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Jean-Michel Combes

Publications and source records attributed to Jean-Michel Combes.

7 recordsLinked to original sources
arXiv2013

Local Wegner and Lifshitz tails estimates for the density of states for continuous random Schrödinger operators

We introduce and prove local Wegner estimates for continuous generalized Anderson Hamiltonians, where the single-site random variables are independent but not necessarily identically distributed. In particular, we get Wegner estimates with a constant that goes to zero as we approach the bottom of the spectrum. As an application, we show that the (differentiated) density of states exhibits the same Lifshitz tails upper bound as the integrated density of states.

math-ph↗
arXiv2009

Poisson Statistics for Eigenvalues of Continuum Random Schrödinger Operators

We show absence of energy levels repulsion for the eigenvalues of random Schrödinger operators in the continuum. We prove that, in the localization region at the bottom of the spectrum, the properly rescaled eigenvalues of a continuum Anderson Hamiltonian are distributed as a Poisson point process with intensity measure given by the density of states. We also obtain simplicity of the eigenvalues. We derive a Minami estimate for continuum Anderson Hamiltonians. We also give a simple and transparent proof of Minami's estimate for the (discrete) Anderson model.

math-ph↗
arXiv2009

Generalized eigenvalue-counting estimates for the Anderson model

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.

math-ph↗
arXiv2006

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

We prove that the integrated density of states (IDS) of random Schrödinger operators with Anderson-type potentials on $L^2 (\R^d)$, for $d \geq1$, is locally Hölder continuous at all energies with the same Hölder exponent $0<α\leq1$ as the conditional probability measure for the single-site random variable. As a special case, we prove that if the probability distribution is absolutely continuous with respect to Lebesgue measure with a bounded density, then the IDS is Lipschitz continuous at all energies. The single-site potential $u\in L\_0^\infty (\R^d)$ must be nonnegative and compactly-supported. The unperturbed Hamiltonian must be periodic and satisfy a unique continuation principle. We also prove analogous continuity results for the IDS of random Anderson-type perturbations of the Landau Hamiltonian in two-dimensions. All of these results follow from a new Wegner estimate for local random Hamiltonians with rather general probability measures.

math-ph↗
arXiv2006

Some new estimates on the spectral shift function associated with random Schrödinger operators

We prove some new pointwise-in-energy bounds on the expectations of various spectral shift functions associated with random Schrödinger operators in the continuum having Anderson-type random potentials in both finite-volume and infinite-volume. These estimates are a consequence of our new Wegner estimate for finite-volume random Schrödinger operators. For lattice models, we also obtain a representation of the infinite-volume density of states in terms of a spectral shift function. For continuum models, the corresponding measure is absolutely continuous with respect to the density of states and agrees with it in certain cases. We present a variant of a new spectral averaging result and use it to prove a pointwise upper bound on the SSF for finite-rank perturbations.

math-ph↗
arXiv2000

Absorption in periodic layered structures

Photonic band structure of metal-dielectric and semiconductor-dielectric layered structures are studied in the presence of a strong absorption. It is shown that absorption can enlarge some gaps by as much as 50%.

cond-mat↗