Inverse spectral problems for functional-differential operators with involution

TL;DR

This paper develops a method to solve inverse spectral problems for second-order functional-differential operators with involution, reducing the problem to matrix form and establishing unique determination by five spectra.

Contribution

It introduces a novel approach using spectral mappings to solve inverse problems for FDO with involution, including handling the associated matrix Sturm-Liouville operator.

Findings

The inverse spectral problem is uniquely solvable with five spectra.

Reduction to matrix form simplifies the analysis of FDO with involution.

The method addresses qualitative difficulties caused by the weight in the Sturm-Liouville operator.

Abstract

The main goal of this paper is to propose an approach to inverse spectral problems for functional-differential operators (FDO) with involution. For definiteness, we focus on the second-order FDO with involution-reflection. Our approach is based on the reduction of the problem to the matrix form and on the solution of the inverse problem for the matrix Sturm-Liouville operator by developing the method of spectral mappings. The obtained matrix Sturm-Liouville operator contains the weight, which causes qualitative difficulties in the study of the inverse problem. As a result, we show that the considered FDO with involution is uniquely specified by five spectra of certain regular boundary value problems.