Resonances and inverse problems for energy-dependent potentials on the half-line

TL;DR

This paper investigates the spectral properties of energy-dependent Schr"odinger equations on the half-line, providing eigenvalue and resonance estimates, and solves inverse resonance problems for specific classes of potentials using a Schr"odinger-Dirac correspondence.

Contribution

It offers new estimates for eigenvalues and resonances and solves the inverse resonance problem for energy-dependent potentials using a novel Schr"odinger-Dirac relation.

Findings

Eigenvalue and resonance estimates for complex potentials

Solution of inverse resonance problem for Miura potentials

Reduction of scattering problems to Dirichlet boundary conditions

Abstract

We consider Schr\"{o}dinger equations with linearly energy-depending potentials which are compactly supported on the half-line. We first provide estimates of the number of eigenvalues and resonances for such complex-valued potentials under suitable regularity assumptions. Then, we consider a specific class of energy-dependent Schr\"{o}dinger equations without eigenvalues, defined with Miura potentials and boundary conditions at the origin. We solve the inverse resonance problem in this case and describe sets of iso-resonance potentials and boundary condition parameters. Our strategy consists in exploiting a correspondance between Schr\"{o}dinger and Dirac equations on the half-line. As a byproduct, we describe similar sets for Dirac operators and show that the scattering problem for Schr\"{o}dinger equation or Dirac operator with an arbitrary boundary condition can be reduced to the scattering problem with the Dirichlet boundary condition.