Dirac systems with locally square-integrable potentials: direct and inverse problems for the spectral functions

TL;DR

This paper addresses the inverse and direct spectral problems for Dirac systems with locally square-integrable potentials, providing conditions for spectral functions and exploring connections with Paley-Wiener measures.

Contribution

It introduces new criteria for spectral functions of Dirac systems with locally square-integrable potentials and solves inverse problems on intervals and semiaxes.

Findings

Established necessary and sufficient conditions for spectral functions.

Solved inverse problems for Dirac systems with locally square-integrable potentials.

Revealed connections with Paley-Wiener sampling measures in scalar cases.

Abstract

We solve the inverse problems to recover Dirac systems on an interval or semiaxis from their spectral functions (matrix valued functions) for the case of locally square-integrable potentials. Direct problems in terms of spectral functions are treated as well. Moreover, we present necessary and sufficient conditions on the given distribution matrix valued function to be a spectral function of some Dirac system with a locally square-integrable potential. Interesting connections with Paley-Wiener sampling measures appear in the case of scalar spectral functions.