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 Schrödinger operator $-y^{\prime \prime }+Vy$ with absolute value less than $T$. This paper studies the Weyl asymptotics of perturbations of the Schrödinger operator $-y^{\prime \prime }+\frac{1}{4}{e}^{2t}y$ on $[x_0,\infty )$. In particular, we show that perturbations by functions $\epsilon (t)$ that satisfy $\left|\epsilon (t)\right|\lesssim e^t$ 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.