Inverse scattering theory and trace formulae for one-dimensional Schrödinger problems with singular potentials

Abstract

Inverse scattering theory is extended to one-dimensional Schrödinger 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 Schrödinger problems with singular potentials.

12 pages, no figures, pdf-LaTeX; v3: published version, 24 pages, uses IOP style files, minor changes, introduction extended, references added

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Cited by in corpus (3)

• Fluctuation-induced forces in confined ideal and imperfect Bose gases
• Inverse scattering-theory approach to the exact large-$n$ solutions of $O(n)$ ${\varphi }^4$ models on films and semi-infinite systems bounded by free surfaces
• The three-dimensional $O(n)$ ${\varphi }^4$ model on a strip with free boundary conditions: exact results for a nontrivial dimensional crossover in the limit $n\to \infty$