Semi-exact solutions to position-dependent mass Schrödinger problem with a class of hyperbolic potential V₀tanh(ax)

Abstract.

In this work we report the semi-exact solutions of the position-dependent mass Schrödinger equation (PDMSE) with a class of hyperbolic potential $V_{0}\tanh(a x)$. 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 $V(z)$ 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 $\varepsilon_{1}$ decreases and the $\varepsilon_{6-8}$ ones increase with the increasing potential parameter v, respectively, while the $\varepsilon_{2-5}$ ones first increase and then decrease with increasing v.