Anderson Localization for 2D Discrete Schr?dinger Operators with Random Magnetic Fields

We prove Anderson localization near the bottom of the spectrum for two-dimensional discrete Schrödinger operators with random magnetic fields and no scalar potentials. We suppose the magnetic fluxes vanish in pairs, and the magnetic field strength is bounded from below by a positive constant. Main lemmas are the Lifshitz tail and the Wegner estimate on the integrated density of states. Then, Anderson localization, i.e., pure point spectrum with exponentially decreasing eigenfunctions, is proved by the standard multiscale argument. Communicated by Gian Michele Graf submitted 01/10/02, accepted: 16/04/03     

Localization for the Random Displacement Model at Weak Disorder

This paper is devoted to the study of the random displacement model on ${\mathbb{R}^d}$ . We prove that, in the weak displacement regime, Anderson and dynamical localization hold near the bottom of the spectrum under a generic assumption on the single-site potential and a fairly general assumption on the support of the possible displacements. This result follows from the proof of the existence of Lifshitz tails and of a Wegner estimate for the model under scrutiny.     

Wegner Estimate for Discrete Alloy-type Models

We study discrete alloy-type random Schrödinger operators on ${\ell^2(\mathbb{Z}^d)}We study discrete alloy-type random Schr?dinger operators on l²(\mathbbZ^d){\ell^2(\mathbb{Z}^d)} . Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. If the single site potential is compactly supported and the distribution of the coupling constant is of bounded variation a Wegner estimate holds. The bound is polynomial in the volume of the box and thus applicable as an ingredient for a localisation proof via multiscale analysis.     

Lifshitz Tail for 2D Discrete Schr?dinger Operator with Random Magnetic Field

. Lifshitz tail for 2 dimensional discrete Schrödinger operator with Anderson-type random magnetic field is proved. We first prove local energy estimates for deterministic discrete magnetic Schrödinger operators, and then follow the large deviation argument of Simon [6].     

Wegner estimate for sparse and other generalized alloy type potentials

We prove a Wegner estimate for generalized alloy type models at negative energies (Theorems 8 and 13). The single site potential is assumed to be non-positive. The random potential does not need to be stationary with respect to translations from a lattice. Actually, the set of points to which the individual single site potentials are attached, needs only to satisfy a certain density condition. The distribution of the coupling constants is assumed to have a bounded density only in the energy region where we prove the Wegner estimate.     

An uncertainty principle, Wegner estimates and localization near fluctuation boundaries

We prove a simple uncertainty principle and show that it can be applied to prove Wegner estimates near fluctuation boundaries. This gives new classes of models for which localization at low energies can be proven.     

Wegner Estimate for Random Magnetic Laplacian on {\mathbb{Z}^{2}}

We consider a two-dimensional magnetic Schr?dinger operator on a square lattice with a spatially stationary random magnetic field. We prove Anderson localization near the spectral edges. We use a new approach to establish a Wegner estimate that does not rely on the monotonicity of the energy on the random parameters.     

The Wegner estimate and the integrated density of states for some random operators

The integrated density of states (IDS) for random operators is an important function describing many physical characteristics of a random system. Properties of the IDS are derived from the Wegner estimate that describes the influence of finite-volume perturbations on a background system. In this paper, we present a simple proof of the Wegner estimate applicable to a wide variety of random perturbations of deterministic background operators. The proof yields the correct volume dependence of the upper bound. This implies the local H?lder continuity of the integrated density of states at energies in the unperturbed spectral gap. The proof depends on theL ^p-theory of the spectral shift function (SSF), forp ≥ 1, applicable to pairs of self-adjoint operators whose difference is in the trace idealI _p, for 0p ≤ 1. We present this and other results on the SSF due to other authors. Under an additional condition of the single-site potential, local H?lder continuity is proved at all energies. Finally, we present extensions of this work to random potentials with nonsign definite single-site potentials.     

The Sine-Gordon regime of the Landau–Lifshitz equation with a strong easy-plane anisotropy

It is well-known that the dynamics of biaxial ferromagnets with a strong easy-plane anisotropy is essentially governed by the Sine-Gordon equation. In this paper, we provide a rigorous justification to this observation. More precisely, we show the convergence of the solutions to the Landau–Lifshitz equation for biaxial ferromagnets towards the solutions to the Sine-Gordon equation in the regime of a strong easy-plane anisotropy. Moreover, we establish the sharpness of our convergence result.This result holds for solutions to the Landau–Lifshitz equation in high order Sobolev spaces. We first provide an alternative proof for local well-posedness in this setting by introducing high order energy quantities with better symmetrization properties. We then derive the convergence from the consistency of the Landau–Lifshitz equation with the Sine-Gordon equation by using well-tailored energy estimates. As a by-product, we also obtain a further derivation of the free wave regime of the Landau–Lifshitz equation.     

Lifshitz tails for Anderson models with sign‐indefinite single‐site potentials

We study the spectral minimum and Lifshitz tails for continuum random Schrödinger operators of the form where V₀ is the periodic potential, are i.i.d random variables and u is the sign‐indefinite impurity potential. Recently, this model has been proven to exhibit Lifshitz tails near the bottom of the spectrum under the small support assumption of u and the reflection symmetry assumption of V₀ and u. We here drop the reflection symmetry assumption of V₀ and u. We first give characterizations of the bottom of the spectrum. Then, we show the existence of Lifshitz tails in the regime where the characterization of the bottom of the spectrum is explicit. In particular, this regime covers the reflection symmetry case.     

Continuity properties of the integrated density of states on manifolds

We first analyze the integrated density of states (IDS) of periodic Schrödinger operators on an amenable covering manifold. A criterion for the continuity of the IDS at a prescribed energy is given along with examples of operators with both continuous and discontinuous IDS. Subsequently, alloy-type perturbations of the periodic operator are considered. The randomness may enter both via the potential and the metric. A Wegner estimate is proven which implies the continuity of the corresponding IDS. This gives an example of a discontinuous “periodic” IDS which is regularized by a random perturbation.     

Lifshitz tails for random perturbations of periodic Schrödinger operators

The present paper is a non-exhaustive review of Lifshitz tails for random perturbations of periodic Schrödinger operators. It is not our goal to review the whole literature on Lifshitz tails; we will concentrate on a single model, the continuous Anderson model.     

Error estimates for a semi-implicit numerical scheme solving the Landau-Lifshitz equation with an exchange field

^** Email: Ivan.Cimrak{at}ugent.be We study the Landau–Lifshitz (LL) equation describingthe evolution of spin fields in continuum ferromagnets. We considerthe 3D case when the effective field arising in the LL equationincludes exchange interaction, the most challenging case. Thissetting corresponds to the pure isotropic case without a demagnetizingfield. We derive some regularity results for the exact solutionto the LL with Neumann-type boundary conditions. We modify thenumerical scheme studied by A. Prohl in two dimensions and weprove error estimates for this scheme in three dimensions.     

On the Lipschitz continuity of the integrated density of states for sign-indefinite potentials

The present paper is devoted to the study of spectral properties of random Schrödinger operators. Using a finite section method for Toeplitz matrices, we prove a Wegner estimate for some alloy type models where the single site potential is allowed to change sign. The results apply to the corresponding discrete model, too. In certain disorder regimes we are able to prove the Lipschitz continuity of the integrated density of states and/or localization near spectral edges.     

Spectral localization for quantum Hamiltonians with weak random delta interaction

We consider a negative Laplacian in multi-dimensional Euclidean space (or a multi-dimensional layer) with a weak disorder random perturbation. The perturbation consists of a sum of lattice translates of a delta interaction supported on a compact manifold of co-dimension one and modulated by coupling constants, which are independent identically distributed random variables times a small disorder parameter. We establish that the spectrum of the considered operator is almost surely a fixed set, characterize its minimum, give an initial length scale estimate and the Wegner estimate, and conclude that there is a small zone of a pure point spectrum containing the almost sure spectral bottom. The length of this zone is proportional to the small disorder parameter.     

Multiscale analysis for ergodic schrödinger operators and positivity of Lyapunov exponents

A variant of multiscale analysis for ergodic Schrödinger operators is developed. This enables us to prove positivity of Lyapunov exponents, given initial scale estimates and an initial Wegner estimate. This postivivity is then applied to high-dimensional skew-shifts at small coupling, where initial conditions are checked using the Pastur-Figotin formalism.     

Self-similar solutions for the LSW model with encounters

The LSW model with encounters has been suggested by Lifshitz and Slyozov as a regularization of their classical mean-field model for domain coarsening to obtain universal self-similar long-time behavior. We rigorously establish that an exponentially decaying self-similar solution to this model exist, and show that this solutions is isolated in a certain function space. Our proof relies on setting up a suitable fixed-point problem in an appropriate function space and careful asymptotic estimates of the solution to a corresponding homogeneous problem.     

Vortex dynamics in the presence of excess energy for the Landau–Lifshitz–Gilbert equation

We study the Landau–Lifshitz–Gilbert equation for the dynamics of a magnetic vortex system. We present a PDE-based method for proving vortex dynamics that does not rely on strong well-preparedness of the initial data and allows for instantaneous changes in the strength of the gyrovector force due to bubbling events. The main tools are estimates of the Hodge decomposition of the supercurrent and an analysis of the defect measure of weak convergence of the stress energy tensor. Ginzburg–Landau equations with mixed dynamics in the presence of excess energy are also discussed.     

Improved error estimates for a Maxwell‐Landau‐Lifschitz system

We study Maxwell's system coupled with the Landau‐Lifshitz (LL) equation. We consider the nonlinear dissipative case with a neglected exchange field. For a recent numerical scheme conserving magnitude of the magnetization we derive new error estimates and establish a better rate of convergence. These theoretical results are demonstrated on a numerical example. For computations we use the software package ALBERT. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)