Results 1  10
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21
Localization Near Band Edges For Random Schrödinger Operators
, 1997
"... In this article, we prove exponential localization for wide classes of Schrödinger operators, including those with magnetic fields, at the edges of unperturbed spectral gaps. We assume that the unperturbed operator H 0 has an open gap I 0 j (B \Gamma ; B+ ). The random potential is assumed to be And ..."
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Cited by 80 (12 self)
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In this article, we prove exponential localization for wide classes of Schrödinger operators, including those with magnetic fields, at the edges of unperturbed spectral gaps. We assume that the unperturbed operator H 0 has an open gap I 0 j (B \Gamma ; B+ ). The random potential is assumed to be Andersontype with independent, identically distributed coupling constants. The common density may have either bounded or unbounded support. For either case, we prove that there exists an interval of energies in the unperturbed gap for which the almost sure spectrum of the family H ! j H 0 +V ! is dense pure point with exponentially decaying eigenfunctions. We also prove that the integrated density of states is Lipschitz continuous in the unperturbed spectral gap I 0 .
Localization Bounds for an Electron Gas
, 1998
"... Mathematical analysis of the Anderson localization has been facilitated by the use of suitable fractional moments of the Green function. Related methods permit now a readily accessible derivation of a number of physical manifestations of localization, in regimes of strong disorder, extreme energies ..."
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Cited by 48 (8 self)
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Mathematical analysis of the Anderson localization has been facilitated by the use of suitable fractional moments of the Green function. Related methods permit now a readily accessible derivation of a number of physical manifestations of localization, in regimes of strong disorder, extreme energies, or weak disorder away from the unperturbed spectrum. The present work establishes on this basis exponential decay for the modulus of the two–point function, at all temperatures as well as in the ground state, for a Fermi gas within the one–particle approximation. Different implications, in particular for the Integral Quantum Hall Effect, are reviewed.
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 in general onedimensional random systems
 II. Continuum Schrodinger operators, Commun. Math. Phys
, 1987
"... Dedicated to Walter Thirrίng on his 60 th birthday Abstract. We discuss two ways of extending the recent ideas of localization from discrete Schrodinger operators (Jacobi matrices) to the continuum case. One case allows us to prove localization in the Goldshade, Molchanov, Pastur model for a larger ..."
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Cited by 37 (2 self)
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Dedicated to Walter Thirrίng on his 60 th birthday Abstract. We discuss two ways of extending the recent ideas of localization from discrete Schrodinger operators (Jacobi matrices) to the continuum case. One case allows us to prove localization in the Goldshade, Molchanov, Pastur model for a larger class of functions than previously. The other method studies the model — A + V, where V is a random constant in each (hyper) cube. We extend Wegner's result on the Lipschitz nature of the ids to this model. 1.
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 ...
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
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 !.