Nonlinear Phenomena in Complex Systems, 2019, Vol.22, No.1, pp.84

NONLINEAR PHENOMENA IN COMPLEX SYSTEMS
An Interdisciplinary Journal

2019, Vol.22, No.1, pp.84 - 92

A Schrödinger Potential Involving x^2/3 and Centrifugal-Barrier terms Conditionally Integrable in Terms of the Confluent Hypergeometric Functions
V. A. Manukyan, T. A. Ishkhanyan, and A. M. Ishkhanyan

The solution of the one-dimensional Schrödinger equation for a potential involving an attractive ∼ x^2/3 and a repulsive centrifugal-barrier ∼ x⁻² terms is presented in terms of the non-integer-order Hermite functions. The potential belongs to one of the five biconfluent Heun families. This is a conditionally integrable potential in that the strength of the centrifugal-barrier term is fixed to 91ℏ²/(72m). The general solution of the problem is composed using fundamental solutions each of which presents an irreducible linear combination of two Hermite functions of a scaled and shifted argument. The potential presents an infinitely extended confining well defined on the positive semi-axis and sustains infinitely many bound states.

Key words: integrable Schrödinger potential, energy spectrum, bi-confluent Heun equation

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