  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 $g>0$.
