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Abel Klein

Publications and source records attributed to Abel Klein.

At least 37 records · Page 2Linked to original sources
arXiv2013

Ground state energy of trimmed discrete Schrödinger operators and localization for trimmed Anderson models

We consider discrete Schrödinger operators of the form $H=-Δ+V$ on $\ell^2(\Z^d)$, where $Δ$ is the discrete Laplacian and $V$ is a bounded potential. Given $Γ\subset \Z^d$, the $Γ$-trimming of $H$ is the restriction of $H$ to $\ell^2(\Z^d\setminusΓ)$, denoted by $H_Γ$. We investigate the dependence of the ground state energy $E_Γ(H)=\inf σ(H_Γ)$ on $Γ$. We show that for relatively dense proper subsets $Γ$ of $\Z^d$ we always have $E_Γ(H)>E_\emptyset(H)$. We use this lifting of the ground state energy to establish Wegner estimates and localization at the bottom of the spectrum for $Γ$-trimmed Anderson models, i.e., Anderson models with the random potential supported by the set $Γ$

math-ph↗
arXiv2013

Unique continuation principle for spectral projections of Schr\" odinger operators and optimal Wegner estimates for non-ergodic random Schr\" odinger operators

We prove a unique continuation principle for spectral projections of Schr\" odinger operators. We consider a Schr\" odinger operator $H= -Δ+ V$ on $\mathrm{L}^2(\mathbb{R}^d)$, and let $H_Λ$ denote its restriction to a finite box $Λ$ with either Dirichlet or periodic boundary condition. We prove unique continuation estimates of the type $χ_I (H_Λ) W χ_I (H_Λ) \ge κ\, χ_I (H_Λ) $ with $κ>0$ for appropriate potentials $W\ge 0$ and intervals $I$. As an application, we obtain optimal Wegner estimates at all energies for a class of non-ergodic random Schr\" odinger operators with alloy{-type random potentials (`crooked' Anderson Hamiltonians). We also prove optimal Wegner estimates at the bottom of the spectrum with the expected dependence on the disorder (the Wegner estimate improves as the disorder increases), a new result even for the usual (ergodic) Anderson Hamiltonian. These estimates are applied to prove localization at high disorder for Anderson Hamiltonians in a fixed interval at the bottom of the spectrum.

math-ph↗
arXiv2012

Bounds on the density of states for Schr\" odinger operators

We establish bounds on the density of states measure for Schrödinger operators. These are deterministic results that do not require the existence of the density of states measure, or, equivalently, of the integrated density of states. The results are stated in terms of a "density of states outer-measure" that always exists, and provides an upper bound for the density of states measure when it exists. We prove log-Hölder continuity for this density of states outer-measure in one, two, and three dimensions for Schrödinger operators, and in any dimension for discrete Schrödinger operators.

math-ph↗
arXiv2012

A comprehensive proof of localization for continuous Anderson models with singular random potentials

We study continuous Anderson Hamiltonians with non-degenerate single site probability distribution of bounded support, without any regularity condition on the single site probability distribution. We prove the existence of a strong form of localization at the bottom of the spectrum, which includes Anderson localization (pure point spectrum with exponentially decaying eigenfunctions) with finite multiplicity of eigenvalues, dynamical localization (no spreading of wave packets under the time evolution), decay of eigenfunctions correlations, and decay of the Fermi projections. We also prove log-H\" older continuity of the integrated density of states at the bottom of the spectrum.

math-ph↗
arXiv2012

Absolutely Continuous Spectrum for Random Schroedinger Operators on the Bethe Strip

The Bethe Strip of width $m$ is the cartesian product $\B\times\{1,...,m\}$, where $\B$ is the Bethe lattice (Cayley tree). We prove that Anderson models on the Bethe strip have "extended states" for small disorder. More precisely, we consider Anderson-like Hamiltonians $\;H_λ=\frac12 Δ\otimes 1 + 1 \otimes A + λ\Vv$ on a Bethe strip with connectivity $K \geq 2$, where $A$ is an $m\times m$ symmetric matrix, $\Vv$ is a random matrix potential, and $λ$ is the disorder parameter. Given any closed interval $I\subset (-\sqrt{K}+a_{\mathrm{max}},\sqrt{K}+a_{\mathrm{min}})$, where $a_{\mathrm{min}}$ and $a_{\mathrm{max}}$ are the smallest and largest eigenvalues of the matrix $A$, we prove that for $λ$ small the random Schrödinger operator $\;H_λ$ has purely absolutely continuous spectrum in $I$ with probability one and its integrated density of states is continuously differentiable on the interval $I$.

math-ph↗
arXiv2011

Ballistic Behavior for Random Schrödinger Operators on the Bethe Strip

The Bethe Strip of width $m$ is the cartesian product $\B\times\{1,...,m\}$, where $\B$ is the Bethe lattice (Cayley tree). We consider Anderson-like Hamiltonians $H_λ=\frac12 Δ\otimes 1 + 1 \otimes A+λ\Vv$ on a Bethe strip with connectivity $K \geq 2$, where $A$ is an $m\times m$ symmetric matrix, $\Vv$ is a random matrix potential, and $λ$ is the disorder parameter. Under certain conditions on $A$ and $K$, for which we previously proved the existence of absolutely continuous spectrum for small $λ$, we now obtain ballistic behavior for the spreading of wave packets evolving under $H_λ$ for small $λ$.

math-ph↗
arXiv2009

The Liouville equation for singular ergodic magnetic Schrödinger operators

We study the time evolution of a density matrix in a quantum mechanical system described by an ergodic magnetic Schrödinger operator with singular magnetic and electric potentials, the electric field being introduced adiabatically. We construct a unitary propagator that solves weakly the corresponding time-dependent Schrödinger equation, and solve a Liouville equation in an appropriate Hilbert space.

math-ph↗
arXiv2009

Quantization of the Hall conductance and delocalization in ergodic Landau Hamiltonians

We prove quantization of the Hall conductance for continuous ergodic Landau Hamiltonians under a condition on the decay of the Fermi projections. This condition and continuity of the integrated density of states are shown to imply continuity of the Hall conductance. In addition, we prove the existence of delocalization near each Landau level for these two-dimensional Hamiltonians. More precisely, we prove that for some ergodic Landau Hamiltonians there exists an energy $E$ near each Landau level where a ``localization length'' diverges. For the Anderson-Landau Hamiltonian we also obtain a transition between dynamical localization and dynamical delocalization in the Landau bands, with a minimal rate of transport, even in cases when the spectral gaps are closed.

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↗
arXiv2009

Delocalization for Random Landau Hamiltonians with Unbounded Random Variables

In this note we prove the existence of a localization/delocalization transition for Landau Hamiltonians randomly perturbed by an electric potential with unbounded amplitude. In particular, with probability one, no Landau gaps survive as the random potential is turned on, the gaps close, filling up partly with localized states. A minimal rate of transport is exhibited in the region of delocalization. To do so, we exploit the a priori quantization of the Hall conductance and extend recent Wegner estimates to the case of unbounded random variables.

math-ph↗
arXiv2007

The conductivity measure for the Anderson model

We study the ac-conductivity in linear response theory for the Anderson tight-binding model. We define the electrical ac-conductivity and calculate the linear-response current at zero temperature for arbitrary Fermi energy. In particular, the Fermi energy may lie in a spectral region where extended states are believed to exist.

math-ph↗
arXiv2007

Multiscale Analysis and Localization of Random Operators

A discussion of the method of multiscale analysis in the study of localization of random operators based on lectures given at \emph{Random Schrödinger operators: methods, results, and perspectives}, États de la recherche, Université Paris 13, June 2002

math-ph↗
arXiv2007

Localization for a continuum Cantor-Anderson Hamiltonian

We prove localization at the bottom of the spectrum for a random Schrödinger operator in the continuum with a single-site potential probability distribution supported by a Cantor set of zero Lebesgue measure. This distribution is too singular to be treated by the usual methods. In particular, an "a priori" Wegner estimate is not available. To prove the result we perform a multiscale analysis following the work of Bourgain and Kenig for the Bernoulli-Anderson Hamiltonian, and obtain the required Wegner estimate scale by scale. To do so, we generalize their argument based on Sperner's Lemma by resorting to the LYM inequality for multisets, and combine it with the concept of scale dependent equivalent classes of configurations introduced by Germinet, Hislop and Klein for the study of Poisson Hamiltonians.

math-ph↗
arXiv2007

On Bernoulli Decompositions for Random Variables, Concentration Bounds, and Spectral Localization

As was noted already by A. N. Kolmogorov, any random variable has a Bernoulli component. This observation provides a tool for the extension of results which are known for Bernoulli random variables to arbitrary distributions. Two applications are provided here: i. an anti-concentration bound for a class of functions of independent random variables, where probabilistic bounds are extracted from combinatorial results, and ii. a proof, based on the Bernoulli case, of spectral localization for random Schroedinger operators with arbitrary probability distributions for the single site coupling constants. For a general random variable, the Bernoulli component may be defined so that its conditional variance is uniformly positive. The natural maximization problem is an optimal transport question which is also addressed here.

math.PR↗
arXiv2007

Localization for Schrödinger operators with Poisson random potential

We prove exponential and dynamical localization for the Schrödinger operator with a nonnegative Poisson random potential at the bottom of the spectrum in any dimension. We also conclude that the eigenvalues in that spectral region of localization have finite multiplicity. We prove similar localization results in a prescribed energy interval at the bottom of the spectrum provided the density of the Poisson process is large enough.

math-ph↗