316 research outputs found
Rotation Numbers, Boundary Forces and Gap labelling
We review the Johnson-Moser rotation number and the -theoretical gap
labelling of Bellissard for one-dimensional Schr\"odinger operators. We compare
them with two further gap-labels, one being related to the motion of Dirichlet
eigenvalues, the other being a -theoretical gap label. We argue that the
latter provides a natural generalisation of the Johnson-Moser rotation number
to higher dimensions.Comment: 10 pages, version accepted for publicatio
Topological quantization of boundary forces and the integrated density of states
For quantum systems described by Schr\"odinger operators on the half-space
\RR^{d-1}\times\RR^{leq 0} the boundary force per unit area and unit energy
is topologically quantised provided the Fermi energy lies in a gap of the bulk
spectrum. Under this condition it is also equal to the integrated density of
states at the Fermi energy.Comment: 7 page
The Local Semicircle Law for Random Matrices with a Fourfold Symmetry
We consider real symmetric and complex Hermitian random matrices with the
additional symmetry . The matrix elements are independent
(up to the fourfold symmetry) and not necessarily identically distributed. This
ensemble naturally arises as the Fourier transform of a Gaussian orthogonal
ensemble (GOE). It also occurs as the flip matrix model - an approximation of
the two-dimensional Anderson model at small disorder. We show that the density
of states converges to the Wigner semicircle law despite the new symmetry type.
We also prove the local version of the semicircle law on the optimal scale.Comment: 20 pages, to appear in J. Math. Phy
Spectrum and diffusion for a class of tight-binding models on hypercubes
We propose a class of exactly solvable anisotropic tight-binding models on an
infinite-dimensional hypercube. The energy spectrum is analytically computed
and is shown to be fractal and/or absolutely continuous according to the value
hopping parameters. In both cases, the spectral and diffusion exponents are
derived. The main result is that, even if the spectrum is absolutely
continuous, the diffusion exponent for the wave packet may be anything between
0 and 1 depending upon the class of models.Comment: 5 pages Late
Almost Sure Frequency Independence of the Dimension of the Spectrum of Sturmian Hamiltonians
We consider the spectrum of discrete Schr\"odinger operators with Sturmian
potentials and show that for sufficiently large coupling, its Hausdorff
dimension and its upper box counting dimension are the same for Lebesgue almost
every value of the frequency.Comment: 12 pages, to appear in Commun. Math. Phy
Spectra of Discrete Schr\"odinger Operators with Primitive Invertible Substitution Potentials
We study the spectral properties of discrete Schr\"odinger operators with
potentials given by primitive invertible substitution sequences (or by Sturmian
sequences whose rotation angle has an eventually periodic continued fraction
expansion, a strictly larger class than primitive invertible substitution
sequences). It is known that operators from this family have spectra which are
Cantor sets of zero Lebesgue measure. We show that the Hausdorff dimension of
this set tends to as coupling constant tends to . Moreover, we
also show that at small coupling constant, all gaps allowed by the gap labeling
theorem are open and furthermore open linearly with respect to .
Additionally, we show that, in the small coupling regime, the density of states
measure for an operator in this family is exact dimensional. The dimension of
the density of states measure is strictly smaller than the Hausdorff dimension
of the spectrum and tends to as tends to
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