Continuity of the Lyapunov Exponent for Quasiperiodic Operators with Analytic Potential

We study regularity properties of the Lyapunov exponent L of one-frequency quasiperiodic operators with analytic potential, under no assumptions on the Diophantine class of the frequency. We prove joint continuity of L, in frequency and energy, at every irrational frequency.     

Uniform Spectral Properties of One-Dimensional Quasicrystals, I. Absence of Eigenvalues

We consider discrete one-dimensional Schr?dinger operators with Sturmian potentials. For a full-measure set of rotation numbers including the Fibonacci case, we prove absence of eigenvalues for all elements in the hull. Received: 7 January 1999 / Accepted: 12 May 1999     

Singular Continuous Spectrum for a Class of Substitution Hamiltonians II

We consider discrete one-dimensional Schrödinger operators with aperiodic potentials generated by primitive substitutions. Using the three-block version of Gordon's criterion, we establish purely singular continuous spectrum with probability one provided that the potentials have index greater than three. It is also shown that one cannot use this criterion to prove uniform results.     

Adiabatic invariants and asymptotic behavior of Lyapunov exponents of the Schrödinger equation

We give an upper bound for the high-energy behavior of the Lyapunov exponent of the one-dimensional Schrödinger equation. We relate this behavior to the diffrentiability properties of the potential. As an application, this result provides an upper bound for the asymptotic length of the gaps of the Schrödinger equation.     

Spectral Localization by Gaussian Random Potentials in Multi-Dimensional Continuous Space

A detailed mathematical proof is given that the energy spectrum of a non-relativistic quantum particle in multi-dimensional Euclidean space under the influence of suitable random potentials has almost surely a pure-point component. The result applies in particular to a certain class of zero-mean Gaussian random potentials, which are homogeneous with respect to Euclidean translations. More precisely, for these Gaussian random potentials the spectrum is almost surely only pure point at sufficiently negative energies or, at negative energies, for sufficiently weak disorder. The proof is based on a fixed-energy multi-scale analysis which allows for different random potentials on different length scales.     

Localization for random and quasiperiodic potentials

A survey is made of some recent mathematical results and techniques for Schrödinger operators with random and quasiperiodic potentials. A new proof of localization for random potentials, established in collaboration with H. von Dreifus, is sketched.     

Perturbation expansion for a one-dimensional Anderson model with off-diagonal disorder

The weak disorder expansion for a random Schrödinger equation with off-diagonal disorder in one dimension is studied. The invariant measure, the density of states, and the Lyapunov exponent are computed. The most interesting feature in this model appears at the band center, where the differentiated density of states diverges, while the Lyapunov exponent vanishes. The invariant measure approaches an atomic measure concentrated on zero and infinity. The results extend previous work of Markos to all orders of perturbation theory.     

Singular Continuous Spectrum for a Class of Substitution Hamiltonians

We consider discrete one-dimensional Schrödinger operators with potentials generated by primitive substitutions. A purely singular continuous spectrum with probability one is established provided that the potentials have a local four-block structure.     

Singular continuous spectrum on a cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian

It is rigorously proven that the spectrum of the tight-binding Fibonacci Hamiltonian,H _mn= _{m, n+1}+ _{m, n–1}+ _{m, n} [(n+1)]–[n]) where =(5–1)/2 and [·] means integer part, is a Cantor set of zero Lebesgue measure for all real nonzero, and the spectral measures are purely singular continuous. This follows from a recent result by Kotani, coupled with the vanishing of the Lyapunov exponent in the spectrum.On leave from the Central Research Institute for Physics, Budapest, Hungary.     

The fate of lifshits tails in magnetic fields

We investigate the integrated density of states of the Schrödinger operator in the Euclidean plane with a perpendicular constant magnetic field and a random potential. For a Poisson random potential with a nonnegative, algebraically decaying, single-impurity potential we prove that the leading asymptotic behavior for small energies is always given by the corresponding classical result, in contrast to the case of vanishing magnetic field. We also show that the integrated density of states of the operator restricted to the eingenspace of any Landau level exhibits the same behavior. For the lowest Landau level, this is in sharp contrast to the case of a Poisson random potential with a delta-function impurity potential.     

On the Number of Negative Eigenvalues of a One-Dimensional Schrödinger Operator with Point Interactions

An effective algorithm is provided for determining the number of negative eigenvalues of a one-dimensional Schrödinger operator with point interactions in terms of the intensities and the distances between the interactions.     

Front Propagation Techniques to Calculate the Largest Lyapunov Exponent of Dilute Hard Disk Gases

A kinetic approach is adopted to describe the exponential growth of a small deviation of the initial phase space point, measured by the largest Lyapunov exponent, for a dilute system of hard disks, both in equilibrium and in a uniform shear flow. We derive a generalized Boltzmann equation for an extended one-particle distribution that includes deviations from the reference phase space point. The equation is valid for very low densities n, and requires an unusual expansion in powers of 1/|ln n|. It reproduces and extends results from the earlier, more heuristic clock model and may be interpreted as describing a front propagating into an unstable state. The asymptotic speed of propagation of the front is proportional to the largest Lyapunov exponent of the system. Its value may be found by applying the standard front speed selection mechanism for pulled fronts to the case at hand. For the equilibrium case, an explicit expression for the largest Lyapunov exponent is given and for sheared systems we give explicit expressions that may be evaluated numerically to obtain the shear rate dependence of the largest Lyapunov exponent.     

The Band-Edge Behavior of the Density of Surfacic States

This paper is devoted to the asymptotics of the density of surfacic states near the spectral edges for a discrete surfacic Anderson model. Two types of spectral edges have to be considered: fluctuating edges and stable edges. Each type has its own type of asymptotics. In the case of fluctuating edges, one obtains Lifshitz tails the parameters of which are given by the initial operator suitably 'reduced' to the surface. For stable edges, the surface density of states behaves like the surface density of states of a constant (equal to the expectation of the random potential) surface potential. Among the tools used to establish this are the asymptotics of the surface density of states for constant surface potentials.Mathematics Subject Classifications (2000) 35P20, 46N50, 47B80.     

The fractionary Schrödinger equation,Green functions and ultradistributions

In this work, we generalize previous results about the Fractionary Schrödinger Equation within the formalism of the theory of Tempered Ultradistributions. Several examples of the use of this theory are given. In particular we evaluate the Green function for a free particle in the general case, for an arbitrary order of the derivative index.     

Scattering of radiation by a quasiperiodic two-dimensional medium

We study the scattering of radiation by a medium presenting inhomogeneities distributed in a quasiperiodic way. We show the existence of quasiperiodic solutions of the two-dimensional stationary wave equation, under certain conditions on the index of refraction, using a technique based on Dinaburg-Sinai method for one-dimensional Schrödinger equation with a quasiperiodic potential. Moreover we show that the energy spctrum contains a nonempty absolutely continuous component, with a subset having high degeneracy, provided the inhomogeneities are small enough.     

Statistical mechanics of the nonlinear Schrödinger equation. II. Mean field approximation

We investigate a mean field approximation to the statistical mechanics of complex fields with dynamics governed by the nonlinear Schrödinger equation. Such fields, whose Hamiltonian is unbounded below, may model plasmas, lasers, and other physical systems. Restricting ourselves to one-dimensional systems with periodic boundary conditions, we find in the mean field approximation a phase transition from a uniform regime to a regime in which the system is dominated by solitons. We compute explicitly, as a function of temperature and density (L ² norm), the transition point at which the uniform configuration becomes unstable to local perturbations; static and dynamic mean field approximations yield the same result.     

Nonperturbative Time-Independent Green Function of Matrix Schrödinger Equation. General Formalism and Quasiclassical Representation

A Green function of time-independent multichannel Schrödinger equation is considered in matrix representation beyond a perturbation theory. Nonperturbative Green functions are obtained through the regular in zero and at infinity solutions of the multichannel Schrödinger equation for different cases of symmetry of the full Hamiltonian. The spectral expansions for the nonperturbative Green functions are obtained in simple form through multichannel wave functions. The developed approach is applied to obtain simple analytic equations for the Green functions and transition matrix elements for compound multipotential system within quasiclassical approximation. The limits of strong and weak interchannel interactions are studied.Alexander I. Pegarkov:On leave from Physics Faculty     

Non-Existence of Liouvillian Solutions for a Free Quantum Particle on a Torus Surface,Part 1: Polar States

We consider a quantum particle constrained to the surface of a torus that we parametrize by its azimuthal and polar angle. We show that the corresponding Schrödinger equation does not have closed-form solutions (in the sense of Liouvillian functions) that depend on the polar angle only. It follows that if there are any wavefunctions in closed form, they must contain nondegenerate, special functions.