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Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold

Nalini Anantharaman, Stéphane Nonnenmacher (2007)

Annales de l’institut Fourier

We study the high-energy eigenfunctions of the Laplacian on a compact Riemannian manifold with Anosov geodesic flow. The localization of a semiclassical measure associated with a sequence of eigenfunctions is characterized by the Kolmogorov-Sinai entropy of this measure. We show that this entropy is necessarily bounded from below by a constant which, in the case of constant negative curvature, equals half the maximal entropy. In this sense, high-energy eigenfunctions are at least half-delocalized....

Hall's transformation via quantum stochastic calculus

Paula Cohen, Robin Hudson, K. Parthasarathy, Sylvia Pulmannová (1998)

Banach Center Publications

It is well known that Hall's transformation factorizes into a composition of two isometric maps to and from a certain completion of the dual of the universal enveloping algebra of the Lie algebra of the initial Lie group. In this paper this fact will be demonstrated by exhibiting each of the maps in turn as the composition of two isometries. For the first map we use classical stochastic calculus, and in particular a stochastic analogue of the Dyson perturbation expansion. For the second map we make...

Hamiltonian identification for quantum systems: well-posedness and numerical approaches

Claude Le Bris, Mazyar Mirrahimi, Herschel Rabitz, Gabriel Turinici (2007)

ESAIM: Control, Optimisation and Calculus of Variations

This paper considers the inversion problem related to the manipulation of quantum systems using laser-matter interactions. The focus is on the identification of the field free Hamiltonian and/or the dipole moment of a quantum system. The evolution of the system is given by the Schrödinger equation. The available data are observations as a function of time corresponding to dynamics generated by electric fields. The well-posedness of the problem is proved, mainly focusing on the uniqueness of the...

High frequency limit of the Helmholtz equations.

Jean-David Benamou, François Castella, Theodoros Katsaounis, Benoit Perthame (2002)

Revista Matemática Iberoamericana

We derive the high frequency limit of the Helmholtz equations in terms of quadratic observables. We prove that it can be written as a stationary Liouville equation with source terms. Our method is based on the Wigner Transform, which is a classical tool for evolution dispersive equations. We extend its use to the stationary case after an appropriate scaling of the Helmholtz equation. Several specific difficulties arise here; first, the identification of the source term ( which does not share the...

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