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

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

Результат исследований: Материалы для журналаСтатья

Выдержка

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.

Язык оригиналаАнглийский
Страницы (с-по)407-414
Число страниц8
ЖурналZeitschrift fur Naturforschung - Section A Journal of Physical Sciences
Том73
Номер выпуска5
DOI
СостояниеОпубликовано - 24 мая 2018

Отпечаток

Square root
Confluent Hypergeometric Function
hypergeometric functions
Term
Bound States
trigonometric functions
Hermite Functions
Circular function
Potential Well
Transcendental
Energy Spectrum
Fundamental Solution
Explicit Solution
Energy Levels
General Solution
Electron energy levels
Linear Combination
Well-defined
energy spectra
energy levels

ASJC Scopus subject areas

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

Цитировать

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

В: Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences, Том 73, № 5, 24.05.2018, стр. 407-414.

Результат исследований: Материалы для журналаСтатья

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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.

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