Exact solutions of the sine hyperbolic type potential

We study quantum system with a symmetric sine hyperbolic type potential V(x) = V[sinh ⁴(x) - ksinh ²(x)] , which becomes single or double well depending on whether the potential parameter k is taken as negative or positive. We find that its exact solutions can be written as the confluent Heun functions H_c(α, β, γ, δ, η; z) , in which the energy level E is involved inside the parameter η. The properties of the wave functions, which is strongly relevant for the potential parameter k, are illustrated for a given potential parameter V. It is shown that the wave functions are shrunk to the origin when the negative potential parameter |k| increases, while for a positive k which corresponding to a double well, the wave functions with a certain parity are changed sensitively.