The Schr\"odinger operator with Morse potential on the right half line

TL;DR

This paper analyzes the eigenvalue distribution of the Schr"odinger operator with Morse potential on a half line, deriving Weyl asymptotics and studying zeros of related special functions, with parallels to the Riemann zeta function.

Contribution

It provides the Weyl asymptotics for eigenvalues and characterizes zeros of the Whittaker function as an entire function of the complex variable.

Findings

Eigenvalues follow Weyl asymptotics with explicit formulas

Zeros of the Whittaker function mostly lie on the imaginary axis

Asymptotic count of zeros of modulus at most T is given by a specific formula

Abstract

This paper studies the Schr\"odinger 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 $W_{\kappa, \mu}(x)$ for fixed real parameters $\kappa, x$, with x positive, viewed as an entire function of the complex variable $\mu$. 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 $N(T) = (2/\pi) T \log T + f(u) T + O(1)$. Some parallels are noted with zeros of the Riemann zeta function.