Abstract
We present a conditionally exactly solvable singular potential for the one-dimensional Schrödinger equation which involves the exactly solvable inverse square root potential. Each of the two fundamental solutions that compose the general solution of the problem is given by a linear combination with non-constant coefficients of two confluent hypergeometric functions. Discussing the bound-state wave functions vanishing both at infinity and in the origin, we derive the exact equation for the energy spectrum which is written using two Hermite functions of non-integer order. In specific auxiliary variables this equation becomes a mathematical equation that does not refer to a specific physical context discussed. In the two-dimensional space of these auxiliary variables the roots of this equation draw a countable infinite set of open curves with hyperbolic asymptotes. We present an analytic description of these curves by a transcendental algebraic equation for the involved variables. The intersections of the curves thus constructed with a certain cubic curve provide a highly accurate description of the energy spectrum.
| Original language | English |
|---|---|
| Pages (from-to) | 3786-3790 |
| Number of pages | 5 |
| Journal | Physics Letters, Section A: General, Atomic and Solid State Physics |
| Volume | 380 |
| Issue number | 45 |
| DOIs | |
| Publication status | Published - 25 Nov 2016 |
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Keywords
- Bi-confluent Heun potentials
- Conditionally exactly solvable potentials
- Hermite function
- Inverse square root potential
- Stationary Schrödinger equation
ASJC Scopus subject areas
- Physics and Astronomy(all)
Cite this
A conditionally exactly solvable generalization of the inverse square root potential. / Ishkhanyan, A. M.
In: Physics Letters, Section A: General, Atomic and Solid State Physics, Vol. 380, No. 45, 25.11.2016, p. 3786-3790.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - A conditionally exactly solvable generalization of the inverse square root potential
AU - Ishkhanyan, A. M.
PY - 2016/11/25
Y1 - 2016/11/25
N2 - We present a conditionally exactly solvable singular potential for the one-dimensional Schrödinger equation which involves the exactly solvable inverse square root potential. Each of the two fundamental solutions that compose the general solution of the problem is given by a linear combination with non-constant coefficients of two confluent hypergeometric functions. Discussing the bound-state wave functions vanishing both at infinity and in the origin, we derive the exact equation for the energy spectrum which is written using two Hermite functions of non-integer order. In specific auxiliary variables this equation becomes a mathematical equation that does not refer to a specific physical context discussed. In the two-dimensional space of these auxiliary variables the roots of this equation draw a countable infinite set of open curves with hyperbolic asymptotes. We present an analytic description of these curves by a transcendental algebraic equation for the involved variables. The intersections of the curves thus constructed with a certain cubic curve provide a highly accurate description of the energy spectrum.
AB - We present a conditionally exactly solvable singular potential for the one-dimensional Schrödinger equation which involves the exactly solvable inverse square root potential. Each of the two fundamental solutions that compose the general solution of the problem is given by a linear combination with non-constant coefficients of two confluent hypergeometric functions. Discussing the bound-state wave functions vanishing both at infinity and in the origin, we derive the exact equation for the energy spectrum which is written using two Hermite functions of non-integer order. In specific auxiliary variables this equation becomes a mathematical equation that does not refer to a specific physical context discussed. In the two-dimensional space of these auxiliary variables the roots of this equation draw a countable infinite set of open curves with hyperbolic asymptotes. We present an analytic description of these curves by a transcendental algebraic equation for the involved variables. The intersections of the curves thus constructed with a certain cubic curve provide a highly accurate description of the energy spectrum.
KW - Bi-confluent Heun potentials
KW - Conditionally exactly solvable potentials
KW - Hermite function
KW - Inverse square root potential
KW - Stationary Schrödinger equation
UR - http://www.scopus.com/inward/record.url?scp=84992183895&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84992183895&partnerID=8YFLogxK
U2 - 10.1016/j.physleta.2016.09.035
DO - 10.1016/j.physleta.2016.09.035
M3 - Article
VL - 380
SP - 3786
EP - 3790
JO - Physics Letters, Section A: General, Atomic and Solid State Physics
JF - Physics Letters, Section A: General, Atomic and Solid State Physics
SN - 0375-9601
IS - 45
ER -