The Schrödinger Operator with Morse Potential on the Right Half Line

This paper studies the Schrödinger operator with Morse potential Vk(u) = 1 4e 2u + keu on a right half-line [u0,∞), and determines the Weyl asymptotics of eigenvalues for constant boundary conditions at the endpoint u0. In consequence it obtains information on the location of zeros of the Whittaker function Wκ,μ(x), for fixed real parameters κ, x with x > 0, viewed as an entire function in 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 which lie in the interval |κ| < |k|. We obtain an asymptotic formula for the number of zeros N(T ) = {ρ | Wκ,ρ(x) = 0, Im(ρ)| < T} of the form N(T ) = 2 πT log T+ 2 π (2 log 2−1− log x)T+O(1). Parallels are noted with zeros of the Riemann zeta function.