The boundary control approach to inverse spectral theory

TL;DR

This paper explores various inverse spectral problem methods, highlighting the Boundary Control approach's simplicity and physical intuition, and connecting it with classical and modern theories.

Contribution

It demonstrates that the Boundary Control method offers straightforward, physically motivated proofs and links between different inverse spectral theory approaches.

Findings

Boundary Control approach simplifies proofs of inverse spectral results.

Connections established between dynamical and spectral data.

Derived local Gelfand--Levitan equations.

Abstract

We establish connections between different approaches to inverse spectral problems: the classical Gelfand--Levitan theory, the Krein method, the Simon theory, the approach proposed by Remling and the Boundary Control method. We show that the Boundary Control approach provides simple and physically motivated proofs of the central results of other theories. We demonstrate also the connections between the dynamical and spectral data and derive the local version of the classical Gelfand--Levitan equations.