Semi-exact Solutions of Konwent Potential

In this work we study the quantum system with the symmetric Konwent potential and show how to find its exact solutions. We find that the solutions are given by the confluent Heun function. The eigenvalues have to be calculated numerically because series expansion method does not work due to the variable z >= 1. The properties of the wave functions depending on the potential parameter A are illustrated for given potential parameters V-0 and a. The wave functions are shrunk towards the origin with the increasing vertical bar A vertical bar. In particular, the amplitude of wave function of the second excited state moves towards the origin when the positive parameter A decreases. We notice that the energy levels increase with the increasing potential parameter vertical bar A vertical bar >= 1, but the variation of the energy levels becomes complicated for vertical bar A vertical bar is an element of (0, 1), which possesses a double well. It is seen that the energy levels e increase with vertical bar A vertical bar for the parameter interval vertical bar A vertical bar is an element of (-1,0), while they decrease with vertical bar A vertical bar for the parameter interval A is an element of (0,1).