Eigenvalue Fluctuations for Lattice Anderson Hamiltonians: Unbounded Potentials

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11620%2F19%3A10394508" target="_blank" >RIV/00216208:11620/19:10394508 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=IhY0~TTFXz" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=IhY0~TTFXz</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/14097389X" target="_blank" >10.1137/14097389X</a>

Result language
angličtina
Original language name
Eigenvalue Fluctuations for Lattice Anderson Hamiltonians: Unbounded Potentials
Original language description
We study the statistics of Dirichlet eigenvalues of the random Schrodinger operator -epsilon(-2)Delta((d)) + xi((epsilon))(x), with Delta((d)) the discrete Laplacian on Z(d) and xi((epsilon))(x) uniformly bounded independent random variables, on sets of the form D-epsilon := {x is an element of Z(d) : x epsilon is an element of D} for D subset of R-d bounded, open, and with a smooth boundary. If E xi((epsilon))(x) = U(x epsilon) holds for some bounded and continuous U : D -> R, we show that, as epsilon down arrow 0, the kth eigenvalue converges to the kth Dirichlet eigenvalue of the homogenized operator -Delta + U(x), where Delta is the continuum Dirichlet Laplacian on D. Assuming further that Var(xi((epsilon))(x)) = V (x epsilon) for some positive and continuous V : D -> 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).
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10103 - Statistics and probability

Project
<a href="/en/project/GA16-15238S" target="_blank" >GA16-15238S: Collective behavior of large stochastic systems</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2019
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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