New characterizations of the region of complete localization for random Schr\"odinger operators

We study the region of complete localization in a class of random operators which includes random Schr\"odinger operators with Anderson-type potentials and classical wave operators in random media, as well as the Anderson tight-binding model. We establish new characterizations or criteria for this region of complete localization, given either by the decay of eigenfunction correlations or by the decay of Fermi projections. (These are necessary and sufficient conditions for the random operator to exhibit complete localization in this energy region.) Using the first type of characterization we prove that in the region of complete localization the random operator has eigenvalues with finite multiplicity

Scarring on invariant manifolds for perturbed quantized hyperbolic toral automorphisms

We exhibit scarring for certain nonlinear ergodic toral automorphisms. There are perturbed quantized hyperbolic toral automorphisms preserving certain co-isotropic submanifolds. The classical dynamics is ergodic, hence in the semiclassical limit almost all eigenstates converge to the volume measure of the torus. Nevertheless, we show that for each of the invariant submanifolds, there are also eigenstates which localize and converge to the volume measure of the corresponding submanifold.Comment: 17 page

Characterization of the Anderson metal-insulator transition for non ergodic operators and application

We study the Anderson metal-insulator transition for non ergodic random Schr\"odinger operators in both annealed and quenched regimes, based on a dynamical approach of localization, improving known results for ergodic operators into this more general setting. In the procedure, we reformulate the Bootstrap Multiscale Analysis of Germinet and Klein to fit the non ergodic setting. We obtain uniform Wegner Estimates needed to perform this adapted Multiscale Analysis in the case of Delone-Anderson type potentials, that is, Anderson potentials modeling aperiodic solids, where the impurities lie on a Delone set rather than a lattice, yielding a break of ergodicity. As an application we study the Landau operator with a Delone-Anderson potential and show the existence of a mobility edge between regions of dynamical localization and dynamical delocalization.Comment: 36 pages, 1 figure. Changes in v2: corrected typos, Theorem 5.1 slightly modifie

Semiclassical measures and the Schroedinger flow on Riemannian manifolds

In this article we study limits of Wigner distributions (the so-called semiclassical measures) corresponding to sequences of solutions to the semiclassical Schroedinger equation at times scales $\alpha_{h}$ tending to infinity as the semiclassical parameter $h$ tends to zero (when $\alpha _{h}=1/h$ this is equivalent to consider solutions to the non-semiclassical Schreodinger equation). Some general results are presented, among which a weak version of Egorov's theorem that holds in this setting. A complete characterization is given for the Euclidean space and Zoll manifolds (that is, manifolds with periodic geodesic flow) via averaging formulae relating the semiclassical measures corresponding to the evolution to those of the initial states. The case of the flat torus is also addressed; it is shown that non-classical behavior may occur when energy concentrates on resonant frequencies. Moreover, we present an example showing that the semiclassical measures associated to a sequence of states no longer determines those of their evolutions. Finally, some results concerning the equation with a potential are presented.Comment: 18 pages; Theorems 1,2 extendend to deal with arbitrary time-scales; references adde

Near Sharp Strichartz estimates with loss in the presence of degenerate hyperbolic trapping

We consider an $n$-dimensional spherically symmetric, asymptotically Euclidean manifold with two ends and a codimension 1 trapped set which is degenerately hyperbolic. By separating variables and constructing a semiclassical parametrix for a time scale polynomially beyond Ehrenfest time, we show that solutions to the linear Schr\"odiner equation with initial conditions localized on a spherical harmonic satisfy Strichartz estimates with a loss depending only on the dimension $n$ and independent of the degeneracy. The Strichartz estimates are sharp up to an arbitrary $\beta>0$ loss. This is in contrast to \cite{ChWu-lsm}, where it is shown that solutions satisfy a sharp local smoothing estimate with loss depending only on the degeneracy of the trapped set, independent of the dimension

Taylor approximations of operator functions

This survey on approximations of perturbed operator functions addresses recent advances and some of the successful methods.Comment: 12 page

Widths of the Hall Conductance Plateaus

We study the charge transport of the noninteracting electron gas in a two-dimensional quantum Hall system with Anderson-type impurities at zero temperature. We prove that there exist localized states of the bulk order in the disordered-broadened Landau bands whose energies are smaller than a certain value determined by the strength of the uniform magnetic field. We also prove that, when the Fermi level lies in the localization regime, the Hall conductance is quantized to the desired integer and shows the plateau of the bulk order for varying the filling factor of the electrons rather than the Fermi level.Comment: 94 pages, v2: a revision of Sec. 5; v3: an error in Sec. 7 is corrected, major revisions of Sec. 7 and Appendix E, Sec. 7 is enlarged to Secs. 7-12, minor corrections; v4: major revisions, accepted for publication in Journal of Statistical Physics; v5: minor corrections, accepted versio

An Improved Combes-Thomas Estimate of Magnetic Schr\"{o}dinger Operators

In the present paper, we prove an improved Combes-Thomas estimate, i.e., the Combes-Thomas estimate in trace-class norms, for magnetic Schr\"{o}dinger operators under general assumptions. In particular, we allow unbounded potentials. We also show that for any function in the Schwartz space on the reals the operator kernel decays, in trace-class norms, faster than any polynomial.Comment: 25 pages, some errors correcte

Non-accretive Schrödinger operators and exponential decay of their eigenfunctions

International audienceWe consider non-self-adjoint electromagnetic Schrödinger operators on arbitrary open sets with complex scalar potentials whose real part is not necessarily bounded from below. Under a suitable sufficient condition on the electromagnetic potential, we introduce a Dirichlet realisation as a closed densely defined operator with non-empty resolvent set and show that the eigenfunctions corresponding to discrete eigenvalues satisfy an Agmon-type exponential decay