A singular one-dimensional bound state problem and its degeneracies

Creative Commons License

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

European Physical Journal Plus, cilt.132, sa.8, 2017 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 132 Sayı: 8
  • Basım Tarihi: 2017
  • Doi Numarası: 10.1140/epjp/i2017-11613-7
  • Dergi Adı: European Physical Journal Plus
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • İstanbul Medipol Üniversitesi Adresli: Hayır

Özet

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.