Negative spectrum of Schr\"odinger Operators with Rapidly Oscillating Potentials

TL;DR

This paper investigates the negative spectrum of Schr"odinger operators with rapidly oscillating potentials, establishing conditions for finiteness or infiniteness of negative eigenvalues and deriving asymptotic formulas using a new proof technique.

Contribution

It provides a new proof for the negative spectrum behavior of oscillatory potentials and extends asymptotic eigenvalue formulas to this regime.

Findings

Coupling determines negative eigenvalues finiteness for oscillatory potentials.

Asymptotic formulas for eigenvalues are derived using ground state representation.

Conditions for infinite negative spectrum are characterized.

Abstract

For Schr\"odinger operators with potentials that are asymptotically homogeneous of degree $-2$, the size of the coupling determines whether it has finite or infinitely many negative eigenvalues. In the latter case the asymptotic accumulation of these eigenvalues at zero has been determined by Kirsch and Simon. A similar regime occurs for potentials which are not asymptotically monotone, but oscillatory. In this case when the ratio between the amplitude and frequency of oscillation is asymptotically homogeneous of degree $-1$ the coupling determines the finiteness of the negative spectrum. We present a new proof of this fact by making use of a ground state representation. As a consequence of this approach we derive an asymptotic formula analogous to that of Kirsch and Simon.