Results 1 
9 of
9
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 57 (19 self)
 Add to MetaCart
We investigate the Anderson metalinsulator transition for random Schrödinger operators. We define the strong...
Lower transport bounds for onedimensional continuum Schrödinger operators
"... We prove quantum dynamical lower bounds for onedimensional continuum Schrödinger operators that possess critical energies for which there is slow growth of transfer matrix norms and a large class of compactly supported initial states. This general result is applied to a number of models, including ..."
Abstract

Cited by 14 (5 self)
 Add to MetaCart
(Show Context)
We prove quantum dynamical lower bounds for onedimensional continuum Schrödinger operators that possess critical energies for which there is slow growth of transfer matrix norms and a large class of compactly supported initial states. This general result is applied to a number of models, including the BernoulliAnderson model with a constant singlesite potential.
Imbedded Singular Continuous Spectrum For Schrödinger Operators
 Evolutionary Algorithms in Engineering Applications
, 1997
"... We construct examples of potentials V (x) satisfying jV (x)j where the function h(x) is growing arbitrarily slowly, such that the corresponding Schrödinger operator has imbedded singular continuous spectrum. This solves one of the fifteen "twentyfirst century" problems for Schrödinger ope ..."
Abstract

Cited by 13 (3 self)
 Add to MetaCart
(Show Context)
We construct examples of potentials V (x) satisfying jV (x)j where the function h(x) is growing arbitrarily slowly, such that the corresponding Schrödinger operator has imbedded singular continuous spectrum. This solves one of the fifteen "twentyfirst century" problems for Schrödinger operators posed by Barry Simon in [22]. The construction also provides the first example of a Schrödinger operator for which Möller wave operators exist but are not asymptotically complete due to the presence of singular continuous spectrum.
FourierBessel Functions of Singular Continuous Measures and their
 Many Asymptotics, Electronic Transactions on Numerical Analysis
, 2006
"... Abstract. We study the Fourier transform of polynomials in an orthogonal family, taken with respect to the orthogonality measure. Mastering the asymptotic properties of these transforms, that we call Fourier–Bessel functions, in the argument, the order, and in certain combinations of the two is requ ..."
Abstract

Cited by 11 (8 self)
 Add to MetaCart
(Show Context)
Abstract. We study the Fourier transform of polynomials in an orthogonal family, taken with respect to the orthogonality measure. Mastering the asymptotic properties of these transforms, that we call Fourier–Bessel functions, in the argument, the order, and in certain combinations of the two is required to solve a number of problems arising in quantum mechanics. We present known results, new approaches and open conjectures, hoping to justify our belief that the importance of these investigations extends beyond the application just mentioned, and may involve interesting discoveries. Key words. Singular measures, Fourier transform, orthogonal polynomials, almost periodic Jacobi matrices, FourierBessel functions, quantum intermittency, Julia sets, iterated function systems, generalized dimensions, potential theory. AMS subject classifications. 42C05, 33E20, 28A80, 30E15, 30E20 1. Introduction and examples
Dynamical Analysis of Schrödinger Operators with Growing Sparse
 Potentials, Commun. Math. Phys
, 2005
"... Consider the discrete Schrodinger operators in l2(Z+): H (n) = (n 1) + (n + 1) + V (n) (n); (1.1) where V (n) is some real function, with boundary condition (0)cos + (1)sin = 0; 2 (=2; =2): (1.2) ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
(Show Context)
Consider the discrete Schrodinger operators in l2(Z+): H (n) = (n 1) + (n + 1) + V (n) (n); (1.1) where V (n) is some real function, with boundary condition (0)cos + (1)sin = 0; 2 (=2; =2): (1.2)
GENERALIZED FRACTAL DIMENSIONS ON THE NEGATIVE AXIS FOR NON COMPACTLY SUPPORTED MEASURES
"... Abstract. We study the finiteness of the generalized fractal dimensions D ± µ (q) (also called HentschelProcaccia dimensions) for a non compactly supported measure µ on a complete metric space, and for q < 0. The upper dimensions are shown to be always infinite. We then provide a sufficient cond ..."
Abstract
 Add to MetaCart
Abstract. We study the finiteness of the generalized fractal dimensions D ± µ (q) (also called HentschelProcaccia dimensions) for a non compactly supported measure µ on a complete metric space, and for q < 0. The upper dimensions are shown to be always infinite. We then provide a sufficient condition for the lower dimensions to bemeasures infinite. Optimality of our theorems is proved by constructing explicit measures on R. 1. Introduction and
DYNAMICAL LOWER BOUNDS FOR 1D DIRAC OPERATORS
, 708
"... Abstract. Quantum dynamical lower bounds for continuous and discrete onedimensional Dirac operators are established in terms of transfer matrices. Then such results are applied to various models, including the BernoulliDirac one and, in contrast to the discrete case, critical ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. Quantum dynamical lower bounds for continuous and discrete onedimensional Dirac operators are established in terms of transfer matrices. Then such results are applied to various models, including the BernoulliDirac one and, in contrast to the discrete case, critical
SPECTRAL AND LOCALIZATION PROPERTIES FOR THE ONEDIMENSIONAL BERNOULLI DISCRETE DIRAC OPERATOR
, 2005
"... Abstract. A 1D Dirac tightbinding model is considered and it is shown that its nonrelativistic limit is the 1D discrete Schrödinger model. For random Bernoulli potentials taking two values (without correlations), for typical realizations and for all values of the mass, it is shown that its spectrum ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. A 1D Dirac tightbinding model is considered and it is shown that its nonrelativistic limit is the 1D discrete Schrödinger model. For random Bernoulli potentials taking two values (without correlations), for typical realizations and for all values of the mass, it is shown that its spectrum is pure point, whereas the zero mass case presents dynamical delocalization for specific values of the energy. The massive case presents dynamical localization (excluding some particular values of the energy). Finally, for general potentials the dynamical moments for distinct masses are compared, especially the massless and massive Bernoulli cases. 1.