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Author: Fukushima, Ryoki × Subject: central limit theorem ×

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    Eigenvalue fluctuations for lattice Anderson Hamiltonians
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Biskup, Marek; Fukushima, Ryoki; König, Wolfgang
    We consider the random Schrödinger operator on a large box in the lattice with a large prefactor in front of the Laplacian part of the operator, which is proportional to the square of the diameter of the box. The random potential is assumed to be independent and bounded; its expectation function and variance function is given in terms of continuous bounded functions on the rescaled box. Our main result is a multivariate central limit theorem for all the simple eigenvalues of this operator, after centering and rescaling. The limiting covariances are expressed in terms of the limiting homogenized eigenvalue problem; more precisely, they are equal to the integral of the product of the squares of the eigenfunctions of that problem times the variance function.
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    Eigenvalue fluctuations for lattice Anderson Hamiltonians: Unbounded potentials
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Biskup, Marek; Fukushima, Ryoki; König, Wolfgang
    We consider random Schrödinger operators with Dirichlet boundary conditions outside lattice approximations of a smooth Euclidean domain and study the behavior of its lowest-lying eigenvalues in the limit when the lattice spacing tends to zero. Under a suitable moment assumption on the random potential and regularity of the spatial dependence of its mean, we prove that the eigenvalues of the random operator converge to those of a deterministic Schrödinger operator. Assuming also regularity of the variance, the fluctuation of the random eigenvalues around their mean are shown to obey a multivariate central limit theorem. This extends the authors recent work where similar conclusions have been obtained for bounded random potentials.