Connection conditions and the spectral family under singular potentials

Izumi Tsutsui, Tamás Fülöp, Taksu Cheon

Research output: Contribution to journalArticle

43 Citations (Scopus)

Abstract

To describe a quantum system whose potential is divergent at one point, one must provide proper connection conditions for the wavefunctions at the singularity. Generalizing the scheme used for point interactions in one dimension, we present a set of connection conditions which are well defined even if the wavefunctions and/or their derivatives are divergent at the singularity. Our generalized scheme covers the entire U(2) family of quantizations (self-adjoint Hamiltonians) admitted for the singular system. We use this scheme to examine the spectra of the Coulomb potential V(x) = -e 2/|x| and the harmonic oscillator with square inverse potential V(x) = (mω2/2)x2 + g/x2, and thereby provide a general perspective for these models which have previously been treated with restrictive connection conditions resulting in conflicting spectra. We further show that, for any parity invariant singular potential V(-x) = V(x), the spectrum is determined solely by the eigenvalues of the characteristic matrix U ∈ U (2).

Original languageEnglish
Pages (from-to)275-287
Number of pages13
JournalJournal of Physics A: Mathematical and General
Volume36
Issue number1
DOIs
Publication statusPublished - Jan 10 2003

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)

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