A conditionally exactly solvable generalization of the inverse square root potential

Abstract

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.

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Cited by in corpus (10)

• Solutions of the bi-confluent Heun equation in terms of the Hermite functions
• Schrödinger potentials solvable in terms of the general Heun functions
• Discretization of Natanzon potentials
• Exact solutions of the sextic oscillator from the bi-confluent Heun equation
• Hermite function solutions of the Schrödinger equation for the sextic oscillator
• A conditionally integrable bi-confluent Heun potential involving inverse square root and centrifugal barrier terms
• New conditionally exactly solvable potentials of exponential type
• A conditionally exactly solvable generalization of the potential step
• A conditionally integrable Schrödinger potential of a bi-confluent Heun class
• The third five-parametric hypergeometric quantum-mechanical potential