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An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators, eprint arXiv:mathph/0605029v2
"... et Institut Universitaire de France We prove that the integrated density of states (IDS) of random Schrödinger operators with Andersontype potentials on L 2 (R d), for d ≥ 1, is locally Hölder continuous at all energies with the same Hölder exponent 0 < α ≤ 1 as the conditional probability measure ..."
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Cited by 32 (9 self)
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et Institut Universitaire de France We prove that the integrated density of states (IDS) of random Schrödinger operators with Andersontype potentials on L 2 (R d), for d ≥ 1, is locally Hölder continuous at all energies with the same Hölder exponent 0 < α ≤ 1 as the conditional probability measure for the singlesite random variable. As a special case, we prove that if the probability distribution is absolutely continuous with respect to Lebesgue measure with a bounded density, then the IDS is Lipschitz continuous at all energies. The singlesite potential u ∈ L ∞ 0 (R d) must be nonnegative and compactlysupported. The unperturbed Hamiltonian must be periodic and satisfy a unique continuation principle. We also prove analogous continuity results for the IDS of random Andersontype perturbations of the Landau Hamiltonian in twodimensions. All of these results follow from a new Wegner estimate for local random Hamiltonians with rather general probability measures.
The integrated density of states for random Schrödinger operators
 in “Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday
, 2007
"... Abstract. 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 di ..."
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Cited by 20 (1 self)
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Abstract. 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 of current
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 ..."
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Cited by 14 (5 self)
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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.
Local and Global Continuity of the Integrated Density of States
 COMMUN. PARTIAL DIFFER. EQUATIONS
, 2002
"... The integrated density of states (IDS) N(E) is the distribution function of a nonnegative measure #, the density of states measure (DOS). This measure ..."
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Cited by 13 (3 self)
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The integrated density of states (IDS) N(E) is the distribution function of a nonnegative measure #, the density of states measure (DOS). This measure
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.
Dynamical delocalization in random Landau Hamiltonians
, 2004
"... We prove the existence of dynamical delocalization for random Landau Hamiltonians near each Landau level. Since typically there is dynamical localization at the edges of each disorderedbroadened Landau band, this implies the existence of at least one dynamical mobility edge at each Landau band, n ..."
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Cited by 12 (6 self)
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We prove the existence of dynamical delocalization for random Landau Hamiltonians near each Landau level. Since typically there is dynamical localization at the edges of each disorderedbroadened Landau band, this implies the existence of at least one dynamical mobility edge at each Landau band, namely a boundary point between the localization and delocalization regimes, which we prove to converge to the corresponding Landau level as either the magnetic field or the disorder goes to zero.
Localization for the Schrödinger operator with a Poisson random potential
, 2006
"... We prove exponential and dynamical localization for the Schrödinger operator with a nonnegative Poisson random potential at the bottom of the spectrum in any dimension. We also conclude that the eigenvalues in that spectral region of localization have finite multiplicity. We prove similar localizat ..."
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Cited by 11 (3 self)
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We prove exponential and dynamical localization for the Schrödinger operator with a nonnegative Poisson random potential at the bottom of the spectrum in any dimension. We also conclude that the eigenvalues in that spectral region of localization have finite multiplicity. We prove similar localization results in a prescribed energy interval at the bottom of the spectrum provided the density of the Poisson process is large enough.
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 ..."
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Cited by 8 (4 self)
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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).
Poisson statistics for eigenvalues of continuum random Schrödinger operators, Analysis and PDE
, 2010
"... Abstract. We show absence of energy levels repulsion for the eigenvalues of random Schrödinger operators in the continuum. We prove that, in the localization region at the bottom of the spectrum, the properly rescaled eigenvalues of a continuum Anderson Hamiltonian are distributed as a Poisson point ..."
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Cited by 8 (5 self)
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Abstract. We show absence of energy levels repulsion for the eigenvalues of random Schrödinger operators in the continuum. We prove that, in the localization region at the bottom of the spectrum, the properly rescaled eigenvalues of a continuum Anderson Hamiltonian are distributed as a Poisson point process with intensity measure given by the density of states. We derive a Minami estimate for continuum Anderson Hamiltonians. We also obtain simplicity of the eigenvalues, 1.
Localization at low energies for attractive Poisson random Schrödinger operators
, 2006
"... We prove exponential and dynamical localization at low energies for the Schrödinger operator with an attractive Poisson random potential in any dimension. We also conclude that the eigenvalues in that spectral region of localization have finite multiplicity. ..."
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Cited by 7 (2 self)
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We prove exponential and dynamical localization at low energies for the Schrödinger operator with an attractive Poisson random potential in any dimension. We also conclude that the eigenvalues in that spectral region of localization have finite multiplicity.