Solutions faibles du problème de Cauchy pour certaines équations de Dirac non linéaires
Solvable nonlinear evolution PDEs in multidimensional space.
Sparse grids for the Schrödinger equation
We present a sparse grid/hyperbolic cross discretization for many-particle problems. It involves the tensor product of a one-particle multilevel basis. Subsequent truncation of the associated series expansion then results in a sparse grid discretization. Here, depending on the norms involved, different variants of sparse grid techniques for many-particle spaces can be derived that, in the best case, result in complexities and error estimates which are independent of the number of particles. Furthermore...
Spectral and scattering theory for symbolic potentials of order zero
Spectral Shift Function for the Perturbations of Schrödinger Operators at High Energy
2000 Mathematics Subject Classification: 35P20, 35J10, 35Q40.We give a complete pointwise asymptotic expansion for the Spectral Shift Function for Schrödinger operators that are perturbations of the Laplacian on Rn with slowly decaying potentials.
Spherical harmonics and maximal estimates for the Schrödinger equation.
Splitting of the Dirac operator in the nonrelativistic limit
Stability of matter for the Hartree-Fock functional of the relativistic electron-positron field.
Stability of the Brown-Ravenhall operator.
Stable blow up dynamics for the critical co-rotational Wave Maps and equivariant Yang-Mills Problems
This note summarizes the results obtained in [30]. We exhibit stable finite time blow up regimes for the energy critical co-rotational Wave Map with the target in all homotopy classes and for the equivariant critical Yang-Mills problem. We derive sharp asymptotics on the dynamics at blow up time and prove quantization of the energy focused at the singularity.
Stationary solutions for a Schrödinger-Poisson system in .
Stationary solutions of the generalized Smoluchowski-Poisson equation
The existence of steady states in the microcanonical case for a system describing the interaction of gravitationally attracting particles with a self-similar pressure term is proved. The system generalizes the Smoluchowski-Poisson equation. The presented theory covers the case of the model with diffusion that obeys the Fermi-Dirac statistic.
Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in
We present a novel approach for bounding the resolvent of for large energies. It is shown here that there exist a large integer and a large number so that relative to the usual weighted -norm, for all . This requires suitable decay and smoothness conditions on . The estimate (2) is trivial when , but difficult for large since the gradient term exactly cancels the natural decay of the free resolvent. To obtain (2), we introduce a conical decomposition of the resolvent and then sum over...
Supersymmetry and Ghosts in Quantum Mechanics
2000 Mathematics Subject Classification: 81Q60, 35Q40.A standard supersymmetric quantum system is defined by a Hamiltonian [^H] = ½([^Q]*[^Q] +[^Q][^Q]*), where the super-charge [^Q] satisfies [^Q]2 = 0, [^Q] commutes with [^H]. So we have [^H] ≥ 0 and the quantum spectrum of [^H] is non negative. On the other hand Pais-Ulhenbeck proposed in 1950 a model in quantum-field theory where the d'Alembert operator [¯] = [(∂2)/( ∂t2)] − Δx is replaced by fourth order operator [¯]([¯] + m2), in order to...
Sur le spectre de l’équation de Dirac (dans ou ) avec champ magnétique
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 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)...
Sur l'hamiltonien effectif associé à l'opérateur de Schrödinger magnétique avec potentiel périodique
Symétrisations indépendantes du temps pour certains opérateurs du type de Schrödinger. I
Si danno condizioni sufficienti e condizioni necessarie affinché il problema di Cauchy per alcuni operatori di tipo Schrödinger sia ben posto in spazi di Sobolev. Gli operatori qui considerati sono operatori di Schrödinger con potenziali vettoriali complessi, una generalizzazione degli operatori di 2-evoluzione nel senso di Petrowsky, e alcuni sistemi tipo Leray-Volevich di operatori lineari a derivate parziali. Il metodo che usiamo in questo articolo è la simmetrizazione degli operatori non dipendenti...
Symmetry operators of a Hartree-type equation with quadratic potential.