Results 1  10
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103
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, ..."
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Cited by 79 (44 self)
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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.
Operators With Singular Continuous Spectrum, V. Sparse Potentials
 TO APPEAR: PROC. AMER. MATH. SOC.
, 1995
"... By presenting simple theorems for the absence of positive eigenvalues for certain onedimensional Schrödinger operators, we are able to construct explicit potentials which yield purely singular continuous spectrum. ..."
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Cited by 79 (10 self)
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By presenting simple theorems for the absence of positive eigenvalues for certain onedimensional Schrödinger operators, we are able to construct explicit potentials which yield purely singular continuous spectrum.
An Adiabatic Theorem without a gap condition
 Commun. Math. Phys
, 1999
"... We prove an adiabatic theorem for the ground state of the Dicke model in a slowly rotating magnetic field and show that for weak electronphoton coupling, the adiabatic time scale is close to the time scale of the corresponding two level system–without the quantized radiation field. There is a corre ..."
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Cited by 57 (7 self)
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We prove an adiabatic theorem for the ground state of the Dicke model in a slowly rotating magnetic field and show that for weak electronphoton coupling, the adiabatic time scale is close to the time scale of the corresponding two level system–without the quantized radiation field. There is a correction to this time scale which is the Lamb shift of the model. The photon field affect the rate of approach to the adiabatic limit through a logarithmic correction originating from an infrared singularity characteristic of QED.
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 56 (19 self)
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We investigate the Anderson metalinsulator transition for random Schrödinger operators. We define the strong...
Power Law Subordinacy and Singular Spectra  I. Half Line Operators
"... We present an extension of the GilbertPearson theory of subordinacy, which relates Hausdorffdimensional spectral properties of onedimensional Schrödinger operators to the behavior of solutions of the corresponding Schrödinger equation. We use this theory to analyze these properties for concrete ..."
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Cited by 55 (8 self)
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We present an extension of the GilbertPearson theory of subordinacy, which relates Hausdorffdimensional spectral properties of onedimensional Schrödinger operators to the behavior of solutions of the corresponding Schrödinger equation. We use this theory to analyze these properties for concrete examples. In particular, for every ff 2 (0; 1) we construct potentials for which the spectral measure is of exact dimension ff.
αContinuity Properties of OneDimensional Quasicrystals
 COMMUN. MATH. PHYS
, 1997
"... We apply the JitomirskayaLast extension of the GilbertPearson theory to discrete onedimensional Schrödinger operators with potentials arising from generalized Fibonacci sequences. We prove for certain rotation numbers that for every value of the coup ling constant, there exists an α > 0 such t ..."
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Cited by 31 (22 self)
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We apply the JitomirskayaLast extension of the GilbertPearson theory to discrete onedimensional Schrödinger operators with potentials arising from generalized Fibonacci sequences. We prove for certain rotation numbers that for every value of the coup ling constant, there exists an α > 0 such that the corresponding operator has purely ffcontinuous spectrum. This result follows from uniform upper and lower bounds for the k \Delta kLnorm of the solutions corresponding to energies from the spectrum of the operator.
Dynamical Upper Bounds On Wavepacket Spreading
 Am. J. Math
, 2001
"... We derive a general upper bound on the spreading rate of wavepackets in the framework of Schrödinger time evolution. Our result consists of showing that a portion of the wavepacket cannot escape outside a ball whose size grows dynamically in time, where the rate of this growth is determined by prope ..."
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Cited by 31 (2 self)
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We derive a general upper bound on the spreading rate of wavepackets in the framework of Schrödinger time evolution. Our result consists of showing that a portion of the wavepacket cannot escape outside a ball whose size grows dynamically in time, where the rate of this growth is determined by properties of the spectral measure and by spatial properties of solutions of an associated time independent Schrödinger equation. We also derive a new lower bound on the spreading rate, which is strongly connected with our upper bound. We apply these new bounds to the Fibonacci Hamiltonian  the most studied onedimensional model of quasicrystals. As a result, we obtain for this model upper and lower dynamical bounds establishing wavepacket spreading rates which are intermediate between ballistic transport and localization. The bounds have the same qualitative behavior in the limit of large coupling.
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 ..."
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Cited by 30 (17 self)
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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.
Dimensional Hausdorff Properties of Singular Continuous Spectra
, 1995
"... We present an extension of the GilbertPearson theory of subordinacy, which relates dimensional Hausdorff spectral properties of onedimensional Schrödinger operators to the behavior of solutions of the corresponding Schrödinger equation. We use this theory to analyze the dimensional Hausdorff prope ..."
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Cited by 28 (6 self)
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We present an extension of the GilbertPearson theory of subordinacy, which relates dimensional Hausdorff spectral properties of onedimensional Schrödinger operators to the behavior of solutions of the corresponding Schrödinger equation. We use this theory to analyze the dimensional Hausdorff properties for several examples having singularcontinuous spectrum, including sparse barrier potentials, the almost Mathieu operator and the Fibonacci Hamiltonian.
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 ..."
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Cited by 25 (17 self)
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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.