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34
Localization for random perturbations of periodic Schrödinger operators
 RANDOM OPER. STOCHASTIC EQUATIONS
, 1996
"... We prove localization for Andersontype random perturbations of periodic Schrödinger operators on R d near the band edges. General, possibly unbounded, single site potentials of fixed sign and compact support are allowed in the random perturbation. The proof is based on the following methods: (i) ..."
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Cited by 59 (20 self)
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We prove localization for Andersontype random perturbations of periodic Schrödinger operators on R d near the band edges. General, possibly unbounded, single site potentials of fixed sign and compact support are allowed in the random perturbation. The proof is based on the following methods: (i) A study of the band shift of periodic Schrodinger operators under linearly coupled periodic perturbations. (ii) A proof of the Wegner estimate using properties of the spatial distribution of eigenfunctions of finite box hamiltonians. (iii) An improved multiscale method together with a result of de Branges on the existence of limiting values for resolvents in the upper half plane, allowing for rather weak disorder assumptions on the random potential. (iv) Results from the theory of generalized eigenfunctions and spectral averaging. The paper aims at high accessibility in providing details for all the main steps in the proof.
An invitation to random Schrödinger operators
, 2007
"... This review is an extended version of my mini course at the États de la recherche: Opérateurs de Schrödinger aléatoires at the Université Paris 13 in June 2002, a summer school organized by Frédéric Klopp. These lecture notes try to give some of the basics of random Schrödinger operators. They are m ..."
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Cited by 51 (8 self)
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This review is an extended version of my mini course at the États de la recherche: Opérateurs de Schrödinger aléatoires at the Université Paris 13 in June 2002, a summer school organized by Frédéric Klopp. These lecture notes try to give some of the basics of random Schrödinger operators. They are meant for nonspecialists and require only minor previous knowledge about functional analysis and probability theory. Nevertheless this survey includes complete proofs of Lifshitz tails and Anderson localization. Copyright by the author. Copying for academic purposes is permitted.
Multiscale analysis implies strong dynamical localization
, 1999
"... We prove that a strong form of dynamical localization follows from a variable energy multiscale analysis. This abstract result is applied to a number of models for wave propagation in disordered media. ..."
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Cited by 34 (6 self)
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We prove that a strong form of dynamical localization follows from a variable energy multiscale analysis. This abstract result is applied to a number of models for wave propagation in disordered media.
Localization for onedimensional, continuum, BernoulliAnderson models
 Duke Math. J
"... We use scattering theoretic methods to prove strong dynamical and exponential localization for onedimensional, 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 ..."
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Cited by 28 (12 self)
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We use scattering theoretic methods to prove strong dynamical and exponential localization for onedimensional, 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 singlesite 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.
Twoparameter spectral averaging and localization for nonmonotonic random Schrödinger operators
 TRANS. AMER. MATH. SOC
, 2001
"... ..."
Explicit Finite Volume Criteria For Localization In Continuous Random Media And Applications
 GEOM. FUNCT. ANAL
, 2003
"... We give finite volume criteria for localization of quantum or classical waves in continuous random media. We provide explicit conditions, depending on the parameters of the model, for starting the bootstrap multiscale analysis. A simple application yields localization for Anderson Hamiltonians on th ..."
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Cited by 23 (11 self)
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We give finite volume criteria for localization of quantum or classical waves in continuous random media. We provide explicit conditions, depending on the parameters of the model, for starting the bootstrap multiscale analysis. A simple application yields localization for Anderson Hamiltonians on the continuum at the bottom of the spectrum in an interval of size O() for large , where stands for the disorder parameter. A more sophisticated application proves localization for twodimensional random Schrödinger operators in a constant magnetic field (random Landau Hamiltonians) up to a distance O( B ) from the Landau levels, where B is the strength of the magnetic field.
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
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 19 (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
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...