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 ′′ + Vy with absolute value less than T . This paper studies the Weyl asymptotics of perturbations of the Schrödinger operator −y ′′ + 1 4 e 2t y on [x 0 , ∞). In particular, we show that perturbations by functions ε(t) that satisfy |ε(t)| 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.

Riemann Hypothesis, Modified Morse Potential and Supersymmetric Quantum Mechanics

“…In many physical representations of the Riemann zeta function one has a physical way of expressing the Riemann hypothesis [28][29] [30][31][32] [33]. In the representation in terms of the Riemann potential one can state the hypothesis so that only for a Morse-like parameter A = 1/2 does the ground state wave function in the momentum presentation have nodes or zeros.…”

“…Note that the representation in momentum space is important as there is a theorem that the ground state wave function in position space has no nodes or zeros. Although the connection to supersymmetric quantum mechanics to the Riemann Zeta function and the Morse potential has been made in [31][32] [33] our approach is somewhat different in that we concentrate on the ground state in momentum space to form the hypothesis.…”