Research Article
Constructions of the Soluble Potentials for the Nonrelativistic
Quantum System by Means of the Heun Functions
Shishan Dong ,
1
G. Yáñez-Navarro,
2
M. A. Mercado Sanchez ,
3
C. Mejía-García ,
2
Guo-Hua Sun,
4
and Shi-Hai Dong
3
1
Information and Engineering College, Dalian University, Dalian 116622, China
2
Escuela Superior de F
´
ısica y Matem
´
aticas, Instituto Polit
´
ecnico Nacional, UPALM, CDMX 07738, Mexico
3
Laboratorio de Informaci
´
on Cu
´
antica, CIDETEC, Instituto Polit
´
ecnicoNacional,UPALM,CDMX07700,Mexico
4
Catedr
´
aticaCONACyT,CIC,InstitutoPolit
´
ecnicoNacional,UPALM,CDMX07700,Mexico
Correspondence should be addressed to Shi-Hai Dong; dongsh2@yahoo.com
Received 25 September 2017; Revised 25 April 2018; Accepted 3 May 2018; Published 3 June 2018
Ac
ademic Editor: Won Sang Chung
Copyright © 2018 Shishan Dong et al. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e
publication of this article was funded by SCOAP
3
.
e Schr
¨
odinger equation
()+
2
()=0where
2
=
2
−()isrewrittenasamorepopularformofasecondorderdierential
equation by taking a similarity transformation () =()()with =().eSchr
¨
odinger invariant
𝑆
()can be calculated
directly by the Schwarzian derivative {,}and the invariant ()of the dierential equation
𝑧𝑧
+()
𝑧
+() = 0.Wend
an important relation for a moving particle as ∇
2
=−
𝑆
()and thus explain the reason why the Schr
¨
odinger invariant
𝑆
()keeps
constant. As an illustration, we take the typical Heun’s dierential equation as an object to construct a class of soluble potentials and
generalize the previous results by taking dierent transformation =
()as before. We get a more general solution ()through
integrating (
)
2
=
1
2
+
1
+
1
directly and it includes all possibilities for those parameters. Some particular cases are discussed
in detail. e results are also compared with those obtained by Bose, Lemieux, Batic, Ishkhanyan, and their coworkers. It should
be recognized that a subtle and dierent choice of the transformation ()also related to will lead to dicult connections to the
results obtained from other dierent approaches.
1. Introduction
e exact solution of the Schr
¨
odinger equation with phys-
ical potentials has played an important role in quantum
mechanics. is is due to the fact that an exact knowledge
of the energy spectrum and the wave functions of the one-
dimensional Schr
¨
odinger equation turns out to be very useful
in many applied problems. Generally speaking, for a given
external eld, one of our main tasks is to show how to solve
the dierential equation through choosing suitable variables
and then nd its exact solutions expressed by some special
functions. Here we focus on how to construct a class of
thesolvablepotentialswithintheframeworkofthenonrel-
ativistic Schr
¨
odinger equation. Up to now, a lot of similar
works have been carried out with this stimulation [1–9]. For
example, Lemieux and Bose presented an advanced analysis
of the variable transformation and have listed 8 potentials
[3] (written in French). Batic, Williams, and Nowakowski
have discussed the general potential allowing reduction of
the Schr
¨
odinger equation to the general Heun’s dierential
equation by an energy-independent transformation. In par-
ticular, eorem 4.1 in [8] gives the most general form of this
potential including 15 parameters instead of 14 parameters,
which contain 10 parameters (
0,1,2,3,4
and
0,1,2,3,4
)and
original parameters ,,,,, involved in the general
Heun equation but with a constraint also named as Fuchsian
relation (11). In that work [8], they nd that their results
cover all results obtained by Bose, Lemieux, Natanzon, and
Iwata [1–4, 10]. Iwata studied the soluble potentials in terms
of the hypergeometric functions [10]. It is not surprising to
see this since the special functions considered by them are
only the special cases of the general Heun function. Just
Hindawi
Advances in High Energy Physics
Volume 2018, Article ID 9824538, 8 pages
https://doi.org/10.1155/2018/9824538
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