The bound-state solutions of the Schrödinger equation for an exponential-type potential with the centrifugal term are presented approximately. It is shown that the complicated normalization wavefunctions can be expressed by the generalized hypergeometric functions 2 F 1 (a, b; c; z). To show the accuracy of our results, we calculate the energy eigenvalues numerically for arbitrary quantum numbers n and l with two different values of the parameter σ . It is found that the results are in good agreement with those obtained by another method for short-range potential. Two special cases for s-wave case (l = 0) and σ = 1 are also studied briefly.

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We solve the quantum system with the symmetric Razavy cosine type potential and find that its exact solutions are given by the confluent Heun function. The eigenvalues are calculated numerically. The properties of the wave functions, which depend on the potential parameter a, are illustrated for a given potential parameter ξ. It is shown that the wave functions are shrunk to the origin when the potential parameter a increases. We note that the energy levels ϵi (i∈[1,3]) decrease with the increasing potential parameter a but the energy levels ϵi (i∈[4,7]) first increase and then decrease with the increasing a.

We present exact solutions of solitonic profile mass Schrödinger equation with a modified Pöschl–Teller potential. We find that the solutions can be expressed analytically in terms of confluent Heun functions. However, the energy levels are not analytically obtainable except via numerical calculations. The properties of the wave functions, which depend on the values of potential parameter [Formula: see text] are illustrated graphically. We find that the potential changes from single well to a double well when parameter [Formula: see text] changes from minus to positive. Initially, the crest of wave function for the ground state diminishes gradually with increasing [Formula: see text] and then becomes negative. We notice that the parities of the wave functions for [Formula: see text] also change.