7,248 research outputs found
Semiclassical analysis and sensitivity to initial conditions
We present several recent results concerning the transition between quantum
and classical mechanics, in the situation where the underlying dynamical system
has an hyperbolic behaviour. The special role of invariant manifolds will be
emphasized, and the long time evolution will show how the quantum
non-determinism and the classical chaotic sensitivity to initial conditions can
be compared, and in a certain sense overlap
Semiclassical approximation and noncommutative geometry
We consider the long time semiclassical evolution for the linear
Schr\"odinger equation. We show that, in the case of chaotic underlying
classical dynamics and for times up to , the
symbol of a propagated observable by the corresponding von Neumann-Heisenberg
equation is, in a sense made precise below, precisely obtained by the
push-forward of the symbol of the observable at time . The corresponding
definition of the symbol calls upon a kind of Toeplitz quantization framework,
and the symbol itself is an element of the noncommutative algebra of the
(strong) unstable foliation of the underlying dynamics
Recent results in semiclassical approximation with rough potentials
This is an extended abstract for the conference "Microlocal2011 : Microlocal
Methods in Mathematical Physics and Global Analysis Universit\"at T\"ubingen,
June 14 - 18, 2011
The Schr\"odinger Equation in the Mean-Field and Semiclassical Regime
In this paper, we establish (1) the classical limit of the Hartree equation
leading to the Vlasov equation, (2) the classical limit of the -body linear
Schr\"{o}dinger equation uniformly in N leading to the N-body Liouville
equation of classical mechanics and (3) the simultaneous mean-field and
classical limit of the N-body linear Schr\"{o}dinger equation leading to the
Vlasov equation. In all these limits, we assume that the gradient of the
interaction potential is Lipschitz continuous. All our results are formulated
as estimates involving a quantum analogue of the Monge-Kantorovich distance of
exponent 2 adapted to the classical limit, reminiscent of, but different from
the one defined in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343
(2016), 165-205]. As a by-product, we also provide bounds on the quadratic
Monge-Kantorovich distances between the classical densities and the Husimi
functions of the quantum density matrices.Comment: 33 page
Quantum singular complete integrability
We consider some perturbations of a family of pairwise commuting linear
quantum Hamiltonians on the torus with possibly dense pure point spectra. We
prove that the Rayleigh-Schr{\"o}dinger perturbation series converge near each
unperturbed eigenvalue under the form of a convergent quantum Birkhoff normal
form. Moreover the family is jointly diagonalised by a common unitary operator
explicitly constructed by a Newton type algorithm. This leads to the fact that
the spectra of the family remain pure point. The results are uniform in the
Planck constant near . The unperturbed frequencies satisfy a small
divisors condition %(Bruno type condition (including the Diophantine case) and
we explicitly estimate how this condition can be released when the family tends
to the unperturbed one
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