Consider a one-dimensional Schroedinger operator which is a short range perturbation of a finite-... more Consider a one-dimensional Schroedinger operator which is a short range perturbation of a finite-gap operator. We give necessary and sufficient conditions on the left, right reflection coefficient such that the difference of the potentials has finite support to the left, right, respectively. Moreover, we apply these results to show a unique continuation type result for solutions of the Korteweg-de Vries
We develop direct and inverse scattering theory for Jacobi operators with steplike coefficients w... more We develop direct and inverse scattering theory for Jacobi operators with steplike coefficients which are asymptotically close to different finite-gap quasi-periodic coefficients on different sides. We give a complete characterization of the scattering data, which allow unique solvability of the inverse scattering problem in the class of perturbations with finite first moment.
We solve the Cauchy problem for the modified Korteweg--de Vries equation with steplike quasi-peri... more We solve the Cauchy problem for the modified Korteweg--de Vries equation with steplike quasi-periodic, finite-gap initial conditions under the assumption that the perturbations have a given number of derivatives and moments finite.
Consider a one-dimensional Schrödinger operator which is a shortrange perturbation of a quasi-per... more Consider a one-dimensional Schrödinger operator which is a shortrange perturbation of a quasi-periodic, finite-gap operator. We give necessary and sufficient conditions on the left, right reflection coefficient such that the difference of the potentials has finite support to the left, right, respectively. Moreover, we apply these results to show a unique continuation type result for solutions of the Korteweg-de Vries equation in this context. By virtue of the Miura transform an analogous result for the modified Korteweg-de Vries equation is also obtained.
We show that for a one-dimensional Schr\"odinger operator with a potential whose first momen... more We show that for a one-dimensional Schr\"odinger operator with a potential whose first moment is integrable the scattering matrix is in the unital Wiener algebra of functions with integrable Fourier transforms. Then we use this to derive dispersion estimates for solutions of the associated Schr\"odinger and Klein-Gordon equations. In particular, we remove the additional decay conditions in the case where a resonance is present at the edge of the continuous spectrum.
We consider scattering theory for one-dimensional Jacobi operators with respect to steplike quasi... more We consider scattering theory for one-dimensional Jacobi operators with respect to steplike quasi-periodic finite-gap backgrounds and show how the transmission coefficient can be reconstructed from minimal scattering data. This generalizes the Poisson-Jensen formula for the classical constant background case.
We derive dispersion estimates for solutions of the one-dimensional discrete perturbed Schr\"... more We derive dispersion estimates for solutions of the one-dimensional discrete perturbed Schr\"odinger and wave equations. In particular, we improve upon previous works and weaken the conditions on the potentials.
ABSTRACT We find the high energy asymptotics for the singular Weyl-Titchmarsh m-functions and the... more ABSTRACT We find the high energy asymptotics for the singular Weyl-Titchmarsh m-functions and the associated spectral measures of perturbed spherical Schroedinger operators (also known as Bessel operators). We apply this result to establish an improved local Borg-Marchenko theorem for Bessel operators as well as uniqueness theorems for the radial quantum scattering problem with nontrivial angular momentum.
ABSTRACT Given a one-dimensional weighted Dirac operator we can define a spectral measure by virt... more ABSTRACT Given a one-dimensional weighted Dirac operator we can define a spectral measure by virtue of singular Weyl-Titchmarsh-Kodaira theory. Using the theory of de Branges spaces we show that the spectral measure uniquely determines the Dirac operator up to a gauge transformation. Our result applies in particular to radial Dirac operators and extends the classical results for Dirac operators with one regular endpoint. Moreover, our result also improves the currently known results for canonical (Hamiltonian) systems. If one endpoint is limit circle case, we also establish corresponding two-spectra results.
ABSTRACT We introduce a novel approach for defining a δ′-interaction on a subset of the real line... more ABSTRACT We introduce a novel approach for defining a δ′-interaction on a subset of the real line of Lebesgue measure zero which is based on Sturm–Liouville differential expression with measure coefficients. This enables us to establish basic spectral properties (e.g., self-adjointness, lower semiboundedness and spectral asymptotics) of Hamiltonians with δ′-interactions concentrated on sets of complicated structures.
Methods of Spectral Analysis in Mathematical Physics, 2009
We provide a detailed recursive construction of the Ablowitz-Ladik (AL) hierarchy and its zero-cu... more We provide a detailed recursive construction of the Ablowitz-Ladik (AL) hierarchy and its zero-curvature formalism. The two-coefficient AL hierarchy under investigation can be considered a complexified version of the discrete nonlinear Schrödinger equation and its hierarchy of nonlinear evolution equations.
We investigate trace formulas for Jacobi operators which are trace class perturbations of quasi-p... more We investigate trace formulas for Jacobi operators which are trace class perturbations of quasi-periodic finite-gap operators using Krein's spectral shift theory. In particular we establish the conserved quantities for the solutions of the Toda hierarchy in this class.
... Organizers Anton Arnold, Josef Hofbauer, Christian Schmeiser, Alois Steindl, Peter Szmolyan, ... more ... Organizers Anton Arnold, Josef Hofbauer, Christian Schmeiser, Alois Steindl, Peter Szmolyan, Gerald Teschl, Josef Teichmann. ... Algebro-geometric finite-band solutions of the Ablowitz-Ladik hierarchy by Johanna Michor Imperial College London Coauthors: Fritz Gesztesy ...
We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy with comp... more We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy with complex-valued initial data and prove unique solvability globally in time for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Ablowitz-Ladik hierarchy, which is of independent interest as it solves the inverse algebro-geometric spectral problem for general (non-unitary) Ablowitz-Ladik Lax operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin-type equations), this yields the construction of global algebro-geometric solutions of the time-dependent Ablowitz-Ladik hierarchy.
ABSTRACT We derive the long-time asymptotics for the Toda shock problem using the nonlinear steep... more ABSTRACT We derive the long-time asymptotics for the Toda shock problem using the nonlinear steepest descent analysis for oscillatory Riemann--Hilbert factorization problems. We show that the half plane of space/time variables splits into five main regions: The two regions far outside where the solution is close to free backgrounds. The middle region, where the solution can be asymptotically described by a two band solution, and two regions separating them, where the solution is asymptotically given by a slowly modulated two band solution. In particular, the form of this solution in the separating regions verifies a conjecture from Venakides, Deift, and Oba from 1991.
Consider a one-dimensional Schroedinger operator which is a short range perturbation of a finite-... more Consider a one-dimensional Schroedinger operator which is a short range perturbation of a finite-gap operator. We give necessary and sufficient conditions on the left, right reflection coefficient such that the difference of the potentials has finite support to the left, right, respectively. Moreover, we apply these results to show a unique continuation type result for solutions of the Korteweg-de Vries
We develop direct and inverse scattering theory for Jacobi operators with steplike coefficients w... more We develop direct and inverse scattering theory for Jacobi operators with steplike coefficients which are asymptotically close to different finite-gap quasi-periodic coefficients on different sides. We give a complete characterization of the scattering data, which allow unique solvability of the inverse scattering problem in the class of perturbations with finite first moment.
We solve the Cauchy problem for the modified Korteweg--de Vries equation with steplike quasi-peri... more We solve the Cauchy problem for the modified Korteweg--de Vries equation with steplike quasi-periodic, finite-gap initial conditions under the assumption that the perturbations have a given number of derivatives and moments finite.
Consider a one-dimensional Schrödinger operator which is a shortrange perturbation of a quasi-per... more Consider a one-dimensional Schrödinger operator which is a shortrange perturbation of a quasi-periodic, finite-gap operator. We give necessary and sufficient conditions on the left, right reflection coefficient such that the difference of the potentials has finite support to the left, right, respectively. Moreover, we apply these results to show a unique continuation type result for solutions of the Korteweg-de Vries equation in this context. By virtue of the Miura transform an analogous result for the modified Korteweg-de Vries equation is also obtained.
We show that for a one-dimensional Schr\"odinger operator with a potential whose first momen... more We show that for a one-dimensional Schr\"odinger operator with a potential whose first moment is integrable the scattering matrix is in the unital Wiener algebra of functions with integrable Fourier transforms. Then we use this to derive dispersion estimates for solutions of the associated Schr\"odinger and Klein-Gordon equations. In particular, we remove the additional decay conditions in the case where a resonance is present at the edge of the continuous spectrum.
We consider scattering theory for one-dimensional Jacobi operators with respect to steplike quasi... more We consider scattering theory for one-dimensional Jacobi operators with respect to steplike quasi-periodic finite-gap backgrounds and show how the transmission coefficient can be reconstructed from minimal scattering data. This generalizes the Poisson-Jensen formula for the classical constant background case.
We derive dispersion estimates for solutions of the one-dimensional discrete perturbed Schr\"... more We derive dispersion estimates for solutions of the one-dimensional discrete perturbed Schr\"odinger and wave equations. In particular, we improve upon previous works and weaken the conditions on the potentials.
ABSTRACT We find the high energy asymptotics for the singular Weyl-Titchmarsh m-functions and the... more ABSTRACT We find the high energy asymptotics for the singular Weyl-Titchmarsh m-functions and the associated spectral measures of perturbed spherical Schroedinger operators (also known as Bessel operators). We apply this result to establish an improved local Borg-Marchenko theorem for Bessel operators as well as uniqueness theorems for the radial quantum scattering problem with nontrivial angular momentum.
ABSTRACT Given a one-dimensional weighted Dirac operator we can define a spectral measure by virt... more ABSTRACT Given a one-dimensional weighted Dirac operator we can define a spectral measure by virtue of singular Weyl-Titchmarsh-Kodaira theory. Using the theory of de Branges spaces we show that the spectral measure uniquely determines the Dirac operator up to a gauge transformation. Our result applies in particular to radial Dirac operators and extends the classical results for Dirac operators with one regular endpoint. Moreover, our result also improves the currently known results for canonical (Hamiltonian) systems. If one endpoint is limit circle case, we also establish corresponding two-spectra results.
ABSTRACT We introduce a novel approach for defining a δ′-interaction on a subset of the real line... more ABSTRACT We introduce a novel approach for defining a δ′-interaction on a subset of the real line of Lebesgue measure zero which is based on Sturm–Liouville differential expression with measure coefficients. This enables us to establish basic spectral properties (e.g., self-adjointness, lower semiboundedness and spectral asymptotics) of Hamiltonians with δ′-interactions concentrated on sets of complicated structures.
Methods of Spectral Analysis in Mathematical Physics, 2009
We provide a detailed recursive construction of the Ablowitz-Ladik (AL) hierarchy and its zero-cu... more We provide a detailed recursive construction of the Ablowitz-Ladik (AL) hierarchy and its zero-curvature formalism. The two-coefficient AL hierarchy under investigation can be considered a complexified version of the discrete nonlinear Schrödinger equation and its hierarchy of nonlinear evolution equations.
We investigate trace formulas for Jacobi operators which are trace class perturbations of quasi-p... more We investigate trace formulas for Jacobi operators which are trace class perturbations of quasi-periodic finite-gap operators using Krein's spectral shift theory. In particular we establish the conserved quantities for the solutions of the Toda hierarchy in this class.
... Organizers Anton Arnold, Josef Hofbauer, Christian Schmeiser, Alois Steindl, Peter Szmolyan, ... more ... Organizers Anton Arnold, Josef Hofbauer, Christian Schmeiser, Alois Steindl, Peter Szmolyan, Gerald Teschl, Josef Teichmann. ... Algebro-geometric finite-band solutions of the Ablowitz-Ladik hierarchy by Johanna Michor Imperial College London Coauthors: Fritz Gesztesy ...
We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy with comp... more We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy with complex-valued initial data and prove unique solvability globally in time for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Ablowitz-Ladik hierarchy, which is of independent interest as it solves the inverse algebro-geometric spectral problem for general (non-unitary) Ablowitz-Ladik Lax operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin-type equations), this yields the construction of global algebro-geometric solutions of the time-dependent Ablowitz-Ladik hierarchy.
ABSTRACT We derive the long-time asymptotics for the Toda shock problem using the nonlinear steep... more ABSTRACT We derive the long-time asymptotics for the Toda shock problem using the nonlinear steepest descent analysis for oscillatory Riemann--Hilbert factorization problems. We show that the half plane of space/time variables splits into five main regions: The two regions far outside where the solution is close to free backgrounds. The middle region, where the solution can be asymptotically described by a two band solution, and two regions separating them, where the solution is asymptotically given by a slowly modulated two band solution. In particular, the form of this solution in the separating regions verifies a conjecture from Venakides, Deift, and Oba from 1991.
Uploads
Papers by Gerald Teschl