On recovering quadratic pencils with singular coefficients and entire functions in the boundary conditions

TL;DR

This paper addresses the inverse spectral problem for quadratic Sturm--Liouville pencils with singular coefficients, demonstrating that a complete subspectrum can uniquely recover the pencil and providing an algorithm for the solution.

Contribution

It introduces new conditions under which a subspectrum suffices for unique recovery and develops an algorithm for solving the inverse problem with singular coefficients.

Findings

A subspectrum generating a complete system suffices for recovery.

An explicit algorithm for the inverse problem is constructed.

Results extend to partial inverse problems.

Abstract

In the paper, we study an inverse spectral problem for quadratic pencils of the Sturm--Liouville operators with singular coefficients and entire functions in the boundary conditions. We prove that a subspectrum is sufficient for recovering the pencil if this subspectrum generates a complete functional system. As well, we obtain an algorithm solving the inverse problem and alternative conditions on the subspectrum. Finally, these results are applied to studying a partial inverse problem.