Symmetric Morse potential is exactly solvable

Morse potential V_M(x)= g^2exp (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 0le x<infty and glueing it with the left half of its mirror image V_M(-x), -infty<xle0, the symmetric Morse potential V(x)= g^2exp (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,ν(x) and its linear combination with W'_k,ν(x) as a function of ν with fixed k and x. This quantum mechanical system seems to offer an interesting example for discussing the Hilbert-Pólya conjecture on the pure imaginary zeros of Riemann zeta function on Re(s)=tfrac12.