ANDERSON LOCALIZATION FOR A MULTI-PARTICLE QUANTUM GRAPH

We study a multi-particle quantum graph with random potential. Taking the approach of multiscale analysis, we prove exponential and strong dynamical localization of any order in the Hilbert–Schmidt norm near the spectral edge. Apart from the results on multi-particle systems, we also prove Lifshitz-type asymptotics for single-particle systems. This shows in particular that localization for single-particle quantum graphs holds under a weaker assumption on the random potential than previously known.

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LOCALIZATION ON A QUANTUM GRAPH WITH A RANDOM POTENTIAL ON THE EDGES

Reviews in Mathematical Physics ◽

10.1142/s0129055x07003140 ◽

2007 ◽

Vol 19 (09) ◽

pp. 923-939 ◽

Cited By ~ 17

Author(s):

PAVEL EXNER ◽

MARIO HELM ◽

PETER STOLLMANN

Keyword(s):

Multiscale Analysis ◽

Random Potential ◽

Schrödinger Operators ◽

Quantum Graph ◽

Dynamical Localization ◽

Random Schrödinger Operators ◽

Schrodinger Operators ◽

Cubic Lattice ◽

Lattice Quantum

We prove spectral and dynamical localization on a cubic-lattice quantum graph with a random potential. We use multiscale analysis and show how to obtain the necessary estimates in analogy to the well-studied case of random Schrödinger operators.

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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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An Eigensystem Approach to Anderson Localization for Multi-particle Systems

Annales Henri Poincaré ◽

10.1007/s00023-021-01051-2 ◽

2021 ◽

Author(s):

Bjoern Bringmann ◽

Dana Mendelson

Keyword(s):

Single Particle ◽

Anderson Localization ◽

Particle Systems ◽

Scale Analysis ◽

Multi Scale ◽

Multi Scale Analysis

AbstractThis paper revisits the proof of Anderson localization for multi-particle systems. We introduce a multi-particle version of the eigensystem multi-scale analysis by Elgart and Klein, which had previously been used for single-particle systems.

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