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

Let $N(T;V)$ denote the number of eigenvalues of the Schr"odinger operator $-y^{\mathrm{\prime }\mathrm{\prime }}+Vy$ 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 $\epsilon (t)$ that satisfy $\mid \epsilon (t)\mid \lesssim e^t$ 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.