Results 1  10
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26
The Integrated Density Of States For Some Random Operators With Nonsign Definite Potentials
 J. Funct. Anal
, 2001
"... We study the integrated density of states of random Andersontype additive and multiplicative perturbations of deterministic background operators for which the singlesite potential does not have a fixed sign. Our main result states that, under a suitable assumption on the regularity of the random v ..."
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Cited by 43 (6 self)
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We study the integrated density of states of random Andersontype additive and multiplicative perturbations of deterministic background operators for which the singlesite potential does not have a fixed sign. Our main result states that, under a suitable assumption on the regularity of the random variables, the integrated density of states of such random operators is locally Hölder continuous at energies below the bottom of the essential spectrum of the background operator for any nonzero disorder, and at energies in the unperturbed spectral gaps, provided the randomness is sufficiently small. The result is based on a proof of a Wegner estimate with the correct volume dependence. The proof relies upon the L function for p 1 [9], and the vector field methods of [20]. We discuss the application of this result to Schrödinger operators with random magnetic fields and to bandedge localization.
The Wegner Estimate And The Integrated Density Of States For Some Random Operators
, 2001
"... The integrated density of states (IDS) for random operators is an important function describing many physical characteristics of a random system. Properties of the IDS are derived from the Wegner estimate that describes the influence of finitevolume perturbations on a background system. In this pap ..."
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Cited by 27 (10 self)
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The integrated density of states (IDS) for random operators is an important function describing many physical characteristics of a random system. Properties of the IDS are derived from the Wegner estimate that describes the influence of finitevolume perturbations on a background system. In this paper, we present a simple proof of the Wegner estimate applicable to a wide variety of random perturbations of deterministic background operators. The proof yields the correct volume dependence of the upper bound. This implies the local Hölder continuity of the integrated density of states at energies in the unperturbed spectral gap. The proof depends on the L  theory of the spectral shift function (SSF), for p 1, applicable to pairs of...
Spectral Localization by Gaussian Random Potentials in MultiDimensional Continuous Space
, 2000
"... this paper is to contribute to the understanding of spectral localization for random Schrdinger operators in multidimensional Euclidean space ..."
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Cited by 18 (4 self)
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this paper is to contribute to the understanding of spectral localization for random Schrdinger operators in multidimensional Euclidean space
The spectral shift operator
 IN MATHEMATICAL RESULTS IN QUANTUM MECHANICS
, 1999
"... We introduce the concept of a spectral shift operator and use it to derive Krein’s spectral shift function for pairs of selfadjoint operators. Our principal tools are operatorvalued Herglotz functions and their logarithms. Applications to Krein’s trace formula and to the BirmanSolomyak spectral ..."
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Cited by 16 (7 self)
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We introduce the concept of a spectral shift operator and use it to derive Krein’s spectral shift function for pairs of selfadjoint operators. Our principal tools are operatorvalued Herglotz functions and their logarithms. Applications to Krein’s trace formula and to the BirmanSolomyak spectral averaging formula are discussed.
The L^ptheory of the spectral shift function, the Wegner estimate, and the integrated density of states for some random operators
, 2001
"... We develop the L^ptheory of the spectral shift function, for p # 1, applicable to pairs of selfadjoint operators whose difference is in the trace ideal I p , for 0 < p # 1. This result is a key ingredient of a new, short proof of the Wegner estimate applicable to a wide variety of additive and ..."
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Cited by 13 (5 self)
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We develop the L^ptheory of the spectral shift function, for p # 1, applicable to pairs of selfadjoint operators whose difference is in the trace ideal I p , for 0 < p # 1. This result is a key ingredient of a new, short proof of the Wegner estimate applicable to a wide variety of additive and multiplicative random perturbations of deterministic background operators. The proof yields the correct volume dependence of the upper bound. This implies the local Hölder continuity of the integrated density of states at energies in the unperturbed spectral gap. Under an additional condition of the singlesite potential, local Hölder continuity is proved at all energies. This new Wegner estimate, together with other, standard results, establishes exponential localization for a new family of models for additive and multiplicative perturbations.
INTEGRATED DENSITY OF STATES AND WEGNER ESTIMATES FOR RANDOM SCHRÖDINGER OPERATORS
, 2003
"... We survey recent results on spectral properties of random Schrödinger operators. The focus is set on the integrated density of states (IDS). First we present a proof of the existence of a selfaveraging IDS which is general enough to be applicable to random Schrödinger and LaplaceBeltrami operators ..."
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Cited by 12 (2 self)
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We survey recent results on spectral properties of random Schrödinger operators. The focus is set on the integrated density of states (IDS). First we present a proof of the existence of a selfaveraging IDS which is general enough to be applicable to random Schrödinger and LaplaceBeltrami operators on manifolds. Subsequently we study more specific models in Euclidean space, namely of alloy type, and concentrate on the regularity properties of the IDS. We discuss the role of the integrated density of states and its regularity properties for the spectral analysis of random Schrödinger operators, particularly in relation to localisation. Proofs of the central results are given in detail. Whenever there are alternative proofs, the different approaches are compared.
Localization In One Dimensional Random Media: A Scattering Theoretic Approach
 COMM. MATH. PHYS
, 2000
"... We use scattering theoretic methods to prove exponential localization for random displacement models in one dimension. The operators we consider model both quantum and classical wave propagation. Our main tools are the reflection and transmission coefficients for compactly supported single site per ..."
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Cited by 10 (6 self)
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We use scattering theoretic methods to prove exponential localization for random displacement models in one dimension. The operators we consider model both quantum and classical wave propagation. Our main tools are the reflection and transmission coefficients for compactly supported single site perturbations. We show that randomly displaced, nonreflectionless single sites lead to localization.
Wegner Estimate for Sparse and Other Generalized Alloy Type Potentials
, 2002
"... We prove a Wegner estimate for generalized alloy type models at negative energies (Theorems 8 and 13). The single site potential is assumed to be non positive. The random potential does not need to be stationary with respect to translations from a lattice. Actually, the set of points to which t ..."
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Cited by 10 (2 self)
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We prove a Wegner estimate for generalized alloy type models at negative energies (Theorems 8 and 13). The single site potential is assumed to be non positive. The random potential does not need to be stationary with respect to translations from a lattice. Actually, the set of points to which the individual single site potentials are attached, needs only to satisfy a certain density condition. The distribution of the coupling constants is assumed to have a bounded density only in the energy region where we prove the Wegner estimate.
Lipschitz continuity of the integrated density of states for signindefinite potentials. preliminary version
, 2003
"... ABSTRACT. The present paper is devoted to the study of spectral properties of random Schrödinger operators. Using a finite section method for Toeplitz matrices, we prove a Wegner estimate for some alloy type models where the single site potential is allowed to change sign. The results apply to the c ..."
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Cited by 8 (3 self)
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ABSTRACT. The present paper is devoted to the study of spectral properties of random Schrödinger operators. Using a finite section method for Toeplitz matrices, we prove a Wegner estimate for some alloy type models where the single site potential is allowed to change sign. The results apply to the corresponding discrete model, too. In certain disorder regimes we are able to prove the Lipschitz continuity of the integrated density of states and/or localization near spectral edges. 1.
Strategies in localization proofs for onedimensional random Schrödinger operators
, 2002
"... Recent results on localization, both exponential and dynamical, for various models of onedimensional, continuum, random Schrödinger operators are reviewed. This includes Anderson models with indefinite single site potentials, the BernoulliAnderson model, the Poisson model, and the random displacem ..."
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Cited by 8 (4 self)
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Recent results on localization, both exponential and dynamical, for various models of onedimensional, continuum, random Schrödinger operators are reviewed. This includes Anderson models with indefinite single site potentials, the BernoulliAnderson model, the Poisson model, and the random displacement model. Among the tools which are used to analyse these models are generalized spectral averaging techniques and results from inverse spectral and scattering theory. A discussion of open problems is included.