The boundary control approach to inverse spectral theory
S. A. Avdonin, V. S. Mikhaylov
TL;DR
The paper addresses inverse spectral problems for the Schrödinger operator $H=-\dfrac{d^2}{dx^2}+q(x)$ on $L^2(\mathbb{R}_+)$ with Dirichlet boundary at 0, and unifies classical approaches (Gelfand--Levitan, Krein, Simon, Remling) with the Boundary Control method. It develops the BC framework, connecting dynamic boundary data to spectral information, and provides explicit representations for the kernel $c^T$ and the response $r$ in terms of a regularized spectral measure $d\sigma$, establishing locality and invertibility of the key operators $W^T$ and $C^T$. The authors derive Krein-type equations and a local Gelfand--Levitan system within the BC setting, showing how the potential on $[0,T]$ can be reconstructed from boundary controls. This work demonstrates that the BC method yields simple, physically motivated proofs and a versatile, linear, and locality-preserving route to inverse spectral problems, bridging dynamical and spectral data and enabling local reconstruction.
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.