A Conditionally Integrable Bi-confluent Heun Potential Involving Inverse Square Root and Centrifugal Barrier Terms

Tigran A. Ishkhanyan, Vladimir P. Krainov, Artur M. Ishkhanyan

Research output: Contribution to journalArticle

Abstract

We present a conditionally integrable potential, belonging to the bi-confluent Heun class, for which the Schrödinger equation is solved in terms of the confluent hypergeometric functions. The potential involves an attractive inverse square root term ~x-1/2 with arbitrary strength and a repulsive centrifugal barrier core ~x-2 with the strength fixed to a constant. This is a potential well defined on the half-axis. Each of the fundamental solutions composing the general solution of the Schrödinger equation is written as an irreducible linear combination, with non-constant coefficients, of two confluent hypergeometric functions. We present the explicit solution in terms of the non-integer order Hermite functions of scaled and shifted argument and discuss the bound states supported by the potential. We derive the exact equation for the energy spectrum and approximate that by a highly accurate transcendental equation involving trigonometric functions. Finally, we construct an accurate approximation for the bound-state energy levels.

LanguageEnglish
Pages407-414
Number of pages8
JournalZeitschrift fur Naturforschung - Section A Journal of Physical Sciences
Volume73
Issue number5
DOIs
Publication statusPublished - 24 May 2018

Fingerprint

Confluent Hypergeometric Function
Square root
Bound States
Schrödinger Equation
Hermite Functions
hypergeometric functions
Circular function
Potential Well
Transcendental
Term
Energy Spectrum
Fundamental Solution
Explicit Solution
Energy Levels
General Solution
Linear Combination
Well-defined
trigonometric functions
Arbitrary
Coefficient

Keywords

  • 02.30.Gp special functions
  • 02.30.Ik Integrable systems
  • 03.65.Ge Solutions of wave equations: bound states

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Physical and Theoretical Chemistry

Cite this

A Conditionally Integrable Bi-confluent Heun Potential Involving Inverse Square Root and Centrifugal Barrier Terms. / Ishkhanyan, Tigran A.; Krainov, Vladimir P.; Ishkhanyan, Artur M.

In: Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences, Vol. 73, No. 5, 24.05.2018, p. 407-414.

Research output: Contribution to journalArticle

@article{dd409155c004426d9b7c5e4b7694006e,
title = "A Conditionally Integrable Bi-confluent Heun Potential Involving Inverse Square Root and Centrifugal Barrier Terms",
abstract = "We present a conditionally integrable potential, belonging to the bi-confluent Heun class, for which the Schr{\"o}dinger equation is solved in terms of the confluent hypergeometric functions. The potential involves an attractive inverse square root term ~x-1/2 with arbitrary strength and a repulsive centrifugal barrier core ~x-2 with the strength fixed to a constant. This is a potential well defined on the half-axis. Each of the fundamental solutions composing the general solution of the Schr{\"o}dinger equation is written as an irreducible linear combination, with non-constant coefficients, of two confluent hypergeometric functions. We present the explicit solution in terms of the non-integer order Hermite functions of scaled and shifted argument and discuss the bound states supported by the potential. We derive the exact equation for the energy spectrum and approximate that by a highly accurate transcendental equation involving trigonometric functions. Finally, we construct an accurate approximation for the bound-state energy levels.",
keywords = "02.30.Gp special functions, 02.30.Ik Integrable systems, 03.65.Ge Solutions of wave equations: bound states",
author = "Ishkhanyan, {Tigran A.} and Krainov, {Vladimir P.} and Ishkhanyan, {Artur M.}",
year = "2018",
month = "5",
day = "24",
doi = "10.1515/zna-2017-0314",
language = "English",
volume = "73",
pages = "407--414",
journal = "Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences",
issn = "0932-0784",
publisher = "Verlag der Zeitschrift fur Naturforschung",
number = "5",

}

TY - JOUR

T1 - A Conditionally Integrable Bi-confluent Heun Potential Involving Inverse Square Root and Centrifugal Barrier Terms

AU - Ishkhanyan, Tigran A.

AU - Krainov, Vladimir P.

AU - Ishkhanyan, Artur M.

PY - 2018/5/24

Y1 - 2018/5/24

N2 - We present a conditionally integrable potential, belonging to the bi-confluent Heun class, for which the Schrödinger equation is solved in terms of the confluent hypergeometric functions. The potential involves an attractive inverse square root term ~x-1/2 with arbitrary strength and a repulsive centrifugal barrier core ~x-2 with the strength fixed to a constant. This is a potential well defined on the half-axis. Each of the fundamental solutions composing the general solution of the Schrödinger equation is written as an irreducible linear combination, with non-constant coefficients, of two confluent hypergeometric functions. We present the explicit solution in terms of the non-integer order Hermite functions of scaled and shifted argument and discuss the bound states supported by the potential. We derive the exact equation for the energy spectrum and approximate that by a highly accurate transcendental equation involving trigonometric functions. Finally, we construct an accurate approximation for the bound-state energy levels.

AB - We present a conditionally integrable potential, belonging to the bi-confluent Heun class, for which the Schrödinger equation is solved in terms of the confluent hypergeometric functions. The potential involves an attractive inverse square root term ~x-1/2 with arbitrary strength and a repulsive centrifugal barrier core ~x-2 with the strength fixed to a constant. This is a potential well defined on the half-axis. Each of the fundamental solutions composing the general solution of the Schrödinger equation is written as an irreducible linear combination, with non-constant coefficients, of two confluent hypergeometric functions. We present the explicit solution in terms of the non-integer order Hermite functions of scaled and shifted argument and discuss the bound states supported by the potential. We derive the exact equation for the energy spectrum and approximate that by a highly accurate transcendental equation involving trigonometric functions. Finally, we construct an accurate approximation for the bound-state energy levels.

KW - 02.30.Gp special functions

KW - 02.30.Ik Integrable systems

KW - 03.65.Ge Solutions of wave equations: bound states

UR - http://www.scopus.com/inward/record.url?scp=85043286167&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85043286167&partnerID=8YFLogxK

U2 - 10.1515/zna-2017-0314

DO - 10.1515/zna-2017-0314

M3 - Article

VL - 73

SP - 407

EP - 414

JO - Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences

T2 - Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences

JF - Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences

SN - 0932-0784

IS - 5

ER -