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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 оригинала | Английский |
|---|---|
| Страницы (с-по) | 407-414 |
| Число страниц | 8 |
| Журнал | Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences |
| Том | 73 |
| Номер выпуска | 5 |
| DOI | |
| Состояние | Опубликовано - 24 мая 2018 |
Отпечаток
ASJC Scopus subject areas
- Mathematical Physics
- Physics and Astronomy(all)
- Physical and Theoretical Chemistry
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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 журнала › Статья
}
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
AN - SCOPUS:85043286167
VL - 73
SP - 407
EP - 414
JO - 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 -