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

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.