A conditionally exactly solvable generalization of the inverse square root potential

A. M. Ishkhanyan

Research output: Contribution to journalArticlepeer-review

14 Citations (Scopus)


We present a conditionally exactly solvable singular potential for the one-dimensional Schrödinger equation which involves the exactly solvable inverse square root potential. Each of the two fundamental solutions that compose the general solution of the problem is given by a linear combination with non-constant coefficients of two confluent hypergeometric functions. Discussing the bound-state wave functions vanishing both at infinity and in the origin, we derive the exact equation for the energy spectrum which is written using two Hermite functions of non-integer order. In specific auxiliary variables this equation becomes a mathematical equation that does not refer to a specific physical context discussed. In the two-dimensional space of these auxiliary variables the roots of this equation draw a countable infinite set of open curves with hyperbolic asymptotes. We present an analytic description of these curves by a transcendental algebraic equation for the involved variables. The intersections of the curves thus constructed with a certain cubic curve provide a highly accurate description of the energy spectrum.

Original languageEnglish
Pages (from-to)3786-3790
Number of pages5
JournalPhysics Letters, Section A: General, Atomic and Solid State Physics
Issue number45
Publication statusPublished - 25 Nov 2016


  • Bi-confluent Heun potentials
  • Conditionally exactly solvable potentials
  • Hermite function
  • Inverse square root potential
  • Stationary Schrödinger equation

ASJC Scopus subject areas

  • Physics and Astronomy(all)

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