Semiclassical quantization of circular billiard in homogeneous magnetic field: Berry-Tabor approach.
Semiclassical scattering by the Coulomb potential
Semi-classical Schrödinger equations with harmonic potential and nonlinear perturbation
Semiclassical spectral estimates for Toeplitz operators
Let be a compact Kähler manifold with integral Kähler class and a holomorphic Hermitian line bundle whose curvature is the symplectic form of . Let be a Hamiltonian, and let be the Toeplitz operator with multiplier acting on the space . We obtain estimates on the eigenvalues and eigensections of as , in terms of the classical Hamilton flow of . We study in some detail the case when is an integral coadjoint orbit of a Lie group.
Semiclassical states of nonlinear Schrödinger equations with bounded potentials
Using some perturbation results in critical point theory, we prove that a class of nonlinear Schrödinger equations possesses semiclassical states that concentrate near the critical points of the potential .
Short waves through thin interfaces and 2-microlocal measures
Singular Bohr–Sommerfeld rules for 2D integrable systems
Spectral projection, residue of the scattering amplitude and Schrödinger group expansion for barrier-top resonances
We study the spectral projection associated to a barrier-top resonance for the semiclassical Schrödinger operator. First, we prove a resolvent estimate for complex energies close to such a resonance. Using that estimate and an explicit representation of the resonant states, we show that the spectral projection has a semiclassical expansion in integer powers of , and compute its leading term. We use this result to compute the residue of the scattering amplitude at such a resonance. Eventually, we...
Spectral theory of damped quantum chaotic systems
We investigate the spectral distribution of the damped wave equation on a compact Riemannian manifold, especially in the case of a metric of negative curvature, for which the geodesic flow is Anosov. The main application is to obtain conditions (in terms of the geodesic flow on and the damping function) for which the energy of the waves decays exponentially fast, at least for smooth enough initial data. We review various estimates for the high frequency spectrum in terms of dynamically defined...
Spectre de l'opérateur de Schrödinger magnétique avec symétrie d'ordre six
Stability and semiclassics in self-generated fields
We consider non-interacting particles subject to a fixed external potential and a self-generated magnetic field . The total energy includes the field energy and we minimize over all particle states and magnetic fields. In the case of spin-1/2 particles this minimization leads to the coupled Maxwell-Pauli system. The parameter tunes the coupling strength between the field and the particles and it effectively determines the strength of the field. We investigate the stability and the semiclassical...
Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains
We prove wellposedness of the Cauchy problem for the nonlinear Schrödinger equation for any defocusing power nonlinearity on a domain of the plane with Dirichlet boundary conditions. The main argument is based on a generalized Strichartz inequality on manifolds with Lipschitz metric.
Study of the behavior of quantum dynamics as
Sur le caractère bien posé des équations de Schrödinger non linéaires
Sur le spectre semi-classique d’un système intégrable de dimension 1 autour d’une singularité hyperbolique
Dans cette article on décrit le spectre semi-classique d’un opérateur de Schrödinger sur avec un potentiel type double puits. La description qu’on donne est celle du spectre autour du maximum local du potentiel. Dans la classification des singularités de l’application moment d’un système intégrable, le double puits représente le cas des singularités non-dégénérées de type hyperbolique.
Sur le spectre semi-classique d’un système intégrable de dimension 1 autour d’une singularité hyperbolique
Dans cet article on décrit le spectre semi-classique d’un opérateur de Schrödinger sur avec un potentiel type double puits. La description qu’on donne est celle du spectre autour du maximum local du potentiel. Dans la classification des singularités de l’application moment d’un système intégrable, le double puits représente le cas des singularités non-dégénérées de type hyperbolique.
Sur les mesures de Wigner.
We study the properties of the Wigner transform for arbitrary functions in L2 or for hermitian kernels like the so-called density matrices. And we introduce some limits of these transforms for sequences of functions in L2, limits that correspond to the semi-classical limit in Quantum Mechanics. The measures we obtain in this way, that we call Wigner measures, have various mathematical properties that we establish. In particular, we prove they satisfy, in linear situations (Schrödinger equations)...
The Cauchy problem for the Gross–Pitaevskii equation
The level crossing problem in semi-classical analysis I. The symmetric case
We describe a microlocal normal form for a symmetric system of pseudo-differential equations whose principal symbol is a real symmetric matrix with a generic crossing of eigenvalues. We use it in order to give a precise description of the microlocal solutions.
The level crossing problem in semi-classical analysis. II. The Hermitian case
This paper is the second part of the paper ``The level crossing problem in semi-classical analysis I. The symmetric case''(Annales de l'Institut Fourier in honor of Frédéric Pham). We consider here the case where the dispersion matrix is complex Hermitian.