Inverse scattering theory and trace formulae for one-dimensional Schr\"odinger problems with singular potentials

TL;DR

This paper extends inverse scattering theory to one-dimensional Schrödinger problems with singular potentials near boundaries, deriving trace formulae linking boundary potential values to spectral data, and illustrating their application.

Contribution

It introduces new trace formulae for Schrödinger problems with boundary singularities, connecting boundary potential values to spectral data, which was not previously established.

Findings

Derived trace formulae for singular potentials.

Applied formulae to various Schrödinger problems with singularities.

Extended inverse scattering theory to boundary-singular cases.

Abstract

Inverse scattering theory is extended to one-dimensional Schr\"odinger problems with near-boundary singularities of the form $v(z\to 0)\simeq -z^{-2}/4+v_{-1}z^{-1}$. Trace formulae relating the boundary value $v_0$ of the nonsingular part of the potential to spectral data are derived. Their potential is illustrated by applying them to a number of Schr\"odinger problems with singular potentials.