WKB QUANTIZATION OF THE MORSE HAMILTONIAN AND PERIODIC MEROMORPHIC FUNCTIONS

We show that the well known fact concerning a coincidence between the leading-order WKB and exact quantum mechanical results for an energy s pectrum of the Morse Hamiltonian is due to the following property of t he bound-state wavefunctions in the complex plane. The logarithmic der ivatives of the bound-state eigenfunctions of the Morse Hamiltonian ar e periodic functions with a pure imaginary period. We show that the Mo rse potential is the only potential having this property in the follow ing class of potentials: U(x) = Sigma(n=0)(n=2m) u(n)e(-anx).