Dirac operators with exponentially decaying entropy

TL;DR

This paper demonstrates that for one-dimensional Dirac operators with exponentially decaying entropy, the Weyl function extends meromorphically into a specific complex strip, and in cases of rapid decay, it is meromorphic everywhere, with poles uniquely identifying the operator.

Contribution

The authors establish the meromorphic extension of the Weyl function for Dirac operators with exponential entropy decay and show that rapid decay leads to a globally meromorphic Weyl function, uniquely determining the operator.

Findings

Weyl function extends meromorphically into a horizontal strip for exponentially decaying entropy.

Rapid entropy decay implies the Weyl function is meromorphic on the entire complex plane.

Poles of the Weyl function uniquely determine the Dirac operator in the case of rapid decay.

Abstract

We prove that the Weyl function of the one-dimensional Dirac operator on the half-line $\mathbb{R}_+$ with exponentially decaying entropy extends meromorphically into the horizontal strip $\{0\ge \mbox{Im}\,z > -\delta\}$ for some $\delta > 0$ depending on the rate of decay. If the entropy decreases very rapidly then the corresponding Weyl function turns out to be meromorphic in the whole complex plane. In this situation we show that poles of the Weyl function (scattering resonances) uniquely determine the operator.