Weyl Asymptotics for Perturbations of Morse Potential and Connections to the Riemann Zeta Function

Abstract

Let $N(T;V)$ denote the number of eigenvalues of the Schr\"odinger operator $-y'' + Vy$ with absolute value less than $T$. This paper studies the Weyl asymptotics of perturbations of the Schr\"odinger operator $-y'' + \frac{1}{4}e^{2t}y$ on $[x_0,\infty)$. In particular, we show that perturbations by functions $\varepsilon(t)$ that satisfy $\left|\varepsilon(t)\right|\lesssim e^{t}$ do not change the Weyl asymptotics very much. Special emphasis is placed on connections to the asymptotics of the zeros of the Riemann zeta function.