The Integrated Density of States for Some Random Operators with Nonsign Definite Potentials

Hislop, Peter D.; Klopp, Frédéric

doi:10.1006/jfan.2002.3947

Cited by 69 publications

(128 citation statements)

References 36 publications

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“…There is another method to prove Wegner estimates for single site potentials that are allowed to change sign which is based on certain vector fields in the parameter space underlying the alloy type model. It was introduced in [Klo95] by F. Klopp and improved by P. Hislop and F. Klopp in [HK02]. Its advantage is that it applies to arbitrary continuous, compactly supported single site potentials (which are not identically equal to zero).…”

Section: Model and Resultsmentioning

confidence: 99%

Wegner Estimates for Sign-Changing Single Site Potentials

Veselić

2010

Math Phys Anal Geom

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Abstract. We study Anderson and alloy type random Schrödinger operators on ℓ 2 (Z d ) and L 2 (R d ). Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. For a certain class of models we prove a Wegner estimate which is linear in the volume of the box and the length of the considered energy interval. The single site potential of the Anderson/alloy type model does not need to have fixed sign, but it needs be of a generalised step function form. The result implies the Lipschitz continuity of the integrated density of states. Model and resultsWe study spectral properties of Schrödinger operators which are given as the sum H = −∆ + V of the negative Laplacian ∆ and a multiplication operator V . The operators can be considered in d-dimensional Euclidean space R d or on the lattice Z d . To be able to treat both cases simultaneously let us use the symboland V is a bounded functionThus H is selfadjoint on the usual Sobolev space W 2,2 (R d ). In the discrete case the Laplacian is given by the rule ∆φwhere φ is a sequence in ℓ 2 (Z d ) and (e 1 , . . . , e d ) is an orthonormal basis which defines the lattice Z d as a subset of R d . The potential is given by a bounded function V : Z d → R, and thus H is a bounded selfadjoint operator.The operators we are considering are random. More precisely, the potential V = V per + V ω decomposes into a part V per which is translation invariant with respect to some sub-lattice nZ d , n ∈ N, i.e. V per (x + k) = V per (x) for all x ∈ R d and all k ∈ nZ d , and a part V ω which is random. The random part of the potential is a stochastic field V ω (x) := k∈Z d ω k u(x − k), x ∈ X d , of alloy or Anderson type. Here u : X d → R is a bounded, compactly supported function, which we call single site potential. The coupling constants ω k , k ∈ Z d form an independent, identically distributed sequence of real random variables. We assume that the random variables are bounded and distributed according to a density f of bounded variation. In the discrete case the random operator H ω = −∆ + V per + V ω is called Anderson model, and in the continuum case H ω is called alloy type model.There is a well defined spectral distribution function N : R → R of the family (H ω ) ω which is closely related to eigenvalue counting functions on finite cubes. To explain this precisely, we need some more notation. Denote by χ the characteristic function of the set [−1/2, 1/2] d ∩ X d . Thus in the continuum case this set is a unit cube, and in the discrete Key words and phrases. random Schrödinger operators, alloy type model, integrated density of states, Wegner estimate, single site potential, non-monotone. March 27, 2018March 27, , 2008.tex.

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Section: Model and Resultsmentioning

confidence: 99%

Wegner Estimates for Sign-Changing Single Site Potentials

Veselić

2010

Math Phys Anal Geom

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“…Using the inequality 25) being valid for all s ∈ R, first for s = t and then for s = 0 we bound the second term on the right-hand side of (2.24) by a time-independent one according to a(…”

Section: Prop A25]) the Same Lines Of Reasoning Implymentioning

confidence: 99%

“…. For more general random vector potentials the integrated density of states is known to be only Hölder continuous in certain energy regimes [25].…”

Section: 2mentioning

confidence: 99%

Energetic and Dynamic Properties of a Quantum Particle in a Spatially Random Magnetic Field with Constant Correlations along one Direction

Leschke

Warzel

Weichlein

2006

Ann. Henri Poincaré

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ABSTRACT. We consider an electrically charged particle on the Euclidean plane subjected to a perpendicular magnetic field which depends only on one of the two Cartesian co-ordinates. For such a "unidirectionally constant" magnetic field (UMF), which otherwise may be random or not, we prove certain spectral and transport properties associated with the corresponding one-particle Schrödinger operator (without scalar potential) by analysing its "energy-band structure". In particular, for an ergodic random UMF we provide conditions which ensure that the operator's entire spectrum is almost surely absolutely continuous. This implies that, along the direction in which the random UMF is constant, the quantum-mechanical motion is almost surely ballistic, while in the perpendicular direction in the plane one has dynamical localisation. The conditions are verified, for example, for Gaussian and Poissonian random UMF's with non-zero mean-values. These results may be viewed as "random analogues" of results first obtained by A.

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“…This theory has been investigated intensively in the last years by mathematical physicists in the context of one-particle Schrödinger operators, in particular, operators with a periodic potential and random Schrödinger operators like Anderson type models. For recent references consult, e.g., [46,13,48,33] and for references before 1992 [27,41,10,61].…”

Section: Introductionmentioning

confidence: 99%

“…For Anderson-type random Schrödinger operators the existence of the density of states (i.e., the absolute continuity of the integrated density of states) is a complicated, still not completely solved problem. Very little known is known about regularity properties of the density of states [12,13,48,33].…”

Section: Introductionmentioning

confidence: 99%

Statistical ensembles and density of states

Kostrykin¹,

Schrader²

2002

Mathematical Results in Quantum Mechanics

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ABSTRACT. We propose a definition of microcanonical and canonical statistical ensembles based on the concept of density of states. This definition applies both to the classical and the quantum case. For the microcanonical case this allows for a definition of a temperature and its fluctuation, which might be useful in the theory of mesoscopic systems. In the quantum case the concept of density of states applies to one-particle Schrödinger operators, in particular to operators with a periodic potential or to random Anderson type models. In the case of periodic potentials we show that for the resulting n-particle system the density of states is [(n − 1)/2] times differentiable, such that like for classical microcanonical ensembles a (positive) temperature may be defined whenever n ≥ 5. We expect that a similar result should also hold for Anderson type models. We also provide the first terms in asymptotic expansions of thermodynamic quantities at large energies for the microcanonical ensemble and at large temperatures for the canonical ensemble. A comparison shows that then both formulations asymptotically give the same results.

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The Integrated Density of States for Some Random Operators with Nonsign Definite Potentials

Cited by 69 publications

References 36 publications

Wegner Estimates for Sign-Changing Single Site Potentials

Wegner Estimates for Sign-Changing Single Site Potentials

Energetic and Dynamic Properties of a Quantum Particle in a Spatially Random Magnetic Field with Constant Correlations along one Direction

Statistical ensembles and density of states

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