https://doi.org/10.1140/epjp/i2016-16176-5
Regular Article
Semi-exact solutions to position-dependent mass Schrödinger problem with a class of hyperbolic potential V0tanh(ax)
1
Information and Engineering College, Dalian University, 116622, Dalian, P. R. China
2
Catedrática CONACyT, Centro de Investigación en Computación, Instituto Politécnico Nacional, UPALM, 07700, Mexico D. F., Mexico
3
ESFM, Instituto Politécnico Nacional, Edificio 9, 07738, Mexico D. F., Mexico
4
CIDETEC, Instituto Politécnico Nacional, 07700, Mexico D. F., Mexico
* e-mail: shishan_dong@163.com
Received:
17
February
2016
Revised:
21
April
2016
Accepted:
3
May
2016
Published online:
30
May
2016
In this work we report the semi-exact solutions of the position-dependent mass Schrödinger equation (PDMSE) with a class of hyperbolic potential 
. The terminology of semi-exact solutions arises from the fact that the wave functions of this quantum system can be expressed by confluent Heun functions, but the energy levels cannot be obtained analytically. The potential 
obtained by some canonical transformations essentially keeps invariant when the potential parameter v is replaced by - v . The properties of the wave functions depending on v are also illustrated graphically. We find that the quasi-symmetric and quasi-antisymmetric wave functions that appear only for very small v are violated completely when v becomes large. This arises from the fact that the parity, which is almost a defined symmetry for very small v, is completely violated for large v. We also notice that the energy level 
decreases and the 
ones increase with the increasing potential parameter v, respectively, while the 
ones first increase and then decrease with increasing v.
© Società Italiana di Fisica and Springer-Verlag Berlin Heidelberg, 2016
