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Uniform semiclassical estimates for the propagation of quantum observables
A. Bouzouina,Didier Robert +1 more
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In this article, it was shown that the semiclassical asymptotic expansion for the propagation of quantum observables, for C ∞ − Hamiltonians growing at most quadratically at infinity, is uniformly dominated by an exponential term whose argument is linear in time.Abstract:
We prove here that the semiclassical asymptotic expansion for the propagation of quantum observables, for C\sp ∞-Hamiltonians growing at most quadratically at infinity, is uniformly dominated at any order by an exponential term whose argument is linear in time. In particular, we recover the Ehrenfest time for the validity of the semiclassical approximation. This extends the result proved in [BGP]. Furthermore, if the Hamiltonian and the initial observables are holomorphic in a complex neighborhood of the phase space, we prove that the quantum observable is an analytic semiclassical observable. Other results about the large time behavior of observables with emphasis on the classical dynamic are also given. In particular, precise Gevrey estimates are established for classically integrable systems.read more
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Entropy and the localization of eigenfunctions.
TL;DR: In this paper, the authors studied the large eigenvalue limit for the eigenfunctions of the Laplacian on a compact Riemannian manifold M of dimension d > 2, and assumed that the geodesic flow (#*)teR> acting on the unit tangent bundle of M, has a chaotic behaviour.
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Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold
TL;DR: In this paper, the authors studied the high-energy eigenfunctions of the Laplacian on a compact Riemannian manifold with Anosov geodesic flow and showed that the Kolmogorov-Sinai entropy is bounded from below by a constant which, in the case of constant negative curvature, equals half the maximal entropy.
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Space-Adiabatic Perturbation Theory
TL;DR: In this article, the authors studied approximate solutions to the Schrodinger equation with the Hamiltonian given as the Weyl quantization of the symbol $H(q,p)$ taking values in the space of bounded operators on the Hilbert space of fast internal degrees of freedom.
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Nonlinear Schrodinger equation with time dependent potential
TL;DR: In this paper, a global well-posedness result for defocusing nonlinear Schrodinger equations with time dependent potentials was proved, motivated by physics and appeared also as a preparation for the analysis of the propagation of wave packets in a nonlinear context.
References
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Book
Mathematical Methods of Classical Mechanics
TL;DR: In this paper, Newtonian mechanics: experimental facts investigation of the equations of motion, variational principles Lagrangian mechanics on manifolds oscillations rigid bodies, differential forms symplectic manifolds canonical formalism introduction to pertubation theory.
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A universal instability of many-dimensional oscillator systems
TL;DR: In this article, the authors demonstrate the mechanism for a universal instability, the Arnold diffusion, which occurs in the oscillating systems having more than two degrees of freedom, which results in an irregular, or stochastic, motion of the system as if the latter were influenced by a random perturbation even though, in fact, the motion is governed by purely dynamical equations.
Book
Spectral asymptotics in the semi-classical limit
Mouez Dimassi,Johannes Sjöstrand +1 more
TL;DR: In this article, the WKB-method, stationary phase and h-pseudodifferential operators have been developed for quantum and classical mechanics, including results on the tunnel effect, the asymptotics of eigenvalues in relation to classical trajectories and normal forms, plus slow perturbations of periodic Schrodinger operators appearing in solid state physics.