Gradual spontaneous breakdown of PI symmetry in a solvable potential

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A new solvable potential is introduced by transforming the Schrödinger equation into the differential equation of the Jacobi polynomials. The z(x) variable transformation function depends on two parameters (C and δ) and can be obtained implicitly as x(z). The remaining two parameters set the strength of the even and odd potential components. Systematic analysis is given for C < 0, δ 0, when the potential contains the Scarf II and Rosen-Morse I potentials as special cases. The normalized bound-state solutions are obtained in closed form both in the real and -symmetric cases. The energy eigenvalues are determined from the roots of a quartic algebraic equation. Despite this implicitly defined spectrum, the spectral structure of the potential can be determined easily. In the -symmetric case, the spontaneous breakdown of symmetry can also be studied in a controllable way, and it is found that with increasing non-Hermiticity, the complexification of the energy eigenvalues is a gradual process that starts with the ground state. The general form of the potential contains the Scarf I, Rosen-Morse II, generalized Pöschl-Teller, Eckart, symmetric Ginocchio and Dutt-Khare-Varshni (DKV) potentials too, and can be identified as a subclass of the general six-parameter Natanzon class. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to Quantum physics with non-Hermitian operators.

Original languageEnglish
Article number444020
JournalJournal of Physics A: Mathematical and Theoretical
Issue number44
Publication statusPublished - Nov 9 2012

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Modelling and Simulation
  • Mathematical Physics
  • Physics and Astronomy(all)

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