Exact solutions of the sine hyperbolic type potential

We study quantum system with a symmetric sine hyperbolic type potential $V\left(x\right)=V_0[{\sinh }^4(x)-k{\sinh }^2(x\left)\right]$, 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 $H_c(\alpha ,\beta ,\gamma ,\delta ,\eta ;z)$, in which the energy level E is involved inside the parameter $\eta$. The properties of the wave functions, which is strongly relevant for the potential parameter k, are illustrated for a given potential parameter $V_0$. 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.