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

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Description:	Peyerimhoff, N (Durham University) Monday 23 March 2015, 13:30-14:30


Created:	2015-03-27 10:00
Collection:	Periodic and Ergodic Spectral Problems
Publisher:	Isaac Newton Institute
Copyright:	Peyerimhoff, N
Language:	eng (English)


Abstract:	Co-authors: Martin Tautenhahn (TU Chemnitz, Germany), Ivan Veselic (Tu Chemnitz, Germany), Karsten Leohardt (MPIPKS, Dresden, Germany) In this talk, we will consider discrete Schroedinger operators on the d-dimensional Euclidean lattice with random potential of alloy-type. The single site potential is exponentially decaying and allowed to be sign changing. The main aim is to prove a Wegner estimate, which is polynomial in the size of the box and linear in the size of the energy interval. Our result generalises earlier ones obtained by Veselic. Our Wegner estimate is of a type which can be used for the multiscale analysis proof of localisation in all energy regions, where the initial scale estimate holds. Concerning localisation, it should be mentioned that Krueger has obtained localisation results for a class of discrete alloy-type models which include ours. This is joint work with Karsten Leonhardt (MPIPKS, Dresden), Martin Tautenhahn (TU Chemnitz), and Ivan Veselic (TU Chemnitz). Bibliography: H. Krueger: Localization for random operators with non-monotone potentials with exponentially decaying correlations, Ann. Henri Poincare 13 (3), 543-598, 2012. I. Veselic: Wegner estimates for discrete alloy-type models, Ann. Henri Poincaree 11 (5), 991-1005, 2010.

Abstract:

Co-authors: Martin Tautenhahn (TU Chemnitz, Germany), Ivan Veselic (Tu Chemnitz, Germany), Karsten Leohardt (MPIPKS, Dresden, Germany)

In this talk, we will consider discrete Schroedinger operators on the d-dimensional Euclidean lattice with random potential of alloy-type. The single site potential is exponentially decaying and allowed to be sign changing. The main aim is to prove a Wegner estimate, which is polynomial in the size of the box and linear in the size of the energy interval. Our result generalises earlier ones obtained by Veselic. Our Wegner estimate is of a type which can be used for the multiscale analysis proof of localisation in all energy regions, where the initial scale estimate holds. Concerning localisation, it should be mentioned that Krueger has obtained localisation results for a class of discrete alloy-type models which include ours.

This is joint work with Karsten Leonhardt (MPIPKS, Dresden), Martin Tautenhahn (TU Chemnitz), and Ivan Veselic (TU Chemnitz).

Bibliography:

H. Krueger: Localization for random operators with non-monotone potentials with exponentially decaying correlations, Ann. Henri Poincare 13 (3), 543-598, 2012.

I. Veselic: Wegner estimates for discrete alloy-type models, Ann. Henri Poincaree 11 (5), 991-1005, 2010.

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