Inverse square singularities and eigenparameter dependent boundary conditions are two sides of the same coin

TL;DR

This paper unifies the treatment of inverse square singularities and eigenparameter dependent boundary conditions in Schrödinger operators, enabling transfer of spectral results between these two frameworks through Darboux-type transformations.

Contribution

It introduces a unified approach to handle inverse square singularities and boundary conditions with rational Herglotz--Nevanlinna functions, establishing transformations between these operator classes.

Findings

Unified treatment of inverse square singularities and boundary conditions.

Darboux-type transformations connecting different operator classes.

Transfer of spectral results between boundary value problems.

Abstract

We show that inverse square singularities can be treated as boundary conditions containing rational Herglotz--Nevanlinna functions of the eigenvalue parameter with "a negative number of poles". More precisely, we treat in a unified manner one-dimensional Schr\"{o}dinger operators with either an inverse square singularity or a boundary condition containing a rational Herglotz--Nevanlinna function of the eigenvalue parameter at each endpoint, and define Darboux-type transformations between such operators. These transformations allow one, in particular, to transfer almost any spectral result from boundary value problems with eigenparameter dependent boundary conditions to those with inverse square singularities, and vice versa.