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64
Moment Analysis for Localization in Random Schrödinger Operators
, 2005
"... We study localization effects of disorder on the spectral and dynamical properties of Schrödinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the ..."
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Cited by 63 (15 self)
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We study localization effects of disorder on the spectral and dynamical properties of Schrödinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonance-diffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. The difficulty is avoided here through the use of a weak-L¹ estimate concerning the boundary-value distribution of resolvents of maximally dissipative operators, combined with standard tools of relative compactness theory.
A characterization of the Anderson metal-insulator transport transition
- Duke Math. J
"... We investigate the Anderson metal-insulator transition for random Schrödinger operators. We define the strong... ..."
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Cited by 56 (19 self)
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We investigate the Anderson metal-insulator transition for random Schrödinger operators. We define the strong...
Multi-scale analysis implies strong dynamical localization
, 1999
"... We prove that a strong form of dynamical localization follows from a variable energy multi-scale analysis. This abstract result is applied to a number of models for wave propagation in disordered media. ..."
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Cited by 48 (7 self)
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We prove that a strong form of dynamical localization follows from a variable energy multi-scale analysis. This abstract result is applied to a number of models for wave propagation in disordered media.
Internal Lifshits Tails For Random Perturbations Of Periodic Schrödinger Operators
- Duke Mathematical J
, 1999
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The integrated density of states for random Schrödinger operators
"... 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 i ..."
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Cited by 43 (4 self)
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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
Anderson Localization for Random Schrödinger Operators with Long Range Interactions
- COMM. MATH. PHYS
, 1998
"... We prove pure point spectrum at all band edges for Schrödinger Operators with a periodic potential plus a random potential of the form V ! (x) = P q i (!)f(x \Gamma i) where f decays at infinity like jxj \Gammam for m ? 3d resp. m ? 2d depending on the regularity of f . Eigenfunctions are shown ..."
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Cited by 42 (18 self)
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We prove pure point spectrum at all band edges for Schrödinger Operators with a periodic potential plus a random potential of the form V ! (x) = P q i (!)f(x \Gamma i) where f decays at infinity like jxj \Gammam for m ? 3d resp. m ? 2d depending on the regularity of f . Eigenfunctions are shown to decay more rapidly than every inverse polynomial. The random variables q i are supposed to be independent and identically distributed. We suppose that their distribution has a bounded density of compact support.
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 30 (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 two-dimensional 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.
Weak Disorder Localization And Lifshitz Tails: Continuous Hamiltonians
- Ann. Henri Poincaré
"... This paper is devoted to the study of band edge localization for continuous random Schrödinger operators with weak random perturbations. We prove that, in the weak disorder regime, small, the spectrum in intervals of size at a non-degenerate simple band edge is exponentially and dynamically localize ..."
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Cited by 28 (0 self)
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This paper is devoted to the study of band edge localization for continuous random Schrödinger operators with weak random perturbations. We prove that, in the weak disorder regime, small, the spectrum in intervals of size at a non-degenerate simple band edge is exponentially and dynamically localized. Upper bounds on the localization length in these energy regions are also obtained. Our results rely on the analysis of Lifshitz tails when the disorder is small; the single site potential need not be of fixed sign.
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 disordered-broadened Landau band, this implies the existence of at least one dynamical mobility edge at each Landau band, n ..."
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Cited by 27 (8 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 disordered-broadened 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.
Multiscale analysis and localization of random operators
- In Random Schrodinger operators: methods, results, and perspectives. Panorama & Synthèse, Société Mathématique de
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