Bound States for Rapidly Oscillatory Schrödinger Operators in Dimension 2

Abstract

We study the eigenvalues of Schrödinger operators $-\Delta_{{R}^2} + V_\epsilon$ on ${R}^2$ with rapidly oscillatory potential: $V_\varepsilon(x) = W(x,x/\varepsilon)$ with $W \in C^\infty_0({R}^2 \times ({R}/(2\pi {Z}))^2,{R})$ satisfying $\int_{ [0,2\pi]^2} W(x,y) dy = 0$. We show that for $\varepsilon$ small enough, such operators have a unique negative eigenvalue, exponentially close to 0.

Locations

• arXiv (Cornell University) - View - PDF
• SIAM Journal on Mathematical Analysis - View