Eigensystem Multiscale Analysis for the Anderson Model via the Wegner Estimate

Alexander Elgart; Abel Klein

doi:10.1007/s00023-020-00926-0

Wegner estimate and localization for alloy-type models with sign-changing exponentially decaying single-site potentials

Reviews in Mathematical Physics ◽

10.1142/s0129055x15500075 ◽

2015 ◽

Vol 27 (04) ◽

pp. 1550007 ◽

Cited By ~ 2

Author(s):

Karsten Leonhardt ◽

Norbert Peyerimhoff ◽

Martin Tautenhahn ◽

Ivan Veselić

Keyword(s):

Multiscale Analysis ◽

Step Function ◽

Single Site ◽

Energy Interval ◽

Wegner Estimate ◽

Alloy Type ◽

Fixed Sign ◽

Wegner Estimates ◽

Single Site Potential ◽

The Continuum

We study Schrödinger operators on L2(ℝd) and ℓ2(ℤd) with a random potential of alloy-type. The single-site potential is assumed to be exponentially decaying but not necessarily of fixed sign. In the continuum setting, we require a generalized step-function shape. Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. In the described situation, a Wegner estimate, which is polynomial in the volume of the box and linear in the size of the energy interval, holds. We apply the established Wegner estimate as an ingredient for a localization proof via multiscale analysis.

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Multiparticle Localization at Low Energy for Multidimensional Continuous Anderson Models

Advances in Mathematical Physics ◽

10.1155/2020/5270541 ◽

2020 ◽

Vol 2020 ◽

pp. 1-15

Author(s):

Trésor Ekanga

Keyword(s):

Probability Distribution ◽

Anderson Model ◽

Multiscale Analysis ◽

Random Potential ◽

Particle Interaction ◽

Low Energy ◽

Dynamical Localization ◽

Hölder Continuous ◽

The Continuum ◽

Number Of Particles

We study the multiparticle Anderson model in the continuum and show that under some mild assumptions on the random external potential and the inter-particle interaction, for any finite number of particles, the multiparticle lower spectral edges are almost surely constant in absence of ergodicity. We stress that this result is not quite obvious and has to be handled carefully. In addition, we prove the spectral exponential and the strong dynamical localization of the continuous multiparticle Anderson model at low energy. The proof based on the multiparticle multiscale analysis bounds needs the values of the external random potential to be independent and identically distributed, whose common probability distribution is at least Log-Hölder continuous.

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The Bootstrap Multiscale Analysis for the Multi-particle Anderson Model

Journal of Statistical Physics ◽

10.1007/s10955-013-0734-8 ◽

2013 ◽

Vol 151 (5) ◽

pp. 938-973 ◽

Cited By ~ 18

Author(s):

Abel Klein ◽

Son T. Nguyen

Keyword(s):

Anderson Model ◽

Multiscale Analysis

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The Integrated Density of States for an Interacting Multiparticle Homogeneous Model and Applications to the Anderson Model

Advances in Mathematical Physics ◽

10.1155/2009/679827 ◽

2009 ◽

Vol 2009 ◽

pp. 1-15 ◽

Cited By ~ 2

Author(s):

Frédéric Klopp ◽

Heribert Zenk

Keyword(s):

Density Of States ◽

Anderson Model ◽

Homogeneous Model ◽

Free Particle ◽

Integrated Density Of States ◽

Interacting Particles ◽

Wegner Estimate ◽

Homogeneous Potential ◽

Interacting Particle ◽

Regularity Properties

For a system of interacting particles moving in the background of a “homogeneous” potential, we show that if the single particle Hamiltonian admits a density of states, so does the interacting -particle Hamiltonian. Moreover, this integrated density of states coincides with that of the free particle Hamiltonian. For the interacting -particle Anderson model, we prove regularity properties of the integrated density of states by establishing a Wegner estimate.

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