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Localization of surface spectra. (English) Zbl 0952.60059
Summary: We study spectral properties of the discrete Laplacian \(H\) on the half-space \(\mathbb{Z}_+^{d+1}=\mathbb{Z}^d\times\mathbb{Z}_+\) with random boundary condition \(\psi(n,-1)=\lambda V(n)\psi(n,0)\); the \(V(n)\) are independent random variables on a probability space \((\Omega,{\mathcal F},P)\) and \(\lambda\) is the coupling constant. It is known that if the \(V(n)\) have densities, then on the interval \([-2(d+1)\), \(2(d+1)] (=\sigma(H_0)\), the spectrum of the Dirichlet Laplacian) the spectrum of \(H\) is \(P\)-a.s. absolutely continuous for all \(\lambda\) [see the first author and Y. Last, “Corrugated surfaces and a.c. spectrum” (subm. for publ.)]. Here we show that if the random potential \(V\) satisfies the assumption of M. Aizenman and S. Molchanov [Commun. Math. Phys. 157, No. 2, 245-278 (1993; Zbl 0782.60044)], then there are constants \(\lambda_d\) and \(\Lambda_d\) such that for \(|\lambda|<\lambda_d\) and \(|\lambda |>\Lambda_d\) the spectrum of \(H\) outside \(\sigma (H_0)\) is \(P\)-a.s. pure point with exponentially decaying eigenfunctions.

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L05 General spectral theory of ordinary differential operators
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