Solution of the inverse spectral problem for a convolution integro-differential operator with Robin boundary conditions

TL;DR

This paper addresses the inverse spectral problem for a specific convolution integro-differential operator with Robin boundary conditions, establishing uniqueness, spectral asymptotics, and a constructive solution method.

Contribution

It introduces a novel inverse problem framework for a convolution integro-differential operator with Robin boundary conditions, providing uniqueness and a constructive solution approach.

Findings

Proved the uniqueness of the inverse spectral problem.

Established that standard asymptotics characterize the spectrum.

Developed a constructive procedure for solving the inverse problem.

Abstract

The operator of double differentiation on a finite interval with Robin boundary conditions perturbed by the composition of a Volterra convolution operator and the differentiation one is considered. We study the inverse problem of recovering the convolution kernel along with a coefficient of the boundary conditions from the spectrum. We prove the uniqueness theorem and that the standard asymptotics is a necessary and sufficient condition for an arbitrary sequence of complex numbers to be the spectrum of such an operator. A constructive procedure for solving the inverse problem is given.