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Internal Lifshits Tails For Random Perturbations Of Periodic Schrödinger Operators
"... . Let H be a \Gammaperiodic Schrodinger operator acting on L 2 (R d ) and consider the random Schrodinger operator H! = H + V! where V! (x) = X fl2\Gamma ! fl V (x \Gamma fl) (here V is a positive potential and (! fl ) fl2\Gamma a collection of positive i.i.d random variables). We prove tha ..."
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Cited by 42 (7 self)
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. Let H be a \Gammaperiodic Schrodinger operator acting on L 2 (R d ) and consider the random Schrodinger operator H! = H + V! where V! (x) = X fl2\Gamma ! fl V (x \Gamma fl) (here V is a positive potential and (! fl ) fl2\Gamma a collection of positive i.i.d random variables). We prove that, at the edge of a gap of H that is not filled in for H! , the integrated density of states of H! has a Lifshits tail behaviour if and only if the integrated density of states of H is nondegenerate. R' esum' e. Soient H un op'erateur de Schrodinger \Gammap'eriodique agissant sur L 2 (R d ), V un potentiel positif et (! fl ) fl2Z d une famille de variables al'eatoires i.i.d positives. Consid'erons l'op'erateur de Schrodinger al'eatoire H! = H + V! o`u V! (x) = X fl2Z d ! fl V (x \Gamma fl). On montre que, au bord d'une lacune spectrale de H qui n'est pas combl'e pour H! la densit'e d"etats int'egr'ee de H! a un comportement asymptotique de Lifshits si et seulement si la densit'e d"...
Existence and uniqueness of the integrated density of states for Schrödinger operators with magnetic fields and unbounded random potentials, Rev
 Math. Phys
"... The object of the present study is the integrated density of states of a quantum particle in multidimensional Euclidean space which is characterized by a Schrödinger operator with a constant magnetic field and a random potential which may be unbounded from above and from below. For an ergodic rando ..."
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Cited by 23 (7 self)
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The object of the present study is the integrated density of states of a quantum particle in multidimensional Euclidean space which is characterized by a Schrödinger operator with a constant magnetic field and a random potential which may be unbounded from above and from below. For an ergodic random potential satisfying a simple moment condition, we give a detailed proof that the infinitevolume limits of spatial eigenvalue concentrations of finitevolume operators with different boundary conditions exist almost surely. Since all these limits are shown to coincide with the expectation of the trace of the spatially localized spectral family of the infinitevolume operator, the integrated density of states is almost surely nonrandom and independent of the chosen boundary condition. Our proof of the
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
Lifshitz Tails For 2Dimensional Random Schrödinger Operators
 J. Anal. Math
, 2000
"... The purpose of this paper is to prove that, for a 2dimensional continuous Anderson model, at an open band edge, the density of states exhibits a Lifshitz behavior. The Lifshitz exponent need not be d/2 = 1 but is determined by the behaviour of the Floquet eigenvalues of a well chosen background per ..."
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Cited by 18 (5 self)
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The purpose of this paper is to prove that, for a 2dimensional continuous Anderson model, at an open band edge, the density of states exhibits a Lifshitz behavior. The Lifshitz exponent need not be d/2 = 1 but is determined by the behaviour of the Floquet eigenvalues of a well chosen background periodic Schrödinger operator.
Internal Lifshitz tails for random Schrödinger operators
"... . We present new results on Lifshitz tails at internal band edges for the density of states of random Schrodinger operators on R 2 . In particular, we show the existence and compute the value of the Lifshitz exponent in the case when the band edge is simple. 0. Introduction In the early 60's (se ..."
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Cited by 16 (1 self)
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. We present new results on Lifshitz tails at internal band edges for the density of states of random Schrodinger operators on R 2 . In particular, we show the existence and compute the value of the Lifshitz exponent in the case when the band edge is simple. 0. Introduction In the early 60's (see [16, 18]), I.M. Lifshitz produced a heuristic showing that, at the uctuational band edges of the spectrum of a random Schrodinger operator, the density of states decays exponentially fast. This diers dramatically from the behavior of the density of states of a periodic Schrodinger operator : in this case, the band edge decay is polynomial. One of the major consequences of this dierence is that the band edge spectral behaviors for these two classes of operators are radically dierent: in the random case, the spectrum is localized and, in the periodic case, the spectrum is extended. To be more specic, let us turn to the random model we will study. As our results will only be valid in d...
Localization of one dimensional, continuum, BernoulliAnderson models
"... Abstract. We use scattering theoretic methods to prove strong dynamical and exponential localization for one dimensional, continuum, Andersontype models with singular distributions; in particular the case of a Bernoulli distribution is covered. The operators we consider model alloys composed of at ..."
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Cited by 14 (4 self)
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Abstract. We use scattering theoretic methods to prove strong dynamical and exponential localization for one dimensional, continuum, Andersontype models with singular distributions; in particular the case of a Bernoulli distribution is covered. The operators we consider model alloys composed of at least two distinct types of randomly dispersed atoms. Our main tools are the reflection and transmission coefficients for compactly supported single site perturbations of a periodic background which we use to verify the necessary hypotheses of multiscale analysis. We show that nonreflectionless single sites lead to a discrete set of exceptional energies away from which localization occurs. 1.
Uniform existence of the integrated density of states for . . . Z^d
"... We give an overview and extension of recent results on ergodic random Schrödinger operators for models on Zd. The operators we consider are defined on combinatorial or metric graphs, with random potentials, random boundary conditions and random metrics taking values in a finite set. We show that n ..."
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Cited by 14 (8 self)
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We give an overview and extension of recent results on ergodic random Schrödinger operators for models on Zd. The operators we consider are defined on combinatorial or metric graphs, with random potentials, random boundary conditions and random metrics taking values in a finite set. We show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable, at least locally. This limit, the integrated density of states (IDS), can be expressed by a closed ShubinPastur type trace formula. The set of points of increase of the IDS supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions. This applies to several examples, including various periodic operators and percolation models.
The Fate of Lifshits Tails in Magnetic Fields
 J. Stat. Phys
, 1995
"... We investigate the integrated density of states of the Schrodinger operator in the Euclidean plane with a perpendicular constant magnetic field and a random potential. For a Poisson random potential with a nonnegative algebraically decaying singleimpurity potential we prove that the leading asympt ..."
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Cited by 13 (6 self)
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We investigate the integrated density of states of the Schrodinger operator in the Euclidean plane with a perpendicular constant magnetic field and a random potential. For a Poisson random potential with a nonnegative algebraically decaying singleimpurity potential we prove that the leading asymptotic behaviour for small energies is always given by the corresponding classical result in contrast to the case of vanishing magnetic field. We also show that the integrated density of states of the operator restricted to the eigenspace of any Landau level exhibits the same behaviour. For the lowest Landau level, this is in sharp contrast to the case of a Poisson random potential with a deltafunction impurity potential. Key words: Random Schrodinger operators, magnetic fields, Lifshits tails Version of February, 22 1995 To appear in Journal of Statistical Physics 1 Introduction Random Schrodinger operators are differential operators on L 2 (IR d ) formally given by \Gamma 1 2 r 2...
Localisation for Random Perturbations of Periodic Schrödinger Operators with Regular Floquet Eigenvalues
, 2000
"... We prove a localisation theorem for continuous ergodic Schrödinger operators H! := H0 + V! , where the random potential V! is a nonnegative Andersontype random perturbation of the periodic operator H0 . We consider a lower spectral band edge of (H0 ), say E = 0, at a gap which is preserved by ..."
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Cited by 12 (4 self)
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We prove a localisation theorem for continuous ergodic Schrödinger operators H! := H0 + V! , where the random potential V! is a nonnegative Andersontype random perturbation of the periodic operator H0 . We consider a lower spectral band edge of (H0 ), say E = 0, at a gap which is preserved by the perturbation V! . Assuming that all Floquet eigenvalues of H0 , which reach the spectral edge 0 as a minimum, have there a positive definite Hessian, we conclude that there exists an interval I containing 0 such that H! has only pure point spectrum in I for almost all !.
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.