Global continuity of the integrated density of states for random Landau Hamiltonians

Jean Michel Combes; Peter D. Hislop; Frédéric Klopp; Georgi Raikov

doi:10.1081/PDE-200033731

Global continuity of the integrated density of states for random Landau Hamiltonians

Jean Michel Combes
, Peter D. Hislop
, Frédéric Klopp
, Georgi Raikov

Mathematics

Research output: Contribution to journal › Article › peer-review

16 Scopus citations

Abstract

We prove that the integrated density of states (IDS) for the randomly perturbed Landau Hamiltonian is Hölder continuous at all energies with any Hölder exponent < q < 1. The random Anderson-type potential is constructed with a non-negative, compactly supported single-site potential u. The distribution of the iid random variables is required to be absolutely continuous with a bounded, compactly upported density. This extends a previous result Combes et al. [Combes, J. M., Hislop, P. D., Klopp, F. (2003a). Hölder continuity of the integrated density of states for some random operators at all energies. Int. Math. Res. Notices 2003: 179-209] that was restricted to constant magnetic fields having rational flux through the unit square. We also prove that the IDS is Hölder continuous as a function of the nonzero magnetic field strength.

Original language	English
Pages (from-to)	1187-1213
Number of pages	27
Journal	Communications in Partial Differential Equations
Volume	29
Issue number	7-8
DOIs	https://doi.org/10.1081/PDE-200033731
State	Published - Jul 2004

Bibliographical note

Funding Information:
The first author is supported by CPT, CNRS, Luminy, Marseille, France; the second author is supported in part by NSF Grant DMS-0202656; the third author is supported in part by FNS 2000 ‘‘Programme Jeunes Chercheurs’’, and by the Chilean Science Foundation Fondecyt under Grant 7020737; the fourth author is Supported in part by Chilean Science Foundation Fondecyt under Grants 1020737 and 7020737.

Funding

The first author is supported by CPT, CNRS, Luminy, Marseille, France; the second author is supported in part by NSF Grant DMS-0202656; the third author is supported in part by FNS 2000 ‘‘Programme Jeunes Chercheurs’’, and by the Chilean Science Foundation Fondecyt under Grant 7020737; the fourth author is Supported in part by Chilean Science Foundation Fondecyt under Grants 1020737 and 7020737.

Funders	Funder number
Chilean Science Foundation Fondecyt	7020737, 1020737
National Science Foundation (NSF)	DMS-0202656
Center for Protein Therapeutics, University at Buffalo
CNRS Centre National de la Recherche Scientifique

Keywords

Irrational flux
Landau levels
Localization
Magnetic fields
Random Schrödinger operators

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1081/PDE-200033731

Cite this

@article{1b78e4da38aa4b6cbd4862982c7fa810,

title = "Global continuity of the integrated density of states for random Landau Hamiltonians",

abstract = "We prove that the integrated density of states (IDS) for the randomly perturbed Landau Hamiltonian is H{\"o}lder continuous at all energies with any H{\"o}lder exponent < q < 1. The random Anderson-type potential is constructed with a non-negative, compactly supported single-site potential u. The distribution of the iid random variables is required to be absolutely continuous with a bounded, compactly upported density. This extends a previous result Combes et al. [Combes, J. M., Hislop, P. D., Klopp, F. (2003a). H{\"o}lder continuity of the integrated density of states for some random operators at all energies. Int. Math. Res. Notices 2003: 179-209] that was restricted to constant magnetic fields having rational flux through the unit square. We also prove that the IDS is H{\"o}lder continuous as a function of the nonzero magnetic field strength.",

keywords = "Irrational flux, Landau levels, Localization, Magnetic fields, Random Schr{\"o}dinger operators",

author = "Combes, \{Jean Michel\} and Hislop, \{Peter D.\} and Fr{\'e}d{\'e}ric Klopp and Georgi Raikov",

note = "Funding Information: The first author is supported by CPT, CNRS, Luminy, Marseille, France; the second author is supported in part by NSF Grant DMS-0202656; the third author is supported in part by FNS 2000 {\textquoteleft}{\textquoteleft}Programme Jeunes Chercheurs{\textquoteright}{\textquoteright}, and by the Chilean Science Foundation Fondecyt under Grant 7020737; the fourth author is Supported in part by Chilean Science Foundation Fondecyt under Grants 1020737 and 7020737.",

year = "2004",

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language = "English",

volume = "29",

pages = "1187--1213",

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AU - Combes, Jean Michel

AU - Hislop, Peter D.

AU - Klopp, Frédéric

AU - Raikov, Georgi

N1 - Funding Information: The first author is supported by CPT, CNRS, Luminy, Marseille, France; the second author is supported in part by NSF Grant DMS-0202656; the third author is supported in part by FNS 2000 ‘‘Programme Jeunes Chercheurs’’, and by the Chilean Science Foundation Fondecyt under Grant 7020737; the fourth author is Supported in part by Chilean Science Foundation Fondecyt under Grants 1020737 and 7020737.

PY - 2004/7

Y1 - 2004/7

N2 - We prove that the integrated density of states (IDS) for the randomly perturbed Landau Hamiltonian is Hölder continuous at all energies with any Hölder exponent < q < 1. The random Anderson-type potential is constructed with a non-negative, compactly supported single-site potential u. The distribution of the iid random variables is required to be absolutely continuous with a bounded, compactly upported density. This extends a previous result Combes et al. [Combes, J. M., Hislop, P. D., Klopp, F. (2003a). Hölder continuity of the integrated density of states for some random operators at all energies. Int. Math. Res. Notices 2003: 179-209] that was restricted to constant magnetic fields having rational flux through the unit square. We also prove that the IDS is Hölder continuous as a function of the nonzero magnetic field strength.

AB - We prove that the integrated density of states (IDS) for the randomly perturbed Landau Hamiltonian is Hölder continuous at all energies with any Hölder exponent < q < 1. The random Anderson-type potential is constructed with a non-negative, compactly supported single-site potential u. The distribution of the iid random variables is required to be absolutely continuous with a bounded, compactly upported density. This extends a previous result Combes et al. [Combes, J. M., Hislop, P. D., Klopp, F. (2003a). Hölder continuity of the integrated density of states for some random operators at all energies. Int. Math. Res. Notices 2003: 179-209] that was restricted to constant magnetic fields having rational flux through the unit square. We also prove that the IDS is Hölder continuous as a function of the nonzero magnetic field strength.

KW - Irrational flux

KW - Landau levels

KW - Localization

KW - Magnetic fields

KW - Random Schrödinger operators

UR - https://www.scopus.com/pages/publications/8744313700

UR - https://www.scopus.com/pages/publications/8744313700#tab=citedBy

U2 - 10.1081/PDE-200033731

DO - 10.1081/PDE-200033731

M3 - Article

AN - SCOPUS:8744313700

SN - 0360-5302

VL - 29

SP - 1187

EP - 1213

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

IS - 7-8

ER -

Global continuity of the integrated density of states for random Landau Hamiltonians

Abstract

Bibliographical note

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Keywords

ASJC Scopus subject areas

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