41 research outputs found
KAM for the quantum harmonic oscillator
In this paper we prove an abstract KAM theorem for infinite dimensional
Hamiltonians systems. This result extends previous works of S.B. Kuksin and J.
P\"oschel and uses recent techniques of H. Eliasson and S.B. Kuksin. As an
application we show that some 1D nonlinear Schr\"odinger equations with
harmonic potential admits many quasi-periodic solutions. In a second
application we prove the reducibility of the 1D Schr\"odinger equations with
the harmonic potential and a quasi periodic in time potential.Comment: 54 pages. To appear in Comm. Math. Phy
Twistless KAM tori
A selfcontained proof of the KAM theorem in the Thirring model is discussed.Comment: 7 pages, 50 K, Plain Tex, generates one figure named gvnn.p
Double exponential stability of quasi-periodic motion in Hamiltonian systems
We prove that generically, both in a topological and measure-theoretical
sense, an invariant Lagrangian Diophantine torus of a Hamiltonian system is
doubly exponentially stable in the sense that nearby solutions remain close to
the torus for an interval of time which is doubly exponentially large with
respect to the inverse of the distance to the torus. We also prove that for an
arbitrary small perturbation of a generic integrable Hamiltonian system, there
is a set of almost full positive Lebesgue measure of KAM tori which are doubly
exponentially stable. Our results hold true for real-analytic but more
generally for Gevrey smooth systems
Holder continuity of absolutely continuous spectral measures for one-frequency Schrodinger operators
We establish sharp results on the modulus of continuity of the distribution
of the spectral measure for one-frequency Schrodinger operators with
Diophantine frequencies in the region of absolutely continuous spectrum. More
precisely, we establish 1/2-Holder continuity near almost reducible energies
(an essential support of absolutely continuous spectrum). For
non-perturbatively small potentials (and for the almost Mathieu operator with
subcritical coupling), our results apply for all energies.Comment: 16 page
Semitoric integrable systems on symplectic 4-manifolds
Let M be a symplectic 4-manifold. A semitoric integrable system on M is a
pair of real-valued smooth functions J, H on M for which J generates a
Hamiltonian S^1-action and the Poisson brackets {J,H} vanish. We shall
introduce new global symplectic invariants for these systems; some of these
invariants encode topological or geometric aspects, while others encode
analytical information about the singularities and how they stand with respect
to the system. Our goal is to prove that a semitoric system is completely
determined by the invariants we introduce
Normal Form for the Schr\"odinger equation with analytic non--linearities
In this paper we discuss a class of normal forms of the completely resonant
non--linear Schr\"odinger equation on a torus. We stress the geometric and
combinatorial constructions arising from this study. Further analytic
considerations and applications to quasi--periodic solutions will appear in a
forthcoming article. This paper replaces a previous version correcting some
mistakes.Comment: 52 pages, 2 figure
Foliations of Isonergy Surfaces and Singularities of Curves
It is well known that changes in the Liouville foliations of the isoenergy
surfaces of an integrable system imply that the bifurcation set has
singularities at the corresponding energy level. We formulate certain
genericity assumptions for two degrees of freedom integrable systems and we
prove the opposite statement: the essential critical points of the bifurcation
set appear only if the Liouville foliations of the isoenergy surfaces change at
the corresponding energy levels. Along the proof, we give full classification
of the structure of the isoenergy surfaces near the critical set under our
genericity assumptions and we give their complete list using Fomenko graphs.
This may be viewed as a step towards completing the Smale program for relating
the energy surfaces foliation structure to singularities of the momentum
mappings for non-degenerate integrable two degrees of freedom systems.Comment: 30 pages, 19 figure
Growth of Sobolev norms and controllability of Schr\"odinger equation
In this paper we obtain a stabilization result for the Schr\"odinger equation
under generic assumptions on the potential. Then we consider the Schr\"odinger
equation with a potential which has a random time-dependent amplitude. We show
that if the distribution of the amplitude is sufficiently non-degenerate, then
any trajectory of system is almost surely non-bounded in Sobolev spaces
Pure Point Spectrum of the Floquet Hamiltonian for the Quantum Harmonic Oscillator Under Time Quasi- Periodic Perturbations
We prove that the quantum harmonic oscillator is stable under spatially
localized, time quasi-periodic perturbations on a set of Diophantine
frequencies of positive measure. This proves a conjecture raised by
Enss-Veselic in their 1983 paper \cite{EV} in the general quasi-periodic
setting. The motivation of the present paper also comes from construction of
quasi-periodic solutions for the corresponding nonlinear equation
Resonance tongues in the quasi-periodic Hill-Schrödinger equation with three frequencies
n this article we investigate numerically the spectrum of some representative
examples of discrete one-dimensional Schrödinger operators with quasi-periodic potential
in terms of a perturbative constant b and the spectral parameter a. Our examples
include the well-known Almost Mathieu model, other trigonometric potentials with a single
quasi-periodic frequency and generalisations with two and three frequencies. We computed
numerically the rotation number and the Lyapunov exponent to detect open and collapsed
gaps, resonance tongues and the measure of the spectrum. We found that the case with one
frequency was significantly different from the case of several frequencies because the latter
has all gaps collapsed for a sufficiently large value of the perturbative constant and thus the
spectrum is a single spectral band with positive Lyapunov exponent. In contrast, in the cases
with one frequency considered, gaps are always dense in the spectrum, although some gaps
may collapse either for a single value of the perturbative constant or for a range of values. In
all cases we found that there is a curve in the (a, b)-plane which separates the regions where
the Lyapunov exponent is zero in the spectrum and where it is positive. Along this curve,
which is b = 2 in the Almost Mathieu case, the measure of the spectrum is zero.Peer ReviewedPostprint (published version