  Morse potential $V_M(x)= g^2\exp (2x)-g(2h+1)\exp(x)$ is defined on the full
line, $-\infty<x<\infty$ and it defines an exactly solvable 1-d quantum
mechanical system with finitely many discrete eigenstates. By taking its right
half $0\le x<\infty$ and glueing it with the left half of its mirror image
$V_M(-x)$, $-\infty<x\le0$, the symmetric Morse potential $V(x)= g^2\exp
(2|x|)-g(2h+1)\exp(|x|)$ is obtained. The quantum mechanical system of this
piecewise analytic potential has infinitely many discrete eigenstates with the
corresponding eigenfunctions given by the Whittaker W function. The eigenvalues
are the square of the zeros of the Whittaker function $W_{k,\nu}(x)$ and its
linear combination with $W'_{k,\nu}(x)$ as a function of $\nu$ with fixed $k$
and $x$. This quantum mechanical system seems to offer an interesting example
for discussing the Hilbert-P\'olya conjecture on the pure imaginary zeros of
Riemann zeta function on Re$(s)=\tfrac12$.
