The noncommutative Dirac equation for some wave functions.
The nonrelativistic limit of the nonlinear Dirac equation
The projected single-particle Dirac operator for Coulombic potentials.
The quantum 3D superparticle.
The quantum stability problem for time-periodic perturbations of the harmonic oscillator
The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave packet approach
We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter . The high-frequency (or: semi-classical) parameter is . We let and go to zero simultaneously. We assume that the zero energy is non-trapping for the underlying classical flow. We also assume that the classical trajectories starting from the origin satisfy a transversality condition, a generic assumption.Under these assumptions, we prove that the solution radiates in the outgoing...
The relation between Maxwell, Dirac, and the Seiberg-Witten equations.
The return of the quartic oscillator. The complex WKB method
The role of locality in perturbation theory
The scattering problem for a noncommutative nonlinear Schrödinger equation.
The Schrödinger equation on non-stationary domains.
The second-order Klein-Gordon field equation.
The Semiclassical Electron in a Magnetic Field and Lattice. Some Problems of Low Dimensional "Periodic" Topology.
The smooth continuation method in optimal control with an application to quantum systems
The motivation of this article is double. First of all we provide a geometrical framework to the application of the smooth continuation method in optimal control, where the concept of conjugate points is related to the convergence of the method. In particular, it can be applied to the analysis of the global optimality properties of the geodesic flows of a family of Riemannian metrics. Secondly, this study is used to complete the analysis of two-level dissipative quantum systems, where the system...
The smooth continuation method in optimal control with an application to quantum systems
The motivation of this article is double. First of all we provide a geometrical framework to the application of the smooth continuation method in optimal control, where the concept of conjugate points is related to the convergence of the method. In particular, it can be applied to the analysis of the global optimality properties of the geodesic flows of a family of Riemannian metrics. Secondly, this study is used to complete the analysis of two-level dissipative quantum systems, where the system...
The spaces of quantum states for some harmonic oscillator potentials on the punctured plane C∖{0}
We consider equivariant solutions of Schrödinger equations on C∖{0} with harmonic oscillator potentials. We determine the spaces of equivariant quantum states in three cases: for an isotropic and anisotropic harmonic oscillator potential centered at 0, and for a potential not centered at 0.
The spectral cutting problem
The spectral properties of the Schrödinger operator in nonseparable Hilbert spaces
The spectrum of Schrödinger operators with random magnetic fields
We shall consider the Schrödinger operators on with the magnetic field given by a nonnegative constant field plus random magnetic fields of the Anderson type or of the Poisson-Anderson type. We shall investigate the spectrum of these operators by the method of the admissible potentials by Kirsch-Martinelli. Moreover, we shall prove the lower Landau levels are infinitely degenerated eigenvalues when the constant field is sufficiently large, by estimating the growth order of the eigenfunctions...
The symmetry algebra and conserved Currents for Klein-Gordon equation on quantum Minkowski space
The symmetry operators for Klein-Gordon equation on quantum Minkowski space are derived and their algebra is studied. The explicit form of the Leibniz rules for derivatives and variables for the case Z=0 is given. It is applied then with symmetry operators to the construction of the conservation law and the explicit form of conserved currents for Klein-Gordon equation.