Gelfand–Levitan condition for Dirac operators

Abstract

We discuss how to generalise a Dirac operator such that the solution of a Dirac equation is of bounded variation rather than continuous. We build the spectral theory for generalised 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 makes it possible to recover a Dirac operator from a function that plays the same role as the spectral measure, which is known as the Gelfand–Levitan condition.