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Discrete Schrödinger Operators with Random Alloy-type Potential

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Spectral Analysis of Quantum Hamiltonians

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 224))

Abstract

We review recent results on localization for discrete alloy-type models based on the multiscale analysis and the fractional moment method, respectively. Th e discrete alloy-type model is a family of Schrödinger operators \( H_\omega = - \Delta + V_\omega\) on \( l^{^2 } (\mathbb{Z}^d )\) where Δ is the discrete Laplacian and \(V_\omega\) the multiplication by the function \(V_\omega (x) = \sum _{k \in\mathbb{Z}^d } \omega _k u(x - k) \cdot\)Here \(\omega _k,k \in \mathbb{Z}^d \) are i.i.d. random variables and \( u \in l^1 (\mathbb{Z}^d ;\mathbb{R}) \) is a so-called single-site potential. Since u may change sign, certain properties of \(H\omega\) depend in a non-monotone way on the random parameters \( \omega _k\). This requires new methods at certain stages of the localization proof.

Mathematics Subject Classification (2000). 82B44, 60H25, 35J10.

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Author information

Authors and Affiliations

  1. 448 Department of Mathematics McBryde Hall, Virginia Tech., Blacksburg, VA, 24061, USA

    Alexander Elgart

  2. Caltech Department of Mathematics, 253-37, 1200 E California Blvd, Pasadena, CA, 91125, USA

    Helge Krüger

  3. Fakultät für Mathematik, Technische Universität Chemnitz, D-09107, Chemnitz, Germany

    Martin Tautenhahn & Ivan Veselić

Authors
  1. Alexander Elgart
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  2. Helge Krüger
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  3. Martin Tautenhahn
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  4. Ivan Veselić
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Correspondence to Alexander Elgart .

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Editors and Affiliations

  1. , Departamento de Física, P. Universidad Católica de Chile, Santiago, 22, Chile

    Rafael Benguria

  2. , Departamento de Matemáticas, Universidad de Chile, Las Palmeras, Nuñoa 3425, Santiago, Chile

    Eduardo Friedman

  3. , Departamento de Matemática, Universidad de Chile, Las Palmeras, Nuñoa 3425, Santiago, Chile

    Marius Mantoiu

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Elgart, A., Krüger, H., Tautenhahn, M., Veselić, I. (2012). Discrete Schrödinger Operators with Random Alloy-type Potential. In: Benguria, R., Friedman, E., Mantoiu, M. (eds) Spectral Analysis of Quantum Hamiltonians. Operator Theory: Advances and Applications, vol 224. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0414-1_6

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