Partial inverse problems for quadratic differential pencils on a graph with a loop

TL;DR

This paper investigates the inverse spectral problem for quadratic Sturm-Liouville operators on a graph with a loop, focusing on recovering unknown coefficients on one edge from spectral data, with proven uniqueness and constructive methods.

Contribution

It introduces new uniqueness theorems and constructive solutions for partial inverse problems on graphs with loops, advancing spectral analysis techniques.

Findings

Proved uniqueness of coefficient recovery on a graph edge from spectral data.

Developed constructive algorithms for solving partial inverse problems.

Extended inverse problem theory to graphs with loops.

Abstract

In this paper, partial inverse problems for the quadratic pencil of Sturm-Liouville operators on a graph with a loop were studied. These problems consist in recovering the pencil coefficients on one edge of the graph (a boundary edge or the loop) from spectral characteristics, while the coefficients on the other edges are known a priori. We obtain uniqueness theorems and constructive solutions for the partial inverse problems.