Lifshitz Tails in Constant Magnetic Fields

Klopp, Frédéric; Raikov, Georgi

doi:10.1007/s00220-006-0059-4

Cited by 9 publications

(17 citation statements)

References 32 publications

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“…Hence, Π q V per L,ω Π q admits an integrated density of states that we denote by ρ q,L,ω (E) (see [5]). In [5], we have proved…”

Section: Periodic Approximationmentioning

confidence: 99%

“…The improvement over the results in [5] is obtained through a different analysis that borrows ideas and estimates from [1]. The basic idea is to show that, for energies at a distance at most E from 2bq, the single site potential u can be replaced by an effective potential that has a support of size approximately | log E| 1/2 (see section 2 and Lemma 3 therein).…”

Section: Introductionmentioning

confidence: 99%

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Lifshitz tails for alloy-type models in a constant magnetic field

Klopp¹

2010

J. Phys. A: Math. Theor.

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Dedicated to the memory of Pierre Duclos.Abstract. In this paper, we study Lifshitz tails for a 2D Landau Hamiltonian perturbed by a random alloy-type potential constructed with single site potentials decaying at least at a Gaussian speed. We prove that, if the Landau level stays preserved as a band edge for the perturbed Hamiltonian, at the Landau levels, the integrated density of states has a Lifshitz behavior of the type e − log 2 |E−2bq| .Résumé. Dans cette note, nous démontrons qu'en dimension 2, la densité d'états intégrée d'un opérateur de Landau avec un potentiel aléatoire non négatif de type Anderson dont le potentiel de simple site décroît au moins aussi vite qu'une fonction gaussienne admet en chaque niveau de Landau, disons, 2bq, si celui-ci est un bord du spectre, une asymptotique de Lifshitz du type e − log

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“…Hence, Π q V per L,ω Π q admits an integrated density of states that we denote by ρ q,L,ω (E) (see [5]). In [5], we have proved…”

Section: Periodic Approximationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Lifshitz tails for alloy-type models in a constant magnetic field

Klopp¹

2010

J. Phys. A: Math. Theor.

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“…It also presumably should be sufficient to deal with the Anderson model in a constant magnetic field (see e.g. [KlR06,Kl09]). However we note that for continuous models, the proofs of Minami's estimate of [CGK10] and [CGK11] do not readily extend to the gap situation.…”

Section: 1mentioning

confidence: 99%

Enhanced Wegner and Minami Estimates and Eigenvalue Statistics of Random Anderson Models at Spectral Edges

Germinet

Klopp

2012

Ann. Henri Poincaré

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Abstract. We consider the discrete Anderson model and prove enhanced Wegner and Minami estimates where the interval length is replaced by the IDS computed on the interval. We use these estimates to improve on the description of finite volume eigenvalues and eigenfunctions obtained in [GKl10]. As a consequence of the improved description of eigenvalues and eigenfunctions, we revisit a number of results on the spectral statistics in the localized regime obtained in [GKl10,Kl10] and extend their domain of validity, namely :• the local spectral statistics for the unfolded eigenvalues;• the local asymptotic ergodicity of the unfolded eigenvalues; In dimension 1, for the standard Anderson model, the improvement enables us to obtain the local spectral statistics at band edge, that is in the Lifshitz tail regime. In higher dimensions, this works for modified Anderson models.

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“…Wang has given a full asymptotic expansion of the density of states away from the Landau bands [165]. More explicit quantitative results are known on the Lifshitz tails, including the double logarithmic asymptotics at each band edge [96]. The pure point spectrum at a certain distance away from the Landau levels was proven independently by Wang [164], Combes and Hislop [24] and in a single-band approximation by Dorlas, Macris and Pulé [29].…”

Section: Constant Magnetic Fieldmentioning

confidence: 99%

Recent developments in quantum mechanics with magnetic fields

Erdős¹

2007

Proceedings of Symposia in Pure Mathematics

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We present a review on the recent developments concerning rigorous mathematical results on Schrödinger operators with magnetic fields. This paper is dedicated to the sixtieth birthday of Barry Simon. AMS 2000 Subject Classification 81Q10, 81Q70Running title: Recent developments on magnetic fields * Partially supported by EU-IHP Network "Analysis and Quantum" HPRN-CT-2002-0027 and by Harvard University relativistic theory. Although the framework for QED has been clear since the 30's, the mathematical difficulties even to formulate the theory rigorously have not yet been resolved. In the low energy regime, however, massive quantum particles can be described non-relativistically. Electric and magnetic fields, with a good approximation, can be considered decoupled. Since typical magnetic fields in laboratory are relatively weak, as a first approximation one can completely neglect magnetic fields and concentrate only on quantum point particles interacting via electric potentials.The rigorous mathematical theory of Schrödinger operators has therefore started with studying the operator H = 1 2m p 2 + V (x) on L 2 (R d ) and its multi-particle analogues. Here x ∈ R d is the location of the particle in the d-dimensional configuration space, p = −i∇ x is the momentum operator and m is the mass, that can be set m = 1 2 with convenient units. The Laplace operator describes the kinetic energy of the particle and the real-valued function V (x) is the electric potential. Although both the kinetic and potential energy operators are very simple to understand separately, their sum exhibits a rich variety of complex phenomena which differ from their classical counterparts in many aspects. The mathematical theory of this operator is the most developed and most extensive in mathematical physics: the best recent review is by Simon [152].As a next approximation, classical magnetic fields are included in the theory, but spins are neglected. The kinetic energy operator is modified from p 2 to (p + A) 2 by the minimal substitution rule: p → p − eA and we set the charge to be e = −1. Here A : R d → R d is the magnetic vector potential that generates the magnetic field B according to classical electrodynamics. In d = 2 or d = 3 dimensions B = ∇ × A is a scalar or a vector field, respectively. In d = 1 dimension the vector potential can be removed by a unitary gauge transformation, e iϕ (p + A) 2 e −iϕ = p 2 , ϕ = A, therefore magnetic phenomena in R 1 are absent (they are present in the case of S 1 ).We will call the operator (p + A) 2 + V the magnetic Schrödinger operator. In general, even the kinetic energy part contains noncommuting operators, [(p + A) k , (p + A) ℓ ] = 0, and the theory of (p + A) 2 itself is more complicated than that of p 2 + V . The simplest case of constant magnetic field, B = const, is explicitly solvable. The resulting Landau-spectrum consists, in two dimensions, of infinitely degenerate eigenvalues at energies (2n+1)|B|, n = 0, 1, . . .. Notice that the magnetic spectrum is characteristically different from that ...

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Lifshitz Tails in Constant Magnetic Fields

Cited by 9 publications

References 32 publications

Lifshitz tails for alloy-type models in a constant magnetic field

Lifshitz tails for alloy-type models in a constant magnetic field

Enhanced Wegner and Minami Estimates and Eigenvalue Statistics of Random Anderson Models at Spectral Edges

Recent developments in quantum mechanics with magnetic fields

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