Bloch electron in a magnetic field and the Ising model

The spectral determinant det(H-\epsilon I) of the Azbel-Hofstadter Hamiltonian H is related to Onsager's partition function of the 2D Ising model for any value of magnetic flux \Phi=2\pi P/Q through an elementary cell, where P and Q are coprime integers. The band edges of H correspond to the critical temperature of the Ising model; the spectral determinant at these (and other points defined in a certain similar way) is independent of P. A connection of the mean of Lyapunov exponents to the asymptotic (large Q) bandwidth is indicated.Comment: 4 pages, 1 figure, REVTE

Absolute Continuity of the Integrated Density of States for the Almost Mathieu Operator with Non-Critical Coupling

We show that the integrated density of states of the almost Mathieu operator is absolutely continuous if and only if the coupling is non-critical. We deduce for subcritical coupling that the spectrum is purely absolutely continuous for almost every phase, settling the measure-theoretical case of Problem 6 of Barry Simon's list of Schr\"odinger operator problems for the twenty-first century.Comment: 13 pages, to appear in Inv. Mat

Double butterfly spectrum for two interacting particles in the Harper model

We study the effect of interparticle interaction $U$ on the spectrum of the Harper model and show that it leads to a pure-point component arising from the multifractal spectrum of non interacting problem. Our numerical studies allow to understand the global structure of the spectrum. Analytical approach developed permits to understand the origin of localized states in the limit of strong interaction $U$ and fine spectral structure for small $U$.Comment: revtex, 4 pages, 5 figure

Continuity of the measure of the spectrum for quasiperiodic Schrodinger operators with rough potentials

We study discrete quasiperiodic Schr\"odinger operators on \ell^2(\zee) with potentials defined by $\gamma$-H\"older functions. We prove a general statement that for $\gamma >1/2$ and under the condition of positive Lyapunov exponents, measure of the spectrum at irrational frequencies is the limit of measures of spectra of periodic approximants. An important ingredient in our analysis is a general result on uniformity of the upper Lyapunov exponent of strictly ergodic cocycles.Comment: 15 page

Suppressed spin dephasing for 2D and bulk electrons in GaAs wires due to engineered cancellation of spin-orbit interaction terms

We report a study of suppressed spin dephasing for quasi-one-dimensional electron ensembles in wires etched into a GaAs/AlGaAs heterojunction system. Time-resolved Kerr-rotation measurements show a suppression that is most pronounced for wires along the [110] crystal direction. This is the fingerprint of a suppression that is enhanced due to a strong anisotropy in spin-orbit fields that can occur when the Rashba and Dresselhaus contributions are engineered to cancel each other. A surprising observation is that this mechanisms for suppressing spin dephasing is not only effective for electrons in the heterojunction quantum well, but also for electrons in a deeper bulk layer.Comment: 5 pages, 3 figure

The Ten Martini Problem

We prove the conjecture (known as the ``Ten Martini Problem'' after Kac and Simon) that the spectrum of the almost Mathieu operator is a Cantor set for all non-zero values of the coupling and all irrational frequencies.Comment: 31 pages, no figure

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

Strong resonant tunneling, level repulsion and spectral type for one-dimensional adiabatic quasi-periodic Schr\"{o}dinger operators

In this paper, we consider one dimensional adiabatic quasi-periodic Schr\"{o}dinger operators in the regime of strong resonant tunneling. We show the emergence of a level repulsion phenomenon which is seen to be very naturally related to the local spectral type of the operator: the more singular the spectrum, the weaker the repulsion

On semiclassical dispersion relations of Harper-like operators

We describe some semiclassical spectral properties of Harper-like operators, i.e. of one-dimensional quantum Hamiltonians periodic in both momentum and position. The spectral region corresponding to the separatrices of the classical Hamiltonian is studied for the case of integer flux. We derive asymptotic formula for the dispersion relations, the width of bands and gaps, and show how geometric characteristics and the absence of symmetries of the Hamiltonian influence the form of the energy bands.Comment: 13 pages, 8 figures; final version, to appear in J. Phys. A (2004