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Title: Eigenvalue fluctuations for lattice Anderson Hamiltonians
Authors: Biskup, Marek
Fukushima, Ryoki
König, Wolfgang
Author's alias: 福島, 竜輝
Keywords: random Schrödinger operator
Anderson Hamiltonian
eigenvalue
spectral statistics
homogenization
central limit theorem
Issue Date: 2016
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Journal title: SIAM Journal on Mathematical Analysis
Volume: 48
Issue: 4
Start page: 2674
End page: 2700
Abstract: We study the statistics of Dirichlet eigenvalues of the random Schrödinger operator $-epsilon^{-2}Delta^{(text{rm d}mkern0.5mu)}+xi^{(epsilon)}(x)$, with $Delta^{(text{rm d}mkern0.5mu)}$ the discrete Laplacian on ${Bbb Z}^d$ and $xi^{(epsilon)}(x)$ uniformly bounded independent random variables, on sets of the form $D_epsilon:={xin{Bbb Z}^dcolon xepsilonin D}$ for $Dsubset{Bbb R}^d$ bounded, open, and with a smooth boundary. If ${Bbb E}xi^{(epsilon)}(x)=U(xepsilon)$ holds for some bounded and continuous $Ucolon Dto{Bbb R}$, we show that, as $epsilondownarrow0$, 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 $text{rm Var}(xi^{(epsilon)}(x))=V(xepsilon)$ for some positive and continuous $Vcolon Dto{Bbb R}$, 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 $-Delta+U(x)$.
Rights: © 2016, Society for Industrial and Applied Mathematics
© 2016 M. Biskup, R. Fukushima, W. Ko¨nig
URI: http://hdl.handle.net/2433/241737
DOI(Published Version): 10.1137/14097389X
Appears in Collections:Journal Articles

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