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

Abstract

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.

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Cited by in corpus (21)

• 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
• Expansions of the solutions of the general Heun equation governed by two-term recurrence relations for coefficients
• Hermite function solutions of the Schrödinger equation for the sextic oscillator
• Exact solution of the 1D Dirac equation for the inverse-square-root potential $1/\sqrt{x}$
• A note on the generalized-hypergeometric solutions of general and single-confluent Heun equations
• Constraint polynomial approach -- an alternative to the functional Bethe Ansatz method?
• A conditionally integrable bi-confluent Heun potential involving inverse square root and centrifugal barrier terms
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• On a Solution to the Dirac Equation with a Triangular Potential Well
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• An application of the HeunB function
• A new exactly integrable hypergeometric potential for the Schrödinger equation
• Bi-confluent Heun potentials for a stationary relativistic wave equation for a spinless particle
• Comment on the paper "Third-harmonic generation investigated by a short-range bottomless exponential potential well" by M. Hu, K. Guo, Q. Yu, Z. Zhang [Superlattices and Microstructures, 122 (2018) 538-547]
• Series solutions of confluent Heun equations in terms of incomplete Gamma-functions
• Spectral analysis of Dirac materials in position-dependent magnetic and electric fields via Heun functions
• Series solutions of linear ODEs by Newton-Raphson method on quotient $D$-modules
• Generalized-hypergeometric solutions of the biconfluent Heun equation
• A Schrödinger potential involving $x^{\frac{2}{3}}$ and centrifugal-barrier terms conditionally integrable in terms of the confluent hypergeometric functions