Spectral identities for Schr\"{o}dinger operators

TL;DR

This paper derives spectral identities linking boundary coefficients and spectral data for 1D Schrödinger operators with specific boundary conditions, offering a simplified analogue of the Gelfand-Levitan integral equation.

Contribution

It introduces a new set of identities connecting boundary coefficients and spectral data for Schrödinger operators with rational Herglotz--Nevanlinna boundary conditions, simplifying the inverse spectral problem.

Findings

Derived identities relate boundary coefficients to spectral data.

Identities serve as a simplified version of the Gelfand-Levitan equation.

Provides tools for inverse spectral analysis of Schrödinger operators.

Abstract

We obtain a system of identities relating boundary coefficients and spectral data for the one-dimensional Schr\"{o}dinger equation with boundary conditions containing rational Herglotz--Nevanlinna functions of the eigenvalue parameter. These identities can be thought of as a kind of mini version of the Gelfand--Levitan integral equation for boundary coefficients only.