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.