Laplacian on Ωε . Conventionally, −∆Ωε is the Friedrichs 2 ∞ extension of the Laplacian operator −∆ in L (Ωε ) with domain C0 (Ωε ). Denote by λ0 > 0 the first eigenvalue of the Dirichlet Laplacian −∆D S in S. Due to the geometrical characteristics of S, λ0 is simple. One of the main results of this work is Theorem 1. For each E > 0, there exists εE > 0 so that the spectrum of −∆D Ωε is 2 absolutely continuous in the interval [0, λ0 /ε + E], for all ε ∈ (0, εE ). In [2], the authors proved this result considering the particular case where the cross section of Ωε is a ball Bε = {y ∈ R2 : |y| < ε} (this fact eliminates the twist effect). Covering the case where Ωε can be simultaneously curved and twisted is our main contribution on the theme. 1 Ahead, we summarize the main steps to prove Theorem 1. In particular, we call attention to Theorem 2 and Corollary 2, which are our main tools to generalize the result of [2]. Then, we present the results related to the existence and location of gaps in σ(−∆D Ωε ). Many details are omitted in this introduction but will be presented in the next sections. Fix a number c > kk 2 /4k∞ . Denote by 1 the identity operator. For technical reasons, we start to study the operator −∆D Ωε + c 1; see Section 4. A change of coordinates shows that −∆D Ωε + c 1 is unitarily equivalent to the operator Tε ψ := − 1 R −1 R 1 (∂sy βε ∂sy )ψ − 2 div(βε ∇y ψ) + c ψ, βε ε βε (1) where R ψ := ψ ′ + h∇y ψ, R yi(τ + α′ )(s), ∂sy (2) ′ div denotes the divergent of a vetor field  in S, ψ := ∂ψ/∂s, ∇y ψ := (∂ψ/∂y1 , ∂ψ/∂y2 ) 0 −1 . The domain dom Tε is a subspace of the Hilbert and R is the rotation matrix 1 0 space L2 (R × S, βε dsdy) where the measure βε dsdy comes from the Riemannian metric (11); see Section 2 for the exact definition of βε and details of this transformation. Since the coefficients of Tε are periodic with respect to s, we utilize the Floquet-Bloch reduction under the Brillouin zone CR := [−π/L, π/L]. More precisely, we show that Tε is ⊕ unitarily equivalent to the operator C Tεθ dθ, where Tεθ ψ := 1 1 R R (−i∂sy + θ)βε−1 (−i∂sy + θ)ψ − 2 div(βε ∇y ψ) + c ψ. βε ε βε (3) Now, the domain of Tεθ is a subspace of L2 ((0, L) × S, βε dsdy) and, in particular, the functions in dom Tεθ satisfy the boundary conditions ψ(0, y) = ψ(L, y) and ψ ′ (0, y) = ψ ′ (L, y) in L2 (S). Furthermore, each Tεθ is self-adjoint. See Lemma 2 in Section 3 for this decomposition. Each Tεθ has compact resolvent and is bounded from below. Thus, σ(Tεθ ) is discrete. Denote by {En (ε, θ)}n∈N the family of all eigenvalues of Tεθ and by {ψn (ε, θ)}n∈N the family of the corresponding normalized eigenfunctions, i.e., Tεθ ψn (ε, θ) = En (ε, θ)ψn (ε, θ), n = 1, 2, 3, · · · , θ ∈ C. We have ∞ σ(−∆D Ωε ) = ∪n=1 {En (ε, C)} , where En (ε, C) := ∪θ∈C {En (ε, θ)} ; (4) each En (ε, C) is called nth band of σ(−∆D Ωε ). We begin with the following result. Lemma 1. {Tεθ : θ ∈ C} is a type A analytic family. This lemma ensures that the functions En (ε, θ) are real analytic in C (its proof is presented in Section 3). Another important point to prove Theorem 1 is to know an asymptotic behavior of the eigenvalues En (ε, θ) as ε tends to 0. For this characterization, for each θ ∈ C, consider the one dimensional self-adjoint operator   k 2 (s) θ 2 ′ 2 w, T w := (−i∂s + θ) w + C(S)(τ + α ) (s) + c − 4 2 acting in L2 (0, L), where the functions in dom T θ satisfy the conditions w(0) = w(L) and w′ (0) = w′ (L). The constant C(S) depends on the cross section S and is defined by (15) in Section 4. For simplicity, write Q := (0, L) × S. Recall λ0 > 0 denotes the first eigenvalue of the Dirichlet Laplacian −∆D S in S. Denote by u0 the corresponding normalized eigenfunction. Consider the closed subspace L := {w(s)u0 (y) : w ∈ L2 (0, L)} ⊂ L2 (Q) and the unitary operator Vε defined by (13) in Section 4. Our main tool to find an asymptotic behavior for En (ε, θ), and then to conclude Theorem 1, is given by Theorem 2. There exists a number K > 0 so that, for all ε > 0 small enough, ( )   λ0 −1 θ −1 θ −1 Tε − 2 1 Vε sup Vε − ((T ) ⊕ 0) ≤ K ε, ε θ∈C (5) where 0 is the null operator on the subspace L⊥ . The spectrum of T θ is purely discrete; denote by κn (θ) its nth eigenvalue counted with multiplicity. Let K be a compact subset of C which contains an open interval and does not contain the points ±π/L and 0. Given E > 0, without lost of generality, we can suppose that, for all θ ∈ K, the spectrum of Tεθ below E + λ0 /ε2 consists of exactly n0 eigenvalues 0 {En (ε, θ)}nn=1 . As a consequence of Theorem 2, Corollary 1. There exists εn0 > 0 so that, for all ε ∈ (0, εn0 ), En (ε, θ) = λ0 + κn (θ) + O(ε), ε2 (6) holds for each n = 1, 2, · · · , n0 , uniformly in K. In [2] the authors found a similar approximation as in Theorem 2 that also holds uniformly for θ in K. However their results were proved with the assumption that the cross section was a ball Bε . In their proofs, they have used results of [11] which do not seem to generalize easily to other cross sections. On the other hand, similar estimates to (5) and (7) were proved in [5, 10, 18] for a larger class of cross sections than only balls, but the results hold only in the case θ = 0. We stressed that in [18] the convergence is established without assuming the existence of a Frenet frame in the reference curve r. With all these tools in hands, we have Proof of Theorem 1: Let E > 0, without loss of generality, we suppose that, for all 0 . θ ∈ K, the spectrum of Tεθ below E +λ0 /ε2 consists of exactly n0 eigenvalues {En (ε, θ)}nn=1 Lemma 1 ensures that En (ε, θ) are real analytic functions. To conclude the theorem, it remains to show that each En (ε, θ) is nonconstant. Consider the functions κn (θ), θ ∈ K. By Theorem XIII.89 in [19], they are nonconstant. By Corollary 2, there exists εE > 0 so that (7) holds true for n = 1, 2, · · · , n0 , uniformly in θ ∈ K, for all ε ∈ (0, εE ). Note that εE > 0 depends on n0 , i.e., the thickness of the tube depends on the length of the energies to be covered. By Section XIII.16 in [19], the conclusion follows. We know that the spectrum of −∆D Ωε coincides with the union of bands; see (4). It is natural to question the existence of gaps in its structure. This subject was studied in [21]. In that work, by considering a curved waveguide in R2 , the author ensured the existence of at least one gap in the spectrum of the Dirichlet Laplacian and found its location. In this work, we prove similar results for the operator −∆D Ωε . 3 At first, it is possible to organize the eigenvalues {En (ε, θ)}n∈N of Tεθ in order to obtain a non-decreasing sequence. We keep the same notation and write E1 (ε, θ) ≤ E2 (ε, θ) ≤ · · · ≤ En (ε, θ) · · · , θ ∈ C. In this step the functions En (ε, θ) are continuous and piece-wise analytic in C (see Chapter 7 in [17]); each En (ε, C) is either a closed interval or a one point set. In this case, similar to Corollary 1, we have Corollary 2. For each n0 ∈ N, there exists εn0 > 0 so that, for all ε ∈ (0, εn0 ), En (ε, θ) = λ0 + κn (θ) + O(ε), ε2 (7) holds for each n = 1, 2, · · · , n0 , uniformly in C. For simplicity of notation, write V (s) := C(S)(τ + α′ )2 (s) + c − k 2 (s) . 4 Theorem 3. Suppose that V (s) is not constant. Then, there exist n1 ∈ N, εn1 +1 > 0 and Cn1 > 0 so that, for all ε ∈ (0, εn1 +1 ), min En1 +1 (ε, θ) − max En1 (ε, θ) = Cn1 + O(ε). θ∈C θ∈C (8) Theorem 3 ensures that at least one gap appears in the spectrum σ(−∆D Ωε ) for ε > 0 small enough. Its proof is based on arguments of [3, 21] and will be presented in Section 5. With the next result, it will be possible to find a location where (8) holds true. However, some adjustments will be necessary. For γ > 0, we use the scales k(s) 7→ γ k(s), (τ + α′ )(s) 7→ γ (τ + α′ )(s) and c 7→ γ 2 c. (9) D Thus, we obtain a new region Ωγ,ε and we consider −∆D Ωγ,ε instead of −∆Ωε . Denote by θ the operators obtained by replacing (9) in (1) and (3), respectively. Denote Tγ,ε and Tγ,ε θ counted with multiplicity. by En (γ, ε, θ) the nth eigenvalue of Tγ,ε Expand the function V (s) as a Fourier series, i.e., V (s) = +∞ X 1 √ νn e2πnis/L L n=−∞ in L2 (0, L), where the sequence {νn }+∞ n=−∞ is called Fourier coefficients of V (s). Since V (s) is a real function, νn = ν −n , for all n ∈ Z. We have the following result. Theorem 4. Suppose that V (s) is not constant, and let n2 ∈ N so that νn2 6= 0. Then, there exist γ > 0 small enough, εn2 +1 > 0 and Cγ,n2 > 0 so that, for all ε ∈ (0, εn2 +1 ), min En2 +1 (γ, ε, θ) − max En2 (γ, ε, θ) = Cγ,n2 + O(ε). θ∈C θ∈C As Theorem 3, the proof of Theorem 4 is based on [21] and will be presented in Section 6. 4 This work is written as follows. In Section 2 we construct with details the tube Ωε where the Dirichlet Laplacian operator is considered. In the same section, we realize a change of coordinates that allows us “straight” Ωε , i.e., to work in the Hilbert space L2 (R×S, βε dsdy). In Section 3 we perform the Floquet-Bloch decomposition and prove Lemma 1. Section 4 is intended at proofs of Theorem 2 and Corollary 2 (Corollary 1 can be proven in a similar way and we omit its proof in this text). Sections 5 and 6 are dedicated to the proofs of Theorems 3 and 4, respectively. A long the text, the symbol K is used to denote different constants and it never depends on θ. 2 Geometry of the domain and change of coordinates Let r : R → R3 be a simple C 3 curve in R3 parametrized by its arc-length parameter s. We suppose that r is periodic, i.e., there exists L > 0 and a nonzero vector u so that r(s + L) = u + r(s), ∀s ∈ R. The curvature of r at the position s is k(s) := kr′′ (s)k. We assume k(s) > 0, for all s ∈ R. Then, r is endowed with the Frenet frame {T (s), N (s), B(s)} given by the tangent, normal and binormal vectors, respectively, moving along the curve and defined by T = r′ ; N = k −1 T ′ ; The Frenet equations are satisfied, that  ′   T  N′  =  B′ B = T × N. is, 0 k −k 0 0 −τ   0 T τ  N , 0 B (10) where τ (s) is the torsion of r(s), actually defined by (10). More generally, we can consider the case where r has pieces of straight lines, i.e., k = 0 identically in these pieces. In this situation, the construction of a C 2 Frenet frame is described in Section 2.1 of [12]. As another alternative, one can assume the Assumption 1 from [6]. For simplicity, we also denote by {T (s), N (s), B(s)} the Frenet frame in those cases. Let α : R → R be an L-periodic and C 1 function so that α(0) = 0, and S an open, bounded, connected and smooth (nonempty) subset of R2 . For ε > 0 small enough and y = (y1 , y2 ) ∈ S, write x(s, y) = r(s) + εy1 Nα (s) + εy2 Bα (s) and consider the domain Ωε = {x(s, y) ∈ R3 : s ∈ R, y = (y1 , y2 ) ∈ S}, where Nα (s) := cos α(s)N (s) + sin α(s)B(s), Bα (s) := − sin α(s)N (s) + cos α(s)B(s). Hence, this tube Ωε is obtained by putting the region εS along the curve r(s), which is simultaneously rotated by an angle α(s) with respect to the cross section at the position s = 0. 5 As already mentioned in the Introduction, let −∆D Ωε be the Friedrichs extension of the Laplacian operator −∆ in L2 (Ωε ) with domain C0∞ (Ωε ). The next step is to perform a change of variables so that Ωε is homeomorphic to the straight cylinder R × S. Consider the mapping Fε : R × S → Ω ε (s, y) 7→ r(s) + εy1 Nα (s) + εy2 Bα (s). In the new variables, the Dirichlet Laplacian −∆D Ωε will be unitarily equivalent to one operator acting in L2 (R × S, βε dsdy); see definition of βε below. The price to be paid is a nontrivial Riemannian metric G = Gαε which is induced by Fε , i.e., G = (Gij ), where e1 = Some calculations show that    e1 J :=  e2  =  e3 where βε (s, y) := 1 − εk(s)hzα , yi, Gij = hei , ej i = Gji , ∂Fε , ∂s e2 = ∂Fε , ∂y1 1 ≤ i, j ≤ 3, e3 = (11) ∂Fε . ∂y2 in the Frenet frame  βε −ε(τ + α′ )hzα⊥ , yi ε(τ + α′ )hzα , yi , 0 ε cos α ε sin α 0 −ε sin α ε cos α zα := (cos α, − sin α), and zα⊥ := (sin α, cos α). (12) The inverse matrix of J is given by   1/βε (τ + α′ )y2 /βε −(τ + α′ )y1 /βε (1/ε) cos α −(1/ε) sin α  . J −1 =  0 0 (1/ε) sin α (1/ε) cos α Note that JJ t = G and det J = | det G|1/2 = ε2 βε . Since k is a bounded function, for ε small enough, βε does not vanish in R × S. Thus, βε > 0 and Fε is a local diffeomorphism. By requiring that Fε is injective (i.e., the tube is not self-intersecting), a global diffeomorphism is obtained. Finally, consider the unitary transformation Jε : L2 (Ωε ) → L2 (R × S, βε dsdy) , u 7→ ε u ◦ Fε and recall the operator Tε given by (1) in the Introduction. After some straightforward −1 D calculations, we can show that Jε (−∆D Ωε )Jε ψ = Tε ψ, where dom Tε = Jε (dom (−∆Ωε )). From now on, we start to study Tε . 3 Floquet-Bloch decomposition Since the coefficients of Tε are periodic with respect to s, in this section we perform the Floquet-Bloch reduction over the Brillouin zone C = [−π/L, π/L]. For simplicity of notation, we write Ω := R × S, Hε := L2 (Ω, βε dsdy), H̃ε := L2 (Q, βε dsdy). Recall that Q = (0, L) × S. 6 R⊕ Lemma 2. There exists a unitary operator Uε : Hε → C H̃ε dθ, so that, Z ⊕ −1 U ε Tε U ε = Tεθ dθ, C where Tεθ ψ := 1 1 R R (−i∂sy + θ)βε−1 (−i∂sy + θ)ψ − 2 div(βε ∇y ψ) + c ψ, βε ε βε and, dom Tεθ = {ψ ∈ H 2 (Q) : ψ(s, y) = 0 on ∂Q\ ({0, L} × S) , ψ(L, y) = ψ(0, y) in L2 (S), ψ ′ (L, y) = ψ ′ (0, y) in L2 (S)}. Furthermore, for each θ ∈ C, Tεθ is self-adjoint. Proof. As in [2], for (θ, s, y) ∈ C × Q define r X L −inLθ−iθs (Uε f )(θ, s, y) := e f (s + Ln, y). 2π n∈Z This transformation is a modification of Theorem XIII.88 in [19]. As a consequence, the domain of the fibers operators Tεθ keep the same. With respect to the proof of this lemma, a detailed proof for periodic strips in the plane can be found in [21]. The argument for periodic waveguides in R3 is analogous and will be omitted in this text. R ψ in its definition Remark 1. Although Tεθ acts in the Hilbert space H̃ε , the operator ∂sy has action given by (2) (see Introduction) and βε is given by (12) (see Section 2). For simplicity, we keep the same notation. Now, we present the proof of Lemma 1 stated in the Introduction. Proof of Lemma 1: For each θ ∈ C, write Tεθ = Tε0 + Vεθ , where, for ψ ∈ dom Tε0 , Vεθ ψ := (Tεθ − Tε0 )ψ =   R R −1 (−2iθ/βε2 )∂sy ψ + −iθ(∂sy βε )/βε + θ2 /βε2 ψ. We affirm that Vεθ is Tε0 -bounded with zero relative bound. In fact, denote Rz = Rz (Tε0 ) = (Tε0 − z1)−1 . Take z ∈ C with img z 6= 0. Since all coefficients of Vεθ are bounded, there exists K > 0, so that, Z kVεθ ψk2H̃ = |Vεθ ψ|2 βε dxdy ε Q   ≤ K hψ, Tε0 ψiH̃ε + kψk2H̃ ε   0 0 ≤ K hRz (Tε − z1)ψ, Tε ψiH̃ε + kψk2H̃ ε   0 0 0 ≤ K hRz Tε ψ, Tε ψiH̃ε + |z|hψ, Rz Tε ψiH̃ε + kψk2H̃ ε   ≤ K kRz Tε0 ψkH̃ε kTε0 ψkH̃ε + |z|hψ, (1 + zRz )ψiH̃ε + kψk2H̃ ε i   h ≤ K kRz kH̃ε kTε0 ψk2H̃ + |z| + |z|2 kRz kH̃ε + 1 kψk2H̃ , ε ε for all ψ ∈ dom Tε0 and all θ ∈ C. In the first inequality we use the Minkovski inequality and the property ab ≤ (a2 + b2 )/2, for all a, b ∈ R. In the third one, we used that Rz Tε0 = 1 + zRz . Since kRz kH̃ε → 0, as img z → ∞, the affirmation is proven. So, the lemma follows. 7 4 Proof of Theorem 2 and Corollary 2 This section is dedicated to prove Theorem 2. Some steps are very similar to that in [10] and require only an adaptation. Because this, most calculations will be omitted here. Since Tεθ > 0 is self-adjoint, there exists a closed sesquilinear form tθε > 0, so that, dom Tεθ ⊂ dom tθε (actually, dom Tεθ is a core of dom tθε ) and tθε (φ, ϕ) = hφ, Tεθ ϕi, ∀φ ∈ dom tθε , ∀ϕ ∈ dom Tεθ ; see Theorem 4.3.1 of [7]. For ϕ ∈ dom Tεθ , the quadratic form tθε (ϕ) := tθε (ϕ, ϕ) acts as Z Z Z
2 1 βε θ R 2 tε (ϕ) = −i∂sy + θ ϕ dsdy + |∇y ϕ| dsdy + c βε |ϕ|2 dsdy. 2 Q βε Q ε Q We are interested in studying tθε (ϕ) for ε > 0 small enough. However, it is necessary R to control the term (1/ε2 ) Q βε |∇y ϕ|2 dsdy, as ε → 0. Since it is related to the transverse oscillations in the waveguide, we make this in the following way. As already mentioned in the Introduction, let u0 be the eigenfunction associated with the first eigenvalue λ0 of the Dirichlet Laplacian −∆D S in S, i.e., Z D −∆S u0 = λ0 u0 , u0 ≥ 0, |u0 |2 dy = 1, λ0 > 0. S Due to the geometrical characteristics of S, λ0 is a simple eigenvalue. We consider the quadratic form Z
2 λ0 1 θ 2 R tε (ϕ) − 2 kϕkH̃ = −i∂sy + θ ϕ dsdy ε ε Q βε Z Z
βε 2 2 |∇y ϕ| − λ0 |ϕ| dsdy + c βε |ϕ|2 dsdy, + 2 ε Q Q R ϕ ∈ dom Tεθ . The subtraction of (λ0 /ε2 ) Q βε |ϕ|2 dsdy is intended to control the divergence of the transverse oscillations, as ε → 0 (see a detailed discussion in Section 1 of [9]). An important point is that, for each ϕ ∈ dom Tεθ , Z Z
βε 2 2 |∇ ϕ| − λ |ϕ| dy ≥ γ (s) |ϕ|2 dy, a.e. s, y 0 ε 2 ε S S where γε (s) → −k 2 (s)/4 uniformly, as ε → 0. The proof of this inequality can be found in [5]. As
a consequence, since kk 2 /4k∞ < c, zero belongs to the resolvent set ρ Tεθ − (λ0 /ε2 )1 , for all ε > 0 small enough. Now, define the unitary operator Vε : L2 (Q) → H̃ε 1/2 . ψ → ψ/βε (13) With this transformation, we start to work in L2 (Q) with the usual measure of R3 . Namely, consider the quadratic form bθε (ψ) := tθε (Vεθ ψ) − 8 λ0 θ 2 kV ψk , ε2 ε H̃ε defined on the subspace dom bθε := Vε−1 (dom Tεθ ) ⊂ L2 (Q). One can show Z i h 2 1 θ R 1/2 R −1/2 )ψ + θψ dsdy bε (ψ) = −i ∂ ψ + β (∂ β sy ε sy ε 2 Q βε Z 2 Z Z
k (s) 2 1 2 2 |∇y ψ| − λ0 |ψ| dsdy − |ψ| dsdy + c |ψ|2 dsdy. + 2 2 Q 4βε Q Q ε The details of the calculations in this change of coordinates can be found in Appendix A of [10]. θ Denote by Bεθ the self-adjoint operator associated with the closure bε of the quadratic θ form bθε . Actually, dom Bεθ ⊂ dom bε and   λ0 θ −1 Tε − 2 1 Vε = Bεθ . Vε ε By replacing the global multiplicative factor βε by 1 in the first and third integral in the expression of bθε (ψ), we arrive now at the quadratic form Z i h 2 R −1/2 R θ βε )ψ + θψ dsdy −i ∂sy ψ + βε1/2 (∂sy dε (ψ) := Q + Z Q
1 |∇y ψ|2 − λ0 |ψ|2 dsdy − 2 ε Z Q k 2 (s) 2 |ψ| dsdy + c 4 Z Q |ψ|2 dsdy, dom dθε = dom bθε . Again, denote by Dεθ the self-adjoint operator associated with the θ closure dε of the quadratic form dθε . We have dom Dεθ = dom Bεθ and 0 ∈ ρ(Bεθ ) ∩ ρ(Dεθ ), for all ε > 0 small enough. To simplify the calculations ahead, we have the following result. Theorem 5. There exists a number K > 0, so that, for all ε > 0 small enough, o n sup k(Bεθ )−1 − (Dεθ )−1 k ≤ K ε. θ∈C The main point in this theorem is that βε → 1 uniformily as ε → 0. Its proof is quite similar to the proof of Theorem 3.1 in [8] and will not be presented here. Consider the closed subspace L := {w(s)u0 (y) : w ∈ L2 (0, L)} of the Hilbert space L2 (Q). Take the orthogonal decomposition L2 (Q) = L ⊕ L⊥ . (14) For ψ ∈ dom Dεθ , we can write ψ(s, y) = w(s)u0 (y) + η(s, y), with w ∈ H 2 (0, L) and η ∈ Dεθ ∩ L⊥ . Furthermore, w(0) = w(L). Define Z C(S) := |h∇y u0 , Ryi|2 dy ≥ 0. (15) S Note that C(S) = 0 if, and only if, S is radial. Recall V (s) = C(S)(τ + α′ )2 (s) + c − k 2 (s)/4 and the one dimensional operator T θ w = (−i∂s + θ)2 w + V (s)w, mentioned in the Introduction. Take dom T θ = {w ∈ L2 (0, L) : wu0 ∈ dom Dεθ } = {w ∈ H 2 (0, L) : w(0) = w(L), w′ (0) = w′ (L)}. In this domain, T θ is self-adjoint and, since kk 2 /4k∞ < c, 0 ∈ ρ(T θ ). 9 Denote by tθ (w) the quadratic form associated with T θ . For w ∈ dom T θ , Z Lh i |(−i∂s + θ)w|2 + V (s)|w|2 ds. tθ (w) = 0 Proof of Theorem 2: The proof is separated in two steps. Step I. Define the one dimensional quadratic form Z Lh i θ θ |(−i∂s + θ)w|2 + (W (s) + c + gε (s)) |w|2 ds, sε (w) := dε (wu0 ) = 0 dom sθε = dom T θ , where Z  h i′  R −1/2 R −1/2 2 |u0 |2 dy ∈ L∞ (0, L). ) − βε1/2 (∂sy βε ) gε (s) = βε βε (∂sy S Actually, sθε is the restriction of dθε on the subspace dom T θ = dom Dεθ ∩ L. Denote by Sεθ the self-adjoint operator associated with the closure sθε of the quadratic form sθε . We have dom Sεθ = dom T θ ⊂ dom sθε . Recall the definition of βε by (12) in Section 2. Some calculations show that |gε (s)| ≤ K ε, ∀s ∈ (0, L), (16) for some K > 0. This fact and the condition kk 2 /4k∞ < c imply 0 ∈ ρ(Sεθ ), for all ε > 0 small enough. Let 0 be the null operator on the subspace L⊥ . In this step, we are going to show that there exists K > 0, so that, for all ε > 0 small enough, o n sup k(Dεθ )−1 − ((Sεθ )−1 ⊕ 0)k ≤ K ε. (17) θ∈C Due to the decomposition (14), for ψ ∈ dom Dεθ , ψ(s, y) = w(s) u0 (y) + η(s, y), w ∈ dom T θ , η ∈ dom Dεθ ∩ L⊥ . Thus, dθε (ψ) can be rewritten as dθε (ψ) = sθε (w) + dθε (wu0 , η) + dθε (η, wu0 ) + dθε (η). We need to check that there are c0 > 0 and functions 0 ≤ q(ε), 0 ≤ p(ε) and c(ε) so that sθε (w), dθε (η) and dθε (w, η) satisfy the following conditions: sθε (w) ≥ c(ε)kwu0 k2L2 (Q) , ∀w ∈ dom T θ , dθε (η) ≥ p(ε)kηk2L2 (Q) , c(ε) ≥ c0 > 0; ∀η ∈ dom Dεθ ∩ L⊥ ; |dθε (w, η)|2 ≤ q(ε)2 sθε (w) dθε (η), ∀ψ ∈ dom Dεθ ; (18) (19) (20) and with p(ε) → ∞, c(ε) = O(p(ε)), q(ε) → 0 as ε → 0. Thus, Proposition 3.1 in [14] guarantees that, for ε > 0 small enough, n o sup k(Dεθ )−1 − ((Sεθ )−1 ⊕ 0)k ≤ p(ε)−1 + K q(ε) c(ε)−1 , θ∈C 10 (21) for some K > 0. We highlight that the main point in this proof is to get functions c(ε), p(ε) and q(ε) that do not depend on θ. Since kk 2 /4k∞ < c and gε (s) → 0 uniformly, there exists c1 > 0, so that, Z L |w|2 ds = c1 kwu0 kL2 (Q) , ∀w ∈ dom T θ , sθε (w) ≥ c1 0 for all ε > 0 small enough. We pick up c(ε) := c1 . Let λ1 > λ0 the second eigenvalue of the Dirichlet Laplacian operator in S. The Min-Max Principle ensures that Z Z
2 2 |∇y η| − λ0 |η| dy ≥ (λ1 − λ0 ) |η|2 dy, a.e. s, ∀η ∈ dom Dεθ ∩ L⊥ . S S Thus, dθε (η) ≥ (λ1 − λ0 ) ε2 )/ε2 . Z Q |η|2 dsdy, ∀η ∈ dom Dεθ ∩ L⊥ . Just to take p(ε) := (λ1 − λ0 The proof of inequality (20) is very similar to that in Appendix B in [10]. Again, it will be omitted here. One can show |dθε (w, η)|2 ≤ K ε2 sθε (w) dθε (η), ∀ψ ∈ dom Dεθ , √ for some K > 0. Take q(ε) := K ε. Since the conditions (18), (19), (20) and (21) are satisfied, (17) holds true. Step II. By (16), for all ε > 0 small enough, Z Z L 2 θ θ |w| ds ≤ K ε |sε (w) − t (w)| ≤ kgε k∞ 0 L 0 |w|2 ds, ∀w ∈ dom T θ , ∀θ ∈ C. By Theorem 3 in [1], for all ε > 0 small enough, n o sup k(Sεθ )−1 − (T θ )−1 k ≤ K ε. θ∈C Taking into account Theorem 5 and the Steps I and II, we conclude the proof of Theorem 2. Remark 2. Let (hε )ε , (mε )ε be two sequences of positive and closed sesquilinear forms in the Hilbert space H with dom hε = dom mε = D, for all ε > 0. Denote by Hε and Mε the self-adjoint operators associated with hε and mε , respectively. Suppose that there exists ζ > 0, so that, hε , mε > ζ, for all ε > 0, and |hε (ϕ) − mε (ϕ)| ≤ j(ε) mε (ϕ), ∀ϕ ∈ D, (22) with j(ε) → 0, as ε → 0. Theorem 3 in [1] implies that there exists a number K > 0, so that, for all ε > 0 small enough, kHε−1 − Mε−1 k ≤ K j(ε). (23) Suppose that dom Hε = dom Mε =: D̃ and that the condition (22) is satisfied for all ϕ ∈ D̃. By applying the same proof of [1], the inequality (23) holds true. The same idea can be applied in Proposition 3.1 in [14]. Because of this, in this section, when working with quadratic forms we have restricted the study to their actions in the domains of their respective associated self-adjoint operators. 11 Proof of Corollary 2: Denote by λn (ε, θ) := En (ε, θ) − (λ0 /ε2 ). Theorem 2 in the Introduction and Corollary 2.3 of [16] imply 1 1 ≤ K ε, − λn (ε, θ) κn (θ) ∀n ∈ N, ∀θ ∈ C, (24) for all ε > 0 small enough. Then, |λn (ε, θ) − kn (θ)| ≤ K ε |λn (ε, θ)| |kn (θ)|, ∀n ∈ N, ∀θ ∈ C, for all ε > 0 small enough. A proof similar to that of Lemma 1 shows that {T θ : θ ∈ C} is a type A analytic family. Thus, the functions kn (θ) are continuous in C and consequently bounded. This fact and the inequality (24) ensure that, for each ñ0 ∈ N, there exists Kñ0 > 0, so that, |λñ0 (ε, θ)| ≤ Kñ0 , ∀θ ∈ C, for all ε > 0 small enough. Finally, for each n0 ∈ N, there exists Kn0 > 0 so that |λn (ε, θ) − kn (θ)| ≤ Kn0 ε, n = 1, 2 · · · , n0 , ∀θ ∈ C, for all ε > 0 small enough. 5 Existence of band gaps; proof of Theorem 3 Again, recall V (s) = C(S)(τ + α′ )2 (s) + c − k 2 (s)/4 and consider the one dimensional operator T w = −w′′ + V (s)w, dom T = H 2 (R). We have denoted by κn (θ) the nth eigenvalue (counted with multiplicity) of the operator T θ . Each κn (θ) is a continuous function in C. By Chapter XIII.16 in [19], we have the following properties: (a) κn (θ) = κn (−θ), for all θ ∈ C, n = 1, 2, 3, · · · . (b) For n odd (resp. even), κn (θ) is strictly monotone increasing (resp. decreasing) as θ increases from 0 to π/L. In particular, κ1 (0) < κ1 (π/L) ≤ κ2 (π/L) < κ2 (0) ≤ · · · ≤ κ2n−1 (0) < κ2n−1 (π/L) ≤ κ2n (π/L) < κ2n (0) ≤ · · · . For each n = 1, 2, 3, · · · , define  [κn (0), κn (π/L)] , for n odd, Bn := [κn (π/L), κn (0)] , for n even, and   (κn (π/L), κn+1 (π/L)) , for n odd so that κn (π/L) 6= κn+1 (π/L), (κn (0), κn+1 (0)) , for n even so that κn (0) = 6 κn+1 (0), Gn :=  ∅, otherwise. 12 By Theorem XIII.90 in [19], one has σ(T ) = ∪∞ n=1 Bn where Bn is called the jth band of σ(T ), and Gn the gap of σ(T ) if Bn 6= ∅. Corollary 2 implies that for each n0 ∈ N, there exists εn0 > 0 so that, for all ε ∈ (0, εn0 ),  λ0 /ε2 + κn (π/L) + O(ε), for n odd, max En (ε, θ) = λ0 /ε2 + κn (0) + O(ε), for n even, θ∈C and min En (ε, θ) = θ∈C  λ0 /ε2 + κn (0) + O(ε), for n odd, λ0 /ε2 + κn (π/L) + O(ε), for n even, hold for each n = 1, 2, · · · , n0 . Thus, we have Corollary 3. For each n0 ∈ N, there exists εn0 +1 > 0 so that, for all ε ∈ (0, εn0 +1 ), min En+1 (ε, θ) − max En (ε, θ) = |Gn | + O(ε), θ∈C θ∈C holds for each n = 1, 2, · · · , n0 , where | · | is the Lebesgue measure. Another important tool to prove Theorem 3 is the following result due to Borg [3]. Theorem 6. (Borg) Suppose that W is a real-valued, piecewise continuous function on [0, L]. Let λ± n be the nth eigenvalue of the following operator counted with multiplicity respectively d2 − 2 + W (s), in L2 (0, L), ds with domain {w ∈ H 2 (0, L); w(0) = ±w(L), w′ (0) = ±w′ (L)}. (25) We suppose that + λ+ n = λn+1 , for all even n, and − λ− n = λn+1 , for all odd n. Then, W is constant on [0, L]. Proof of Theorem 3: For each θ ∈ C, we define the unitary transformation (uθ w)(s) = π/L := u π/L u−1 e−iθs w(s). In particular, consider the operators T̃ 0 := u0 T 0 u−1 π/L T 0 and T̃ π/L whose eigenvalues are given by {νn (0)}n∈N and {νn (π/L)}n∈N , respectively. Furthermore, the domains of these operators are given by (25); T̃ 0 (resp. T̃ π/L ) is called operator with periodic (resp. antiperiodic) boundary conditions. Since V (s) is not constant in [0, L], by Borg’s Theorem, without loss of generality, we can affirm that there exists n1 ∈ N so that νn1 (0) 6= νn1 +1 (0). Now, the result follows by Corollary 3. 6 Location of band gaps; proof of Theorem 4 The proof of Theorem 4 is very similar to the proof of Theorem 1.3 in [21]. Due to this reason, we present only some steps. A more complete proof can be found in that work. We begin with some technical details. Let W ∈ L2 (0, L) be a real function. For µ ∈ C, consider the operators T + w = −w′′ + µ W (s)w and T − w = −w′′ + µ W (s)w, 13 with domains given by dom T + = {w ∈ H 2 (0, L) : w(0) = w(L), w′ (0) = w′ (L)}, dom T − = {w ∈ H 2 (0, L) : w(0) = −w(L), w′ (0) = −w′ (L)}, respectively. Denote by {ln+ (µ)}n∈N and {ln− (µ)}n∈N the eigenvalues of T + and T − , respectively. For µ ∈ R and n ∈ N, define − − + + (µ) − l2n−1 (µ). (µ) − l2n (µ) and δn− (µ) := l2n δn+ (µ) := l2n+1 Now, δ2n−1 (µ) := δn− (µ) and δ2n (µ) := δn+ (µ). n=+∞ Let {ωm }n=−∞ be the Fourier coefficients of W (s). More precisely, one can write W (s) = +∞ X 1 √ ωn e2nπis/L L n=−∞ in L2 (0, L). Since W (s) is a real function, we have ωn = ω−n , for all n ∈ Z. The goal is to find an asymptotic behavior for δn (µ), as µ → 0, in terms of the Fourier coefficients of W (s). Theorem 7. For each n ∈ N, 2 δn (µ) = √ |ωn ||µ| + O(|µ|2 ), L µ → 0, µ ∈ R. A detailed proof of Theorem 7 can be find in [21]; the main tool used by the author in the proof is the analytic perturbation theorem due to Kato and Rellich (see [17]; Chapter VII and Theorem 2.6 in Chapter VIII). Recall the definition of T θ and En (γ, ε, θ) in the Introduction. For each θ ∈ C, define Tγθ w := −w′′ + γ 2 V (s)w, dom Tγθ = dom T θ . Denote by κn (γ, θ) the nth eigenvalue of Tγθ counted with multiplicity. As in Section 5, consider the bands   (κn (γ, π/L), κn+1 (γ, π/L)) , for n odd so that κn (γ, π/L) 6= κn+1 (γ, π/L), (κn (γ, 0), κn+1 (γ, 0)) , for n even so that κn (γ, 0) 6= κn+1 (γ, 0), Gn (γ) :=  ∅, otherwise. and note that |Gn (γ)| = δn (γ), ∀n ∈ N, if we consider µ = γ 2 and W (s) = V (s). We have Corollary 4. For each n3 ∈ N, there exist γ > 0 small enough and εn3 +1 > 0 so that, for all ε ∈ (0, εn3 +1 ), min En3 +1 (γ, ε, θ) − max En3 (γ, ε, θ) = |Gn3 (γ)| + O(ε), θ∈C θ∈C holds for each n = 1, 2, · · · , n3 , where | · | is the Lebesgue measure. 14 (26) n=+∞ Proof of Theorem 4: Recall that we have denoted by {νn }n=−∞ the Fourier coefficients of V (s). Since V (s) is not constant, there exists n2 ∈ N so that νn2 6= 0. By Theorem 7, 2 |Gn2 (γ)| = √ γ 2 |νn2 | + O(γ 4 ), γ → 0. L On the other hand, by Corollary 4, there exists εn2 +1 > 0 so that, for all ε ∈ (0, εn2 +1 ), (26) holds true. Then, by taking Cγ,n2 := |Gn2 (γ)| > 0, theorem is proven. 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