Jean-Michel Combes

An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators

A digital phase control method phase shifts a predetermined number of clock signals having the same frequency and having different phases at high precision and at high resolution as a whole with its phase interval maintained to keep a predetermined interval. The digital phase control method comprises the steps of preparing fourteen first multi-phase clock signals having a fixed phase, of preparing sixteen second multi-phase clock signals, of phase locking a specific clock signal of the fourteen first multi-phase clock signals with a particular clock signal of the sixteen second multi-phase clock signals, and of changing a combination of the specific and the particular clock signals to be phase-locked to phase shift the second multi-phase clock signals. In addition, in order to generate the second multi-phase clock signals, a delay line comprising ring-shaped chained delay buffers may be used.

The Lp-Theory of the Spectral Shift Function,¶the Wegner Estimate, and the Integrated Density¶of States for Some Random Operators

Abstract: We develop the Lp-theory of the spectral shift function, for p≥ 1, applicable to pairs of self-adjoint operators whose difference is in the trace ideal ℐp, for

The Shape Resonance

0 < p≤ 1. This result is a key ingredient of a new, short proof of the Wegner estimate applicable to a wide variety of additive and multiplicative random perturbations of deterministic background operators. The proof yields the correct volume dependence of the upper bound. This implies the local Hölder continuity of the integrated density of states at energies in the unperturbed spectral gap. Under an additional condition of the single-site potential, local Hölder continuity is proved at all energies. This new Wegner estimate, together with other, standard results, establishes exponential localization for a new family of models for additive and multiplicative perturbations.

Landau Hamiltonians with random potentials: localization and the density of states

For a class of Schrödinger operatorsH:=−(ℏ2/2m)Δ+V onL2(ℝn), with potentials having minima embedded in the continuum of the spectrum and non-trapping tails, we show the existence of shape resonances exponentially close to the real axis as ℏ↘0. The resonant energies are given by a convergent perturbation expansion in powers of a parameter exhibiting the expected exponentially small behaviour for tunneling.

Generalized Eigenvalue-Counting Estimates for the Anderson Model

We prove the existence of localized states at the edges of the bands for the two-dimensional Landau Hamiltonian with a random potential, of arbitrary disorder, provided that the magnetic field is sufficiently large. The corresponding eigenfunctions decay exponentially with the magnetic field and distance. We also prove that the integrated density of states is Lipschitz continuous away from the Landau energies. The proof relies on a Wegner estimate for the finite-area magnetic Hamiltonians with random potentials and exponential decay estimates for the finitearea Greens functions. The proof of the decay estimates for the Greens functions uses fundamental results from two-dimensional bond percolation theory.

Connections between quantum dynamics and spectral properties of time-evolution operators

We generalize Minami’s estimate for the Anderson model and its extensions to n eigenvalues, allowing for n arbitrary intervals and arbitrary single-site probability measures with no atoms. As an application, we derive new results about the multiplicity of eigenvalues and Mott’s formula for the ac-conductivity when the single site probability distribution is Hölder continuous.

Krein's formula and one-dimensional multiple-well

Abstract Lower bounds for time averages of mean square displacement are discussed in terms of the Hausdorff dimension of the spectrum.

Spectral stability under tunneling

Abstract It is shown that all the discrete eigenvalues of a one-dimensional Schrodinger operator with a multiple well potential possess an asymptotic expansions in power of h 1 2 when h → 0 . A formula for all the coefficients of these expansions is given. The method uses two main tools: Perturbation by boundary conditions and exponential decay of eigenfunctions which are developed in this article. As a by-product of this work, the exponential localization of eigenvectors when h goes to zero can be proved.

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

We study the spectral properties of multiple well Schrödinger operators on ℝn. We give in particular upper bounds on energy shifts due to tunnel effect and localization properties of wave packets. Our methods are based on Agmon type estimates for resolvents in classically forbidden regions and geometric perturbation theory. Our results are valid also for an infinite number of wells, arbitrary spectral type and in non-semi-classical regimes.

On the location of the resonances for schrodinger

Abstract.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 Hölder continuity of the integrated density of states at energies in the unperturbed spectral gap. The proof depends on theLp-theory of the spectral shift function (SSF), forp ≥ 1, applicable to pairs of self-adjoint operators whose difference is in the trace idealIp, 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 Hölder continuity is proved at all energies. Finally, we present extensions of this work to random potentials with nonsign definite single-site potentials.