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タイトル: | Eigenvalue fluctuations for lattice Anderson Hamiltonians |
著者: | Biskup, Marek Fukushima, Ryoki https://orcid.org/0000-0002-7582-6793 (unconfirmed) König, Wolfgang |
著者名の別形: | 福島, 竜輝 |
キーワード: | random Schrödinger operator Anderson Hamiltonian eigenvalue spectral statistics homogenization central limit theorem |
発行日: | 2016 |
出版者: | Society for Industrial & Applied Mathematics (SIAM) |
誌名: | SIAM Journal on Mathematical Analysis |
巻: | 48 |
号: | 4 |
開始ページ: | 2674 |
終了ページ: | 2700 |
抄録: | 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 $-Delta+U(x)$. |
著作権等: | © 2016, Society for Industrial and Applied Mathematics © 2016 M. Biskup, R. Fukushima, W. Ko¨nig |
URI: | http://hdl.handle.net/2433/241737 |
DOI(出版社版): | 10.1137/14097389X |
出現コレクション: | 学術雑誌掲載論文等 |
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