Results 1  10
of
32
Quantum Dynamics and Decompositions of Singular Continuous Spectra
 J. Funct. Anal
, 1995
"... . We study relations between quantum dynamics and spectral properties, concentrating on spectral decompositions which arise from decomposing measures with respect to dimensional Hausdorff measures. 1. Introduction Let H be a separable Hilbert space, H : H ! H a self adjoint operator, and / 2 H (wi ..."
Abstract

Cited by 79 (11 self)
 Add to MetaCart
. We study relations between quantum dynamics and spectral properties, concentrating on spectral decompositions which arise from decomposing measures with respect to dimensional Hausdorff measures. 1. Introduction Let H be a separable Hilbert space, H : H ! H a self adjoint operator, and / 2 H (with k/k = 1). The spectral measure ¯/ of / (and H ) is uniquely defined by [24]: h/ ; f(H)/i = Z oe(H) f(x) d¯/ (x) ; (1:1) for any measurable (Borel) function f . The time evolution of the state / , in the Schrodinger picture of quantum mechanics, is given by /(t) = e \GammaiHt / : (1:2) The relations between various properties of the spectral measure ¯/ (with an emphasis on "fractal" properties) and the nature of the time evolution have been the subject of several recent papers [7,13,1518,20,22,33,36,39]. Our purpose in this paper is twofold: First, we use a theory, due to Rogers and Taylor [28,29], of decomposing singular continuous measures with respect to Hausdorff measures to i...
Uniform Spectral Properties Of OneDimensional Quasicrystals, IV. QuasiSturmian Potentials
 I. Absence of eigenvalues, Commun. Math. Phys
, 2000
"... We consider discrete onedimensional Schrodinger operators with quasiSturmian potentials. We present a new approach to the trace map dynamical system which is independent of the initial conditions and establish a characterization of the spectrum in terms of bounded trace map orbits. Using this, ..."
Abstract

Cited by 51 (32 self)
 Add to MetaCart
We consider discrete onedimensional Schrodinger operators with quasiSturmian potentials. We present a new approach to the trace map dynamical system which is independent of the initial conditions and establish a characterization of the spectrum in terms of bounded trace map orbits. Using this, it is shown that the operators have purely singular continuous spectrum and their spectrum is a Cantor set of Lebesgue measure zero. We also exhibit a subclass having purely ffcontinuous spectrum. All these results hold uniformly on the hull generated by a given potential.
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... ..."
Abstract

Cited by 41 (17 self)
 Add to MetaCart
We investigate the Anderson metalinsulator transition for random Schrödinger operators. We define the strong...
spectrum, and dynamics for Schrödinger operators on infinite domains
 Duke Math. J
"... 1. Introduction and main results. In this paper we investigate the relations between the rate of decay of solutions of Schrödinger equations, continuity properties of spectral measures of the corresponding operators, and dynamical properties of the corresponding quantum systems. The first main resul ..."
Abstract

Cited by 21 (5 self)
 Add to MetaCart
1. Introduction and main results. In this paper we investigate the relations between the rate of decay of solutions of Schrödinger equations, continuity properties of spectral measures of the corresponding operators, and dynamical properties of the corresponding quantum systems. The first main result of this paper shows that, in great generality, certain upper bounds on the rate of growth of L2 norms of generalized
Fractal dimensions and the phenomenon of intermittency in quantum dynamics
 Duke Math. J
"... We exhibit an intermittency phenomenon in quantum dynamics. More precisely, we derive new lower bounds for the moments of order p associated to the state ψ(t) = e−it H ψ and averaged in time between zero and T. These lower bounds are expressed in terms of generalized fractal dimensions D ± () µψ 1/ ..."
Abstract

Cited by 17 (9 self)
 Add to MetaCart
We exhibit an intermittency phenomenon in quantum dynamics. More precisely, we derive new lower bounds for the moments of order p associated to the state ψ(t) = e−it H ψ and averaged in time between zero and T. These lower bounds are expressed in terms of generalized fractal dimensions D ± () µψ 1/(1 + p/d) of the measure µψ (where d is the space dimension). This improves previous results obtained in terms of Hausdorff and Packing dimension. 1.
Anomalous transport: A mathematical framework
 MR 99b:81046 162
, 1998
"... We develop a mathematical framework allowing to study anomalous transport in homogeneous solids. The main tools characterizing the anomalous transport properties are spectral and diffusion exponents associated to the covariant Hamiltonians describing these media. The diffusion exponents characterize ..."
Abstract

Cited by 15 (6 self)
 Add to MetaCart
We develop a mathematical framework allowing to study anomalous transport in homogeneous solids. The main tools characterizing the anomalous transport properties are spectral and diffusion exponents associated to the covariant Hamiltonians describing these media. The diffusion exponents characterize the spectral measures entering in Kubo’s formula for the conductivity and hence lead to anomalies in Drude’s formula. We give several formulas allowing to calculate these exponents and treat, as an example, Wegner’s norbital model as well as the Anderson model in coherent potential approximation. 1
Generalized Fractal Dimensions: Equivalences and Basic Properties
, 2000
"... Given a positive probability Borel measure , we establish some basic properties of the associated functions (q) and of the generalized fractal dimensions D for q 2 R. We rst give the connections between the generalized fractal dimensions, the Renyi dimensions and the meanq dimensions when q > ..."
Abstract

Cited by 14 (7 self)
 Add to MetaCart
Given a positive probability Borel measure , we establish some basic properties of the associated functions (q) and of the generalized fractal dimensions D for q 2 R. We rst give the connections between the generalized fractal dimensions, the Renyi dimensions and the meanq dimensions when q > 0. We then use these relations to prove some regularity properties for (q); we also provide some estimates for these functions (in particular estimates on their behaviour at 1), as well as for the dimensions corresponding to convolution of two measures. We nally present some calculations for speci c examples. 1
Strictly ergodic subshifts and associated operators
, 2005
"... We consider ergodic families of Schrödinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. It is expected that the majority of these operators have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure. These properties have in ..."
Abstract

Cited by 13 (8 self)
 Add to MetaCart
We consider ergodic families of Schrödinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. It is expected that the majority of these operators have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure. These properties have indeed been established for large classes of operators of this type over the course of the last twenty years. We review the mechanisms leading to these results and briefly discuss analogues for CMV matrices.
Floquet Hamiltonians with pure point spectrum
, 1996
"... . We consider Floquet Hamiltonians of the type K F := \Gammai@ t + H 0 + fiV (!t) where H 0 , a selfadjoint operator acting in a Hilbert space H, has simple discrete spectrum E 1 ! E 2 ! : : : obeying a gap condition of the type inffn \Gammaff (E n+1 \Gamma E n ); n = 1; 2; :::g ? 0 for a given ..."
Abstract

Cited by 12 (5 self)
 Add to MetaCart
. We consider Floquet Hamiltonians of the type K F := \Gammai@ t + H 0 + fiV (!t) where H 0 , a selfadjoint operator acting in a Hilbert space H, has simple discrete spectrum E 1 ! E 2 ! : : : obeying a gap condition of the type inffn \Gammaff (E n+1 \Gamma E n ); n = 1; 2; :::g ? 0 for a given ff ? 0, t 7! V (t) is 2ßperiodic and r times strongly continuously differentiable as a bounded operator on H, ! and fi are real parameters and the periodic boundary condition is imposed in time. We show, roughly, that provided r is large enough, fi small enough and ! nonresonant then the spectrum of K F is pure point. The method we use relies on a successive application of the adiabatic treatment due to Howland and the KAMtype iteration settled by Bellissard and extended by Combescure. Both tools are revisited, adjusted and at some points slightly simplified. 1 Introduction Spectral analysis of Floquet Hamiltonians or, equivalently, Floquet operators [7, 17] is known to be a tool to inve...
PowerLaw Bounds On Transfer Matrices And Quantum Dynamics In One Dimension
"... We present an approach to quantum dynamical lower bounds for discrete onedimensional Schrodinger operators which is based on powerlaw bounds on transfer matrices. It suces to have such bounds for a nonempty set of energies. We apply this result to various models, including the Fibonacci Hamil ..."
Abstract

Cited by 12 (7 self)
 Add to MetaCart
We present an approach to quantum dynamical lower bounds for discrete onedimensional Schrodinger operators which is based on powerlaw bounds on transfer matrices. It suces to have such bounds for a nonempty set of energies. We apply this result to various models, including the Fibonacci Hamiltonian.