The Schrödinger operator with Morse potential on the right half line

Abstract

This paper studies the Schrödinger operator with Morse potential on a right half line [u, \infty) and determines the Weyl asymptotics of eigenvalues for constant boundary conditions. It obtains information on zeros of the Whittaker function $W_{κ, μ}(x)$ for fixed real parameters $κ, x$, with x positive, viewed as an entire function of the complex variable $μ$. In this case all zeros lie on the imaginary axis, with the exception, if $k<0$, of a finite number of real zeros. We obtain an asymptotic formula for the number of zeros of modulus at most T of form $N(T) = (2/π) T \log T + f(u) T + O(1)$. Some parallels are noted with zeros of the Riemann zeta function.