Semiseparable integral operators and explicit solution of an inverse problem for the skew-self-adjoint Dirac-type system

TL;DR

This paper develops explicit formulas for inverting semiseparable integral operators to solve an inverse problem for skew-self-adjoint Dirac-type systems, providing conditions for unique recovery from the Weyl matrix function.

Contribution

It introduces a method to explicitly invert semiseparable operators and solves the inverse problem for a specific class of Weyl functions in Dirac-type systems.

Findings

Explicit inversion formulas for semiseparable integral operators.

Conditions ensuring unique solution of the inverse problem.

Detailed solution for Weyl functions of a specific rational form.

Abstract

Inverse problem to recover the skew-self-adjoint Dirac-type system from the generalized Weyl matrix function is treated in the paper. Sufficient conditions under which the unique solution of the inverse problem exists, are formulated in terms of the Weyl function and a procedure to solve the inverse problem is given. The case of the generalized Weyl functions of the form $\phi(\lambda)\exp\{-2i\lambda D\}$, where $\phi$ is a strictly proper rational matrix function and $D=D^* \geq 0$ is a diagonal matrix, is treated in greater detail. Explicit formulas for the inversion of the corresponding semiseparable integral operators and recovery of the Dirac-type system are obtained for this case.