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Author: König, Wolfgang × Subject: Anderson Hamiltonian ×

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Now showing 1 - 2 of 2
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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.
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    Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) König, Wolfgang; Perkowski, Nicolas; van Zuijlen, Willem
    We consider the parabolic Anderson model (PAM) in ℝ ² with a Gaussian (space) white-noise potential. We prove that the almost-sure large-time asymptotic behaviour of the total mass at time t is given asymptotically by Χ t log t, with the deterministic constant Χ identified in terms of a variational formula. In earlier work of one of the authors this constant was used to describe the asymptotic behaviour principal Dirichlet of the eigenvalue the Anderson operator on the t by t box around zero asymptotically by Χ log t.