Results 1  10
of
10
The integrated density of states for random Schrödinger operators
"... We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is discussed i ..."
Abstract

Cited by 20 (1 self)
 Add to MetaCart
We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is discussed in the second section. One central topic of this survey is the asymptotic behavior of the integrated density of states at the boundary of the spectrum. In particular, we are interested in Lifshitz tails and the occurrence of a classical and a quantum regime. In the last section we discuss regularity properties of the IDS. Our emphasis is on the discussion of fundamental problems and central ideas to handle them. Finally, we discuss further developments and problems
Bounds on the Spectral Shift Function and the Density of States
 COMMUN. MATH. PHYS.
, 2005
"... We study spectra of Schrödinger operators on R d. First we consider a pair of operators which differ by a compactly supported potential, as well as the corresponding semigroups. We prove almost exponential decay of the singular values µn of the difference of the semigroups as n →∞and deduce bounds ..."
Abstract

Cited by 13 (6 self)
 Add to MetaCart
We study spectra of Schrödinger operators on R d. First we consider a pair of operators which differ by a compactly supported potential, as well as the corresponding semigroups. We prove almost exponential decay of the singular values µn of the difference of the semigroups as n →∞and deduce bounds on the spectral shift function of the pair of operators. Thereafter we consider alloy type random Schrödinger operators. The single site potential u is assumed to be nonnegative and of compact support. The distributions of the random coupling constants are assumed to be Hölder continuous. Based on the estimates for the spectral shift function, we prove a Wegner estimate which implies Hölder continuity of the integrated density of states.
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.
Wegner Estimate in One Dimension for Nonoverlapping Single Site Potentials
, 2002
"... In this note we present a proof of the Wegner estimate for random Schrödinger operators of Anderson type that do not have necessarily overlapping single site potentials. We use the spectral averaging procedure via trace formula and spectral shift function. ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
In this note we present a proof of the Wegner estimate for random Schrödinger operators of Anderson type that do not have necessarily overlapping single site potentials. We use the spectral averaging procedure via trace formula and spectral shift function.
FROM UNCERTAINTY PRINCIPLES TO WEGNER ESTIMATES
"... Abstract. We give a shortcut from simple uncertainty principles to Wegner estimates with correct volume term. 1. ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Abstract. We give a shortcut from simple uncertainty principles to Wegner estimates with correct volume term. 1.
The Spectral Shift Function Of A Compactly Supported Potential And Wegner Estimates
, 2003
"... We analyze the spectral shift function (SSF) of a Schrödinger operator due to a compactly supported potential. We give a bound on the integral of the SSF with respect to a bounded compactly supported function. It is based on the control of the singular values of the difference of two Schrödinger sem ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We analyze the spectral shift function (SSF) of a Schrödinger operator due to a compactly supported potential. We give a bound on the integral of the SSF with respect to a bounded compactly supported function. It is based on the control of the singular values of the difference of two Schrödinger semigroups. As an application we improve some earlier results on the regularity of the integrated density of states.
SPECTRAL ANALYSIS OF PERCOLATION HAMILTONIANS
, 2004
"... Abstract. We study the family of Hamiltonians which corresponds to the adjacency operators on a percolation graph. We characterise the set of energies which are almost surely eigenvalues with finitely supported eigenfunctions. This set of energies is a dense subset of the algebraic integers. The int ..."
Abstract
 Add to MetaCart
Abstract. We study the family of Hamiltonians which corresponds to the adjacency operators on a percolation graph. We characterise the set of energies which are almost surely eigenvalues with finitely supported eigenfunctions. This set of energies is a dense subset of the algebraic integers. The integrated density of states has discontinuities precisely at this set of energies. We show that the convergence of the integrated densities of states of finite box Hamiltonians to the one on the whole space holds even at the points of discontinuity. For this we use an equicontinuityfromtheright argument. The same statements hold for the restriction of the Hamiltonian to the infinite cluster. In this case we prove that the integrated density of states can be constructed using local data only. Finally we study some mixed AndersonQuantum percolation models and establish results in the spirit of Wegner, and Delyon and Souillard. 1. Introduction: The Quantum
EXISTENCE OF THE DENSITY OF STATES FOR SOME ALLOY TYPE MODELS WITH SINGLE SITE POTENTIALS OF CHANGING SIGN
, 2002
"... Abstract. We study spectral properties of ergodic random Schrödinger operators on L 2 (R d). The density of states is shown to exist for a certain class of alloy type potentials with single site potentials of changing sign. The Wegner estimate we prove implies Anderson localization under certain add ..."
Abstract
 Add to MetaCart
Abstract. We study spectral properties of ergodic random Schrödinger operators on L 2 (R d). The density of states is shown to exist for a certain class of alloy type potentials with single site potentials of changing sign. The Wegner estimate we prove implies Anderson localization under certain additional assumptions. For some examples we discuss briefly some properties of the common and conditional densities of the random coupling constants used in the proof of the Wegner estimate. Saˇzetak. Analiziraju se spektralna svojstva ergodičkih slučajnih Schrödingerovih operatora na L 2 (R d). Dokazuje se, da gustoća stanja postoji za odredjenu klasu potencijala tipa legure, kod kojih pojedinačni potencijal mijenja predznak. Uz odredjene dodatne uvjete Wegnerova ocjena koju dokazujemo implicira fenomen Andersonove lokalizacije. Na osnovu primjera promatramo neka svojstva zajedničke i uvjetne gustoće slučajnih konstanti veze. Te gustoće se koriste u dokazu Wegnerove ocjene. 1. Alloy type model and the integrated density of states We consider Schrödinger operators with a potential which is a stochastic process ergodic with respect to translations from Zd. Such operators model quantum mechanical Hamiltonians which govern the motion of single electrons in disordered solids. The spectral properties of the Schrödinger operator are related to the dynamical behaviour of the electron wave packets and thus to the charge transport properties of the described solid, cf. e.g. [4,
and localization near fluctuation boundaries
, 2009
"... Abstract We prove a simple uncertainty principle and show that it can be applied to prove Wegner estimates near fluctuation boundaries. This gives new classes of models for which localization at low energies can be proven. 0 ..."
Abstract
 Add to MetaCart
Abstract We prove a simple uncertainty principle and show that it can be applied to prove Wegner estimates near fluctuation boundaries. This gives new classes of models for which localization at low energies can be proven. 0
DOI 10.1007/s1104001090720 From Uncertainty Principles to Wegner Estimates
"... Abstract We give a shortcut from simple uncertainty principles to Wegner estimates with correct volume term. ..."
Abstract
 Add to MetaCart
Abstract We give a shortcut from simple uncertainty principles to Wegner estimates with correct volume term.