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 ″ + V y -{y}^{^{\prime\prime} }+Vy with absolute value less than T T . This article studies the Weyl asymptotics of perturbations of the Schrödinger operator − y ″ + 1 4 e 2 t y -{y}^{^{\prime\prime} }+\frac{1}{4}{e}^{2t}y on [ x 0 , ∞ ) \left[{x}_{0},\infty ) . In particular, we show that perturbations by functions ε ( t ) \varepsilon \left(t) that satisfy ∣ ε ( t ) ∣ ≲ e t | \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.