Bound states for rapidly oscillatory Schr\"odinger operators in dimension 2

TL;DR

This paper investigates the eigenvalues of Schr"odinger operators with rapidly oscillating potentials in two dimensions, revealing the existence of a unique negative eigenvalue close to zero for small oscillation scales.

Contribution

It provides a rigorous analysis of the eigenvalue behavior for Schr"odinger operators with oscillatory potentials in 2D, establishing the existence and proximity of a unique negative eigenvalue.

Findings

Existence of a unique negative eigenvalue for small psilon

Eigenvalue is exponentially close to zero

Analysis specific to 2D Schrd6dinger operators

Abstract

We study the eigenvalues of Schr\"odinger operators on $\mathbb{R}^2$ with rapidly oscillatory potential $V(x) = W(x,x/\varepsilon)$, where $W(x,y) \in C^\infty_0(\mathbb{R}^2 \times \mathbb{T}^2)$ satisfies $\int_{\mathbb{T}^2} W(x,y) dy =0$. We show that for $\varepsilon$ small enough, such operators have a unique negative eigenvalue, that is exponentially close to $0$.