Solutions of the bi-confluent Heun equation in terms of the Hermite functions

Journal Article · Tue Aug 15 04:00:00 EDT 2017 · Annals of Physics

DOI:https://doi.org/10.1016/J.AOP.2017.04.015· OSTI ID:22701510

Ishkhanyan, T. A. ^[1]

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1. Institute for Physical Research, NAS of Armenia, 0203 Ashtarak (Armenia)

We construct an expansion of the solutions of the bi-confluent Heun equation in terms of the Hermite functions. The series is governed by a three-term recurrence relation between successive coefficients of the expansion. We examine the restrictions that are imposed on the involved parameters in order that the series terminates thus resulting in closed-form finite-sum solutions of the bi-confluent Heun equation. A physical application of the closed-form solutions is discussed. We present the five six-parametric potentials for which the general solution of the one-dimensional Schrödinger equation is written in terms of the bi-confluent Heun functions and further identify a particular conditionally integrable potential for which the involved bi-confluent Heun function admits a four-term finite-sum expansion in terms of the Hermite functions. This is an infinite well defined on a half-axis. We present the explicit solution of the one-dimensional Schrödinger equation for this potential and discuss the bound states supported by the potential. We derive the exact equation for the energy spectrum and construct an accurate approximation for the bound-state energy levels. - Highlights: • The bi-confluent Heun functions are expanded in terms of the Hermite functions. • The restrictions leading to closed-form finite-sum solutions are identified. • The five bi-confluent Heun potentials for the Schrödinger equation are presented. • A new conditionally integrable Schrödinger potential is derived. • An accurate approximation for the bound-state energy levels is constructed.

• OSTI ID: 22701510
• Journal Information: Annals of Physics, Journal Name: Annals of Physics Vol. 383; ISSN 0003-4916; ISSN APNYA6
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
BOUND STATE
ENERGY LEVELS
ENERGY SPECTRA
MATHEMATICAL SOLUTIONS
ONE-DIMENSIONAL CALCULATIONS
POTENTIALS
RECURSION RELATIONS
SCHROEDINGER EQUATION
SERIES EXPANSION