Symmetric Morse potential is exactly solvable

Abstract

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$(s)=\tfrac12$.