Gelfand-Levitan condition for Dirac operators

TL;DR

This paper extends the theory of Dirac operators to include solutions of bounded variation, develops their spectral theory, and explores connections with canonical systems and de Branges spaces, providing a framework for operator reconstruction.

Contribution

It introduces a generalized Dirac operator with bounded variation solutions and links it to spectral theory, canonical systems, and the Gelfand-Levitan condition for operator recovery.

Findings

Spectral theory for generalized Dirac operators developed.

Connection established between generalized Dirac operators and canonical systems.

Gelfand-Levitan condition used for operator reconstruction.

Abstract

We discuss how to generalize a Dirac operator such that the solution of a Dirac equation is of bounded variation rather than continuous. We build the spectral theory for generalized Dirac operators and discuss the connection between them and canonical systems. With the help of de Branges' theory, we discuss the de Branges space of such an operator and the norm endowed. On the other hand, the Paley-Wiener theorem gives us a chance to recover a Dirac operator from a function that plays the same role as the spectral measure, which is well-known as the Gelfand-Levitan condition.