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

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