Essentially isospectral transformations and their applications

TL;DR

This paper introduces Darboux-type transformations for Sturm--Liouville problems with rational Herglotz--Nevanlinna boundary conditions, enabling unified derivation of spectral properties, inverse results, and asymptotics.

Contribution

It develops a new class of isospectral transformations for complex boundary conditions, providing a unified approach to spectral analysis and inverse problems in Sturm--Liouville theory.

Findings

Derived asymptotics of eigenvalues and norming constants

Established inverse uniqueness and existence results

Unified treatment of spectral properties for complex boundary conditions

Abstract

We define and study the properties of Darboux-type transformations between Sturm--Liouville problems with boundary conditions containing rational Herglotz--Nevanlinna functions of the eigenvalue parameter (including the Dirichlet boundary conditions). Using these transformations, we obtain various direct and inverse spectral results for these problems in a unified manner, such as asymptotics of eigenvalues and norming constants, oscillation of eigenfunctions, regularized trace formulas, and inverse uniqueness and existence theorems.