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32
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.
A characterization of the Anderson metalinsulator transport transition
 Duke Math. J
"... We investigate the Anderson metalinsulator transition for random Schrödinger operators. We define the strong... ..."
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Cited by 41 (17 self)
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We investigate the Anderson metalinsulator transition for random Schrödinger operators. We define the strong...
Localization of Classical Waves I: Acoustic Waves.
 Commun. Math. Phys
, 1996
"... We consider classical acoustic waves in a medium described by a position dependent mass density %(x). We assume that %(x) is a random perturbation of a periodic function % 0 (x) and that the periodic acoustic operator A 0 = \Gammar \Delta 1 %0 (x) r has a gap in the spectrum. We prove the existe ..."
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Cited by 37 (0 self)
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We consider classical acoustic waves in a medium described by a position dependent mass density %(x). We assume that %(x) is a random perturbation of a periodic function % 0 (x) and that the periodic acoustic operator A 0 = \Gammar \Delta 1 %0 (x) r has a gap in the spectrum. We prove the existence of localized waves, i.e., finite energy solutions of the acoustic equations with the property that almost all of the wave's energy remains in a fixed bounded region of space at all times, with probability one. Localization of acoustic waves is a consequence of Anderson localization for the selfadjoint operators A = \Gammar \Delta 1 %(x) r on L 2 (R d ). We prove that, in the random medium described by %(x), the random operator A exhibits Anderson localization inside the gap in the spectrum of A 0 . This is shown even in situations when the gap is totally filled by the spectrum of the random operator; we can prescribe random environments that ensure localization in almost the wh...
Existence of the density of states for multidimensional continuum Schrödinger operators with Gaussian random potentials
, 1997
"... : A Wegner estimate is proved for quantum systems in multidimensional Euclidean space which are characterized by oneparticle Schrodinger operators with random potentials that admit a certain oneparameter decomposition. In particular, the Wegner estimate applies to systems with rather general Gaus ..."
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Cited by 21 (4 self)
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: A Wegner estimate is proved for quantum systems in multidimensional Euclidean space which are characterized by oneparticle Schrodinger operators with random potentials that admit a certain oneparameter decomposition. In particular, the Wegner estimate applies to systems with rather general Gaussian random potentials. As a consequence, these systems possess an absolutely continuous integrated density of states, whose derivative, the density of states, is locally bounded. An explicit upper bound is derived. 1. Introduction The integrated density of states is a quantity of primary interest in the theory and applications of oneparticle random Schrodinger operators [SE, BEE+, LGP, CL, PF]. For example, the topological support of the associated measure coincides with the almostsure spectrum of the infinitevolume operator. Moreover, its knowledge allows to compute the free energy and hence all basic thermostatic quantities of the corresponding noninteracting manyparticle system. An ...
The Absolute Continuity of the Integrated Density of States for Magnetic Schrödinger Operators with Certain Unbounded Random Potentials
, 2001
"... ..."
Localization of Classical Waves II: Electromagnetic Waves.
 Commun. Math. Phys
, 1997
"... We consider electromagnetic waves in a medium described by a position dependent dielectric constant "(x). We assume that "(x) is a random perturbation of a periodic function " 0 (x) and that the periodic Maxwell operator M 0 = r \Theta 1 " 0 (x) r \Theta has a gap in the spectrum, were r \Thet ..."
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Cited by 20 (0 self)
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We consider electromagnetic waves in a medium described by a position dependent dielectric constant "(x). We assume that "(x) is a random perturbation of a periodic function " 0 (x) and that the periodic Maxwell operator M 0 = r \Theta 1 " 0 (x) r \Theta has a gap in the spectrum, were r \Theta \Psi = r\Theta\Psi. We prove the existence of localized waves, i.e., finite energy solutions of Maxwell's equations with the property that almost all of the wave's energy remains in a fixed bounded region of space at all times. Localization of electromagnetic waves is a consequence of Anderson localization for the selfadjoint operators M = r \Theta 1 "(x) r \Theta . We prove that, in the random medium described by "(x), the random operator M exhibits Anderson localization inside the gap in the spectrum of M 0 . This is shown even in situations when the gap is totally filled by the spectrum of the random operator; we can prescribe random environments that ensure localization in almo...
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
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...
S.: On Bernoulli Decompositions for Random Variables, Concentration Bounds, and Spectral Localization
"... Abstract. As was noted already by A. N. Kolmogorov, any random variable has a Bernoulli component. This observation provides a tool for the extension of results which are known for Bernoulli random variables to arbitrary distributions. Two applications are provided here: i. an anticoncentration bou ..."
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Cited by 13 (2 self)
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Abstract. As was noted already by A. N. Kolmogorov, any random variable has a Bernoulli component. This observation provides a tool for the extension of results which are known for Bernoulli random variables to arbitrary distributions. Two applications are provided here: i. an anticoncentration bound for a class of functions of independent random variables, where probabilistic bounds are extracted from combinatorial results, and ii. a proof, based on the Bernoulli case, of spectral localization for random Schrödinger operators with arbitrary probability distributions for the single site coupling constants. For a general random variable, the Bernoulli component may be defined so that its conditional variance is uniformly positive. The natural maximization problem is an optimal transport question which is also addressed here.