On the Inverse Resonance Problem for Schrodinger Operators

TL;DR

This paper proves that large resonances have diminishing importance in reconstructing complex-valued potentials for Schrödinger operators, establishing conditional stability and uniqueness in the inverse resonance problem.

Contribution

It demonstrates the conditional stability of potential reconstruction from finite resonance data and proves uniqueness for complex-valued potentials.

Findings

Large resonances are less significant for potential reconstruction.

Conditional stability holds for finite resonance data.

Uniqueness is established for complex-valued potentials.

Abstract

We consider Schr\"odinger operators on [0,\infty) with compactly supported, possibly complex-valued potentials in L^1([0,\infty)). It is known (at least in the case of a real-valued potential) that the location of eigenvalues and resonances determines the potential uniquely. From the physical point of view one expects that large resonances are increasingly insignificant for the reconstruction of the potential from the data. In this paper we prove the validity of this statement, i.e., we show conditional stability for finite data. As a by-product we also obtain a uniqueness result for the inverse resonance problem for complex-valued potentials.