The Exact Solutions of a Class of Monotonic Exponential Potential Model

Abstract

We studied a class of exponential potential model $V(x)=-a\,e^{-b\,x}$ ($a>0, b>0$) and found that its solutions are given by the Bessel functions, but the energy spectra $E=-b^2(n+1/2)^2/8$ 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 $J_{\nu }(2\sqrt{2a}/b)=0$ 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.