Wave model of the Sturm-Liouville operator on the half-line

TL;DR

This paper describes the wave spectrum of a Sturm-Liouville operator on the half-line, constructing a functional model that aids in solving inverse problems by reconstructing the original operator from wave data.

Contribution

It provides a detailed description of the wave spectrum for a specific Sturm-Liouville operator and constructs a wave model that facilitates inverse problem solutions.

Findings

Wave spectrum characterized for the operator $L_0$.

Constructed a wave model identical to the original operator.

Model enables reconstruction of the operator from data.

Abstract

The notion of the wave spectrum of a semi-bounded symmetric operator was introduced by one of the authors in 2013. The wave spectrum is a topological space determined by the operator in a canonical way. The definition uses a dynamical system associated with the operator: the wave spectrum is constructed from its reachable sets. In the paper we give a description of the wave spectrum of the operator $L_0=-\frac{d^2}{dx^2}+q$ which acts in the space $L_2(0,\infty)$ and has defect indices $(1,1)$. We construct a functional (wave) model of the operator $L_0^*$ in which the elements of the original $L_2(0,\infty)$ are realized as functions on the wave spectrum. It turns out to be identical to the original $L_0^*$. The latter is fundamental in solving inverse problems: the wave model is determined by their data, which allows for reconstruction of the original.