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.     

Uniform existence of the integrated density of states for models on $${\mathbb{Z}}^d$$

We provide an ergodic theorem for certain Banach-space valued functions on structures over , which allow for existence of frequencies of finite patterns. As an application we obtain existence of the integrated density of states for associated discrete finite-range operators in the sense of convergence of the distributions with respect to the supremum norm. These results apply to various examples including periodic operators, percolation models and nearest-neighbour hopping on the set of visible points. Our method gives explicit bounds on the speed of convergence in terms of the speed of convergence of the underlying frequencies. It uses neither von Neumann algebras nor a framework of random operators on a probability space.      

Integrated Density of States For Random Metrics on Manifolds

We study ergodic random Schrödinger operators on a coveringmanifold, where the randomness enters both via the potentialand the metric. We prove measurability of the random operators,almost sure constancy of their spectral properties, the existenceof a self-averaging integrated density of states and a Pastur–ubintype trace formula. 2000 Mathematics Subjects Classification35J10, 58J35, 82B44.     

The integrated density of states in strong magnetic fields

We consider three-dimensional Schrödinger operators with constant magnetic fields and ergodic electric potentials. We study the strong magnetic field asymptotic behaviour of the integrated density of states, distinguishing between the asymptotics far from the Landau levels, and the asymptotics near a given Landau level.     

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.     

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.     

A remark on the Lifshitz tail for Schrödinger operator with random magnetic field

In this note, we consider the Lifshitz singularity for Schrödinger operator with ergodic random magnetic field. A key estimate is an energy bound for magnetic Schrödinger operators as discussed in Nakamura [8]. Here we remove a technical assumption in [8], namely, the uniform boundedness of the magnetic field.     

Smoothness of density of states for random decaying interaction

In this paper we consider one dimensional random Jacobi operators with decaying independent randomness and show that under some condition on the decay vis-a-vis the distribution of randomness, that the distribution function of the average spectral measures of the associated operators are smooth.     

Integrated Density of States for Ergodic Random Schrödinger Operators on Manifolds

We consider the Riemannian universal covering of a compact manifold M = X/ and assume that is amenable. We show the existence of a (nonrandom) integrated density of states for an ergodic random family of Schrödinger operators on X.     

Solutions of mKdV in Classes of Functions Unbounded at Infinity

Using P. Lax’s concept of a Lax pair we prove global existence and uniqueness for solutions of the initial value problem for mKdV in classes of smooth functions which can be unbounded at infinity, and in particular, may tend to infinity with respect to the space variable. Moreover, we establish the invariance of the spectrum and the unitary type of the Schrödinger operator under the KdV flow and the invariance of the spectrum and the unitary type of the impedance operator under the mKdV flow for potentials in these classes.     

Average density of states for Hermitian Wigner matrices

We consider ensembles of N×N Hermitian Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. Assuming sufficient regularity for the probability density function of the entries, we show that the expectation of the density of states on arbitrarily small intervals converges to the semicircle law, as N tends to infinity.     

Nonsmooth analysis on smooth manifolds

We study infinitesimal properties of nonsmooth (nondifferentiable) functions on smooth manifolds. The eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function.

A subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization problems, nonclassical problems of calculus of variations, and generalized solutions of first-order partial differential equations on manifolds. We also establish criteria for monotonicity and invariance of functions and sets with respect to solutions of differential inclusions.

     

On approximation of the “Membrane” Schrödinger operator by the “Crystal” operator

LetV(x), x=(s ₁,x ₂,x ₃), be a potential periodic inx ₁,x ₂ and exponentially decreasing as |x ₃|→∞, and letV _N (x) be the sum of shiftsV(x?(0,0, Nn ₃)) over all integern ₃. We prove that the spectrum and eigenfunctions (not necessarily in the classL ²) of the Schrödinger operator with potentialV _N , considered in a box, approximate the spectrum and eigenfunctions of the operator with potentialV and, for the negative part of the spectrum, the approximation converges exponentially inN asN→∞.     

Magnetic bottles for the Neumann problem: The case of dimension 3

The main object of this paper is to analyze the recent results obtained on the Neumann realization of the Schrödinger operator in the case of dimension 3 by Lu and Pan. After presenting a short treatment of their spectral analysis of keymodels, we show briefly how to implement the techniques of Helffer-Morame in order to give some localization of the ground state. This leaves open the question of the localization by curvature effect which was solved in the case of dimension 2 in our previous work and will be analysed in the case of dimension 3 in a future paper.     

One-dimensional Schrödinger operator on the half-line: The differential equation for eigenfunctions with respect to the spectral parameter and an analog of the Freud equation

It is shown that the eigenfunctions of the Schrödinger operator on the half-line satisfy an explicitly constructed differential equation with respect to the spectral parameter. Such an equation was earlier obtained for orthogonal polynomials. An analog of the Freud equation is found.     

Uniform existence of the integrated density of states for random Schrödinger operators on metric graphs over

We consider ergodic random Schrödinger operators on the metric graph with random potentials and random boundary conditions taking values in a finite set. We show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable. This limit, the integrated density of states, can be expressed by a closed Shubin–Pastur type trace formula. It supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions. Among other examples we discuss random magnetic fields and percolation models.     

On spectral properties of periodic polyharmonic matrix operators

We consider a matrix operatorH = (-Δ)^l +V inR ⁿ, wheren ≥ 2,l ≥ 1, 4l > n + 1, andV is the operator of multiplication by a periodic inx matrixV(x). We study spectral properties ofH in the high energy region. Asymptotic formulae for Bloch eigenvalues and the corresponding spectral projections are constructed. The Bethe-Sommerfeld conjecture, stating that the spectrum ofH can have only a finite number of gaps, is proved.     

About the notion of non‐T‐resonance and applications to topological multiplicity results for ODEs on differentiable manifolds

By using topological methods, mainly the degree of a tangent vector field, we establish multiplicity results for T‐periodic solutions of parametrized T‐periodic perturbations of autonomous ODEs on a differentiable manifold M. In order to provide insights into the key notion of T‐resonance, we consider the elementary situations and . Doing so, we provide more comprehensive analysis of those cases and find improved conditions. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence and mapping properties of the wave operator for the Schrödinger equation with singular potential

We consider the Schrödinger equation in three-dimensional space with small potential in the Lorentz space and we prove Strichartz-type estimates for the solution to this equation. Moreover, using Cook's method, we prove the existence of the wave operator. In the last section we prove the equivalence between the homogeneous Sobolev spaces and in the case .