Inverse scattering problem for Sturm-Liouville operator on non-compact A-graph. Uniqueness result

TL;DR

This paper investigates the inverse scattering problem for Sturm-Liouville operators on noncompact A-graphs, establishing spectral properties and proving a uniqueness theorem for operator recovery from scattering data.

Contribution

It provides the first uniqueness result for inverse scattering problems on noncompact A-graphs, extending spectral analysis to this class of noncompact structures.

Findings

Spectral properties of Sturm-Liouville operators on noncompact A-graphs are characterized.

A uniqueness theorem for the inverse scattering problem is established.

The results extend inverse spectral theory to noncompact graph structures.

Abstract

In a finite-dimensional Euclidian space we consider a connected metric graph with the following property: each two cycles can have at most one common point. Such graphs are called A-graphs. On noncompact A-graph we consider a scattering problem for Sturm-Liouville differential operator with standard matching conditions in the internal vertices. Transport, spectral and scattering problems for differential operators on graphs appear frequently in mathematics, natural sciences and engineering. In particular, direct and inverse problems for such operators are used to construct and study models in mechanics, nano-electronics, quantum computing and waveguides. The most complete results on (both direct and inverse) spectral problems were achieved in the case of Sturm-Liouville operators on compact graphs, in the noncompact case there are no similar general results. In this paper, we establish some properties of the spectral characteristics and investigate the inverse problem of recovering the operator from the scattering data. A uniqueness theorem for such inverse problem is proved.