Inverse uniqueness results for one-dimensional weighted Dirac operators

TL;DR

This paper proves that the spectral measure uniquely determines a one-dimensional weighted Dirac operator up to gauge transformation, extending classical results and improving known theorems, with applications to radial Dirac operators and systems with limit circle endpoints.

Contribution

It establishes a new inverse uniqueness result for weighted Dirac operators using de Branges space theory, broadening the scope of spectral determination.

Findings

Spectral measure uniquely determines the Dirac operator up to gauge transformation.

Extension of classical results to radial Dirac operators.

Improvement of known results for canonical Hamiltonian systems.

Abstract

Given a one-dimensional weighted Dirac operator we can define a spectral measure by virtue of singular Weyl-Titchmarsh-Kodaira theory. Using the theory of de Branges spaces we show that the spectral measure uniquely determines the Dirac operator up to a gauge transformation. Our result applies in particular to radial Dirac operators and extends the classical results for Dirac operators with one regular endpoint. Moreover, our result also improves the currently known results for canonical (Hamiltonian) systems. If one endpoint is limit circle case, we also establish corresponding two-spectra results.