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Morse potential $V_M(x)= g^2\exp (2x)-g(2h+1)\exp(x)$ is defined on the fullline, $-\infty<x<\infty$ and it defines an exactly solvable 1-d quantummechanical system with finitely many discrete eigenstates. By taking its righthalf $0\le x<\infty$ 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 thispiecewise analytic potential has infinitely many discrete eigenstates with thecorresponding eigenfunctions given by the Whittaker W function. The eigenvaluesare the square of the zeros of the Whittaker function $W_{k,\nu}(x)$ and itslinear 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 examplefor discussing the Hilbert-P\'olya conjecture on the pure imaginary zeros ofRiemann zeta function on Re$(s)=\tfrac12$.

Symmetric Morse potential is exactly solvable