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Extremal theory for spectrum of random discrete Schrödinger operator. II. Distributions with heavy tails. (English) Zbl 1235.82034

Summary: We study the asymptotic structure of the first \(K\) largest eigenvalues \(\lambda_{k,V}\) and the corresponding eigenfunctions \(\psi(\cdot;\lambda_{k,V})\) of a finite-volume Anderson model (discrete Schrödinger operator) \({\mathcal{H}}_{V} = \kappa\Delta_{V} + \xi(\cdot)\) on the multidimensional lattice torus \(V\) increasing to the whole of the lattice \({\mathbb{Z}}^{\nu}\), provided the distribution function \(F(\cdot)\) of the i.i.d. potential \(\xi(\cdot)\) satisfies the condition \(-\log(1- F(t)) = o(t^{3})\) and some additional regularity conditions as \(t \to \infty \). For \(z \in V\), denote by \(\lambda^{0}(z)\) the principal eigenvalue of the “single-peak” Hamiltonian \(\kappa\Delta_{V} + \xi (z)\delta_{z}\) in \(l^{2}(V)\), and let \(\lambda^{0}_{k,V}\) be the \(k\)-th largest value of the sample \(\lambda ^{0}(\cdot)\) in \(V\). We first show that the eigenvalues \(\lambda_{k,V}\) are asymptotically close to \(\lambda^{0}_{k,V}\). We then prove extremal-type limit theorems (i.e., Poisson statistics) for the normalized eigenvalues \((\lambda_{k,V} - B_{V})a _{V}\), where the normalizing constants \(a_{V} > 0\) and \(B_{V}\) are chosen the same as in the corresponding limit theorems for \(\lambda^{0}_{k,V}\). The eigenfunction \(\psi(\cdot;\lambda_{k,V})\) is shown to be asymptotically completely localized (as \(V \uparrow \mathbb{Z}\)) at the sites \(z_{k,V} \in V\) defined by \(\lambda^{0}(z_{k,V}) = \lambda^{0}_{k,V}\). The proofs are based on the finite-rank (in particular, rank-one) perturbation arguments for discrete Schrödinger operator when potential peaks are sparse.
For part I, see [ibid. 131, No. 5, 867–916 (2008; Zbl 1149.82015)].

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis

Citations:

Zbl 1149.82015
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