24 research outputs found
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 tending to
infinity as the semiclassical parameter tends to zero (when 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 -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 and independent of the degeneracy.
The Strichartz estimates are sharp up to an arbitrary 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