Weyl asymptotics for perturbations of Morse potential and connections to the Riemann zeta function

Abstract

Let N(T;V)N\left(T;\hspace{0.33em}V) denote the number of eigenvalues of the Schrödinger operator −y″+Vy-{y}^{^{\prime\prime} }+Vy with absolute value less than TT. This article studies the Weyl asymptotics of perturbations of the Schrödinger operator −y″+14e2ty-{y}^{^{\prime\prime} }+\frac{1}{4}{e}^{2t}y on [x0,∞)\left[{x}_{0},\infty ). In particular, we show that perturbations by functions ε(t)\varepsilon \left(t) that satisfy ∣ε(t)∣≲et| \varepsilon \left(t)| \hspace{0.33em}\lesssim \hspace{0.33em}{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.

Keywords

• morse potential
• perturbation
• riemann zeta function
• weyl asymptotics
• primary: 32l40
• secondary: 34b24
• 11m26