Results 1  10
of
25
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 ..."
Abstract

Cited by 44 (6 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 28 (10 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 19 (4 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 16 (7 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
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.
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 mul ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 10 (6 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
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.
Existence Of The Density Of States For OneDimensional AlloyType Potentials With Small Support
 Mathematical Results in Quantum Mechanics
, 2002
"... We study spectral properties of Schrodinger operators with a random potential of alloy type on L (R) and their restrictions to finite intervals. A Wegner estimates for nonnegative single site potentials with small support is proven. It implies the existence and local uniform boundedness of the de ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
We study spectral properties of Schrodinger operators with a random potential of alloy type on L (R) and their restrictions to finite intervals. A Wegner estimates for nonnegative single site potentials with small support is proven. It implies the existence and local uniform boundedness of the density of states. Our estimate is valid for all bounded energy intervals. Wegner estimates play a key role in an existence proof of pure point spectrum. 1. Model and results We study spectral properties of families of Schrodinger operators on L (R). The considered operators consist of a nonrandom periodic Schrodinger operator plus a random potential of Anderson or alloy type: H# := H 0 + V# , H 0 := #+ V per . (1) Here # is the Laplace operator on R and V per L # (R) is a Zperiodic potential. The random potential V# is a stochastic process of the following form (2) V# (x) = # k u(x k).