Invited Speakers:
Alexander Fedotov University of
Saint Petersburg, Saint-Petersburg, Russia
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On the behavior at infinity of solutions of an almost periodic equation.
We study the behavior at infinity of solutions of a model almost
periodic equation with a ”large” coupling constant using
the monodromization renormalization method. In the course of the
renormalizations, one obtain an infinite series of almost
periodic equations describing a given solution of the input equation
subsequently on larger and larger intervals Il, l =
1,2,3…, the ratio of their length
being determined by the continued fraction for the ratio of the
periods of the input operator. This allows to control the solutions of
the input equation in a constructive way. For a given value of the
ergodic parameter, the behavior of the solutions drastically depends
on the existence of the Lyapunov exponent. Roughly, when the Lyapunov
exponent exists there are solutions that are nicely exponentially
increasing, and when it does not exists, there is an infinite
subsequence of the above intervals such that, on each of the intervals
Ilk
of this subsequence, ( if we ”forget” about the details of
the behavior of the solutions on the distances of order of the length
of Ilk-1, Ilk-2 etc ) the solution first
increases and then begins to decay. The results were obtained in
collaboration with F. Klopp (Univ. Paris 13).
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Abel Klein University of California
at Irvine, Irvine, CA, USA
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Poisson Statistics for Eigenvalues of Continuum Random Schrödinger
Operators.
We prove Poisson statistics for eigenvalues of random Schr\"odinger
operators in the continuum. More specifically, we prove a Minami
estimate for continuum Anderson Hamiltonians in the continuum and
derive Poisson statistics for the eigenvalues in the localization
region at the bottom of the spectrum. We also prove simplicity of the
eigenvalues in that region. (Joint work with J.-M. Combes and F.
Germinet)
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Wolfgang König Universität
Leipzig, Leipzig, Germany
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A two-cities theorem for the parabolic Anderson model.
The parabolic Anderson problem is the Cauchy problem for the heat equation ∂tu(t,z) = Δu(t,z)+ξ(z)u(t,z)
on (0,∞) × Zd with random potential (ξ(z): z ∈ Zd). We consider independent and identically distributed
potentials, such that the distribution function of ξ(z) converges polynomially at infinity, i.e., a heavy-tailed
distribution. If u is initially localised in the origin, i.e., if u(0,x) = δ0(x), we show that, as time goes to
infinity, the solution is completely localised in two points almost surely and in one point with high
probability. We also identify the asymptotic behaviour of the concentration sites in terms of a weak limit
theorem. This is joint work with H. Lacoin, P. Mörters, and N. Sidorova.
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Raphael Krikorian Université
Paris 6, Paris, France
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KAM-Liouville Theory and an extension of a theorem by Dinaburg and Sinai.
We prove that given any irrational frequency α on the one dimensional torus T, the Schrödinger cocycle
associated to an analytic potential on T above the rotation by α, is conjugated to an SO(2,R)-valued cocycle
(and hence bounded) provided the potential is small enough and the rotation number of the cocycle satisfies
a diophantine condition w.r.t. α. When α is fixed, the theorem holds for a set of positive measure of
the energy. This is an extension of a theorem of Dinaburg and Sinai to the case where α is not
diophantine. The technique of the proof is based on a perturbative scheme, reminiscent of KAM
theory but that applies for any α irrational. This is a joint work with Artur Avila and Bassam
Fayad.
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Armen Shirikyan Université de
Cergy-Pontoise, Cergy-Pontoise, France
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Control and mixing for nonlinear PDE's.
Let us consider a differential equation subject to an external
force. It is well known that, in the finite-dimensional case, the
controllability of the equation in question and the mixing property of
the associated random dynamics are essentially equivalent. The
situation is much more delicate in the infinite-dimensional
setting. In this talk, I give an overview of some recent results on
the problem of controllability and mixing for a class of PDE's arising
in mathematical physics and formulate some open questions.
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Simone Warzel Princeton
University, Princeton, NJ, USA.
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On the Joint Distribution of Energy Levels of Random Schroedinger
Operators.
We consider operators with random potentials on graphs, such as the
lattice version of the random Schroedinger operator. The main result
is a general bound on the probabilities of simultaneous occurrence of
eigenvalues in specified distinct intervals, with the corresponding
eigenfunctions being separately localized within prescribed
regions. The bound generalizes the the Wegner estimate on the density
of states. The analysis proceeds through a new multiparameter spectral
averaging principle.
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