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Inverse resonance problem on the line for perturbations of Pöschl-Teller potentials

Valentin Arrigoni

TL;DR

The paper addresses the inverse resonance problem for a one-dimensional Schrödinger operator with a fixed Pöschl-Teller background $V_0(x)=\lambda/\cosh^2(x)$ perturbed by a compactly supported $q$. It develops a transformation-operator framework, showing that perturbed Jost solutions can be represented via kernels $K^\pm$ and that the normalized Jost data $W$ and $S^\pm$ are entire/meromorphic with zeros encoding eigenvalues and resonances; Hadamard factorization and asymptotics then yield unique reconstruction of the perturbation from the zeros (and poles) of a reflection coefficient under various support and spectral data scenarios, including half-bound-state considerations. The work provides precise high-energy resonance asymptotics, revealing two logarithmic branches on which resonances accumulate when $q$ has derivative jumps at the boundary of its support, and it proves absence of large resonances in certain half-planes under additional hypotheses. Overall, the results extend inverse resonance theory to compact perturbations of the Pöschl-Teller potential on the whole line, with explicit asymptotics and constructive uniqueness through the Gelfand-Levitan-Marchenko framework.

Abstract

We study an inverse resonance problem on the line in which we aim at determining a compactly supported and integrable perturbation of a fixed Pöschl-Teller potential. We define the resonances as the poles of the reflection coefficients with a negative imaginary part. Given the zeros and the poles of one of the reflection coefficients, we are able to determine uniquely the perturbation of a Pöschl-Teller potential when its support is to the left or to the right of zero on the line and also in the remaining cases under the addition of other hypothesis or extra spectral data. We also give asymptotics of the resonances and show that they are asymptotically located on two logarithmic branches.

Inverse resonance problem on the line for perturbations of Pöschl-Teller potentials

TL;DR

The paper addresses the inverse resonance problem for a one-dimensional Schrödinger operator with a fixed Pöschl-Teller background perturbed by a compactly supported . It develops a transformation-operator framework, showing that perturbed Jost solutions can be represented via kernels and that the normalized Jost data and are entire/meromorphic with zeros encoding eigenvalues and resonances; Hadamard factorization and asymptotics then yield unique reconstruction of the perturbation from the zeros (and poles) of a reflection coefficient under various support and spectral data scenarios, including half-bound-state considerations. The work provides precise high-energy resonance asymptotics, revealing two logarithmic branches on which resonances accumulate when has derivative jumps at the boundary of its support, and it proves absence of large resonances in certain half-planes under additional hypotheses. Overall, the results extend inverse resonance theory to compact perturbations of the Pöschl-Teller potential on the whole line, with explicit asymptotics and constructive uniqueness through the Gelfand-Levitan-Marchenko framework.

Abstract

We study an inverse resonance problem on the line in which we aim at determining a compactly supported and integrable perturbation of a fixed Pöschl-Teller potential. We define the resonances as the poles of the reflection coefficients with a negative imaginary part. Given the zeros and the poles of one of the reflection coefficients, we are able to determine uniquely the perturbation of a Pöschl-Teller potential when its support is to the left or to the right of zero on the line and also in the remaining cases under the addition of other hypothesis or extra spectral data. We also give asymptotics of the resonances and show that they are asymptotically located on two logarithmic branches.
Paper Structure (10 sections, 34 theorems, 205 equations, 1 figure)

This paper contains 10 sections, 34 theorems, 205 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Pöschl-Teller potentials
  3. Compact perturbations of a Pöschl-Teller potential
  4. Jost solutions and scattering data
  5. An inverse resonance problem
  6. Location of the resonances at infinity for a compactly supported perturbation
  7. Location of resonances in $S_2$
  8. Absence of resonances in $S_1$
  9. Acknowledgements
  10. Appendix: some results on hypergeometric functions

Key Result

Proposition 1

The potential $q$ is an integrable and compactly supported function, with its support satisfying $\mathop{\mathrm{supp}}\nolimits{q} \subset [\alpha,\beta]$. The kernels $K^\pm$ are real-valued, and their supports are included in $\{(x,t) \in \mathbb R^2 ~|~x\leqslant t \leqslant2\beta-x\}$ for $K^+ and Furthermore these functions are continuous and they satisfy the estimates where $\left\lVert

Figures (1)

  • Figure 1: Resonances in the lower half-plane: $\lambda=1/5$ in green, $\lambda = 1/4$ in blue and $\lambda =1$ in red.

Theorems & Definitions (64)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 54 more