Inverse resonance scattering for massless Dirac operators on the real line

TL;DR

This paper investigates inverse problems for massless Dirac operators on the real line, characterizing potentials via reflection coefficient zeros and poles, and analyzing resonance properties and their implications.

Contribution

It provides a unique characterization of potentials using reflection coefficient zeros, describes isoresonance potentials, and estimates resonance distributions.

Findings

Potential is uniquely determined by zeros of reflection coefficients.

Existence of distinct potentials sharing the same resonances.

Asymptotic behavior of resonance counting function is established.

Abstract

We consider massless Dirac operators on the real line with compactly supported potentials. We solve two inverse problems (including characterization): in terms of zeros of reflection coefficient and in terms of poles of reflection coefficients (i.e. resonances). We prove that a potential is uniquely determined by zeros of reflection coefficients and there exist distinct potentials with the same resonances. We describe the set of "isoresonance potentials". Moreover, we prove the following: 1) a zero of the reflection coefficient can be arbitrarily shifted, such that we obtain the sequence of zeros of the reflection coefficient for an other compactly supported potential, 2) the forbidden domain for resonances is estimated, 3) asymptotics of resonances counting function is determined, 4) these results are applied to canonical systems.