Eigenvalue fluctuations for lattice Anderson Hamiltonians

Abstract

We study the statistics of Dirichlet eigenvalues of the random Schrödinger operator $-{\epsilon }^{-2}{\Delta }^{(\text{d})}+{\xi }^{(\epsilon )}(x)$, with ${\Delta }^{(\text{d})}$ the discrete Laplacian on ${\mathbb{Z}}^d$ and ${\xi }^{(\epsilon )}(x)$ uniformly bounded independent random variables, on sets of the form $D_{\epsilon }:=\{x\in {\mathbb{Z}}^d:x\epsilon \in D\}$ for $D\subset {\mathbb{R}}^d$ bounded, open and with a smooth boundary. If $\mathbb{E}{\xi }^{(\epsilon )}(x)=U(x\epsilon )$ holds for some bounded and continuous $U:D\to \mathbb{R}$, we show that, as $\epsilon \downarrow 0$, the $k$-th eigenvalue converges to the $k$-th Dirichlet eigenvalue of the homogenized operator $-\Delta +U(x)$, where $\Delta$ is the continuum Dirichlet Laplacian on $D$. Assuming further that $\text{Var}({\xi }^{(\epsilon )}(x))=V(x\epsilon )$ for some positive and continuous $V:D\to \mathbb{R}$, we establish a multivariate central limit theorem for simple eigenvalues centered by their expectation. The limiting covariance for a given pair of simple eigenvalues is expressed as an integral of $V$ against the product of squares of the corresponding eigenfunctions of $-\Delta +U(x)$.

26 pages, to appear in SIAM J. Math. Anal

References in corpus (4)

• Noise-stability and central limit theorems for effective resistance of random electric networks
• Eigenvalue order statistics for random Schrödinger operators with doubly-exponential tails
• A central limit theorem for the effective conductance: Linear boundary data and small ellipticity contrasts
• Self-intersection local times of random walks: Exponential moments in subcritical dimensions

Cited by in corpus (3)

• Eigenvalue vs perimeter in a shape theorem for self-interacting random walks
• Eigenvalue fluctuations for random elliptic operators in homogenization regime
• Eigenvalue fluctuations for lattice Anderson Hamiltonians: Unbounded potentials