The Exact Solutions of a Class of Monotonic Exponential Potential Model

We studied a class of exponential potential model V(x)=-ae-bx (a> 0 , b> 0) and found that its solutions are given by the Bessel functions, but the energy spectra E= - b²(n+ 1 / 2) ²/ 8 which are derived from the quantization condition do not correspond to any discrete bound states. The energy levels which are calculated by the boundary condition Jν(22a/b)=0 at the origin are in good agreement with the numerical results. We illustrate the wave functions through varying the potential parameters a, b and notice that they are pull back to the origin when the potential parameter a or b increases.