Confining non-analytic exponential potential $V(x)= g^2\exp\,(2|x|)$ and its exact Bessel-function solvability

Abstract

In a previous paper we have shown that Schr\"odinger equation with the non-analytic attractive exponential potential $V(x)= -g^2\exp (-|x|)$ is exactly solvable. It has finitely many discrete eigenstates described by the Bessel function of the first kind $J_{\nu}(z)$ and the eigenvalues are specified by the positive zeros of $J_{\nu}(g)$ and $J'_{\nu}(g)$ as a function of the order $\nu$ with fixed $g>0$. Now we show the corresponding results for the {\em confining\/} non-analytic exponential potential $V(x)= g^2\exp (2|x|)$. This has infinitely many discrete eigenstates described by the modified Bessel function of the second kind $K_{i\nu}(z)$. The eigenvalues are specified by the {\em pure imaginary zeros\/} of $K_{i\nu}(g)$ and $K'_{i\nu}(g)$ as a function of the order with fixed $g>0$.