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

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 $0$.