A singular one-dimensional bound state problem and its degeneracies


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Erman F., Gadella M., Tunalı S., Uncu H.

European Physical Journal Plus, vol.132, no.8, 2017 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 132 Issue: 8
  • Publication Date: 2017
  • Doi Number: 10.1140/epjp/i2017-11613-7
  • Journal Name: European Physical Journal Plus
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Istanbul Medipol University Affiliated: No

Abstract

We give a brief exposition of the formulation of the bound state problem for the one-dimensional system of N attractive Dirac delta potentials, as an N× N matrix eigenvalue problem (ΦA= ωA). The main aim of this paper is to illustrate that the non-degeneracy theorem in one dimension breaks down for the equidistantly distributed Dirac delta potential, where the matrix Φ becomes a special form of the circulant matrix. We then give elementary proof that the ground state is always non-degenerate and the associated wave function may be chosen to be positive by using the Perron-Frobenius theorem. We also prove that removing a single center from the system of N delta centers shifts all the bound state energy levels upward as a simple consequence of the Cauchy interlacing theorem.