« Non linear spectral problems and mean field models»


Paris – IHP - April 4 to April 8, 2005




Amandine Aftalion (Paris 6),

«Vortex patterns for rotating Bose Einstein condensates»




Emmanuelle Amar-Servat (Paris 13),


«A non linear eigenvalue problem arising from Zakharov-Shabat systems»


Thanks to the inverse scattering transform, the focusing non linear Schrödinger equation can be solved formally. The first step of this method consists in a spectral study of Zakharov-Shabat systems. Since these systems are not self-adjoint, the eigenvalues are not confined to the real axis, but can be scattered within the complex plan. In collaboration with A. Tovbis (Central Florida University) I obtained the configuration of the Stokes lines of the problem for which there exists an eigenvalue. We applied the result to examples and localized their eigenvalues



Anton Arnold (Muenster),


«Nonlinear quantum kinetic equations: well-posedness analysis and dispersive effects»




Naoufel Ben Abdallah (Toulouse),


«WKB Schemes for the Schroedinger equation»


The Schroedinger equation is one of the most used models for the simulation of quatum transport in electronic nanostructures. Macroscopic quantities such as particle density or current density are computed as an integral over the energy variable of single state quantities.

Numerically, the integral  is computed  thanks to a suitable numerical integration method and iplies a large number of Schroedinger equations to be solved. An energy grid containing a certain amount of points is  constructed and the wavefunction for each of these points is computed by solving the Schroedinger equation.  For high energies, the single states have a small de Broglie length and oscillate at much smaller space scale than for low energies.

Besides, the macroscopic quantities like  particle density are relatively smooth functions of the position variable. Using the same spatial grid for all the energies to solve the Schroedinger equations with standard finite element or finite difference methods requires a large number of points thus increasing unnecessarily the numerical cost. The approach adapted here is to use the WKB asysmptotic in order to reduce the number of grid points. Indeed, the need for a refined spatial grid is due to the linear or polynomial interpolation underlying the standard finite difference or finite element methods. Therefore, if the oscillation phase is known accurately, the phase factor could be used to interpolate the nodal values of the wave function and a coarser grid can be allowed. In the one dimensional case, this can be done since the WKB asymptotics provide us with an explicit formula for this phase factor. We shall also present the waveguide case where the multidimensional Schroedinger equation can be written as a copupled system of one dimensional Schroedinger equation and for which approximate WKB statistic are used.




Virginie Bonnaillie (Rennes),


«Computing the steady states of an asymptotic model for quantum transport»


We use the asymptotic model derived from the theoretical analysis for resonant tunneling diodes to construct a fast numerical algorithm. We obtain in a short time quantitative results which allow to observe non linear phenomena like hysteresis, and more complex bifurcation diagrams in the presence of multiple wells.

These computed asymptotic solutions can be used as initial data for Newton algorithm in the numerical treatment of the complete quantum model. Alternatively this simple model gives also a good insight of the dependence of the bifurcation diagram with respect to some quantitative data : geometry of the barriers and wells, donor density and the applied bias.




Jean-François Bony (Bordeaux),


« Microlocal solutions of Schrödinger equations at a maximum point of the potential »




Yvan Castin (ENS Paris),


 «Ultracold fermionic atomic gases»


Experimental advances have been made recently in producing degenerate fermionic atomic gases. One tools relies on the possibility to tune the interacting force between the atoms by applying a uniform magnetic field (Feshbach resonance method). This permits the analysis of the transition between molecular Bose-Einstein condensates  and a Cooper pair BCS condensate, via the strong interation regime. After a review of experimental results, theoretical analytical and numerical results will be presented in the mean field approximation as well as on the N body problem.




Frederic Dross,


«Multi-active region semiconductor laser modeling»


A quantum-well semiconductor laser is a complex optoelectronic device in which a guided light-wave is amplified by stimulated electron-hole recombination in a quantum well and the electrons and holes have to be injected into the quantum well. In order to model such a device, one is faced with (i) a quantum mechanical problem of determining the electronic states and transition rates in a quantum well; (ii) a classical electromagnetic problem of wave propagation in a guiding structure; (iii) and a semiclassical problem of transport of carriers in a semiconductor heterostructure. We have recently developed a self-consistent laser diode model compatible with multi-junction lasers. These lasers have applications in high-power and/or high-quantum-efficiency laser diodes. They consist of several active quantum wells monolithically stacked and electrically connected via tunnel junctions. The model includes non-local quantum-well interaction via the electromagnetic field, is thoroughly based on Fermi-Dirac statistics, and self-consistently calculates the tunnel current through the tunnel junctions. This model has been successfully used to design multi-junction lasers with internal quantum efficiency of more than 100 %.



Jurg Fröhlich (Zurich),


  «Thermodynamique et mécanique statistique quantique – 100 ans après Einstein»


Quelques résultats récents concernant le problème de dériver les lois fondamentales de la thermodynamique de la mécanique statistique du non équilibre.



Sandro Graffi (Bologne),


«Mean field approximation of quantum systems  uniform with respect to the Planck constant »



Frédéric Hérau (Reims),


«Uniform bounds and exponential time decay results for the Vlasov-Poisson-Fokker-Planck system»


We consider the non-linear VPFP system with a Coulombian repulsive interaction potential and a generic confining potential in three or more dimensions. Using spectral and kinetic methods we prove existence and uniqueness in weighted spaces and for small data we find an explicit exponential rate of convergence to the equilibrium in terms of the corresponding Witten Laplacian associated to the linear equation.



Giovanni Jona-Lasinio (Rome),


«Spectral and KAM theory for some non linear Schroedinger equations»


We study the stationary solutions for a  class of reversible nonlinear, nolocal Schroedinger equations in arbitrary space dimensions. We then analyse a subclass which includes cases of physical interest. In one space dimension a theorem of Kuksin implies for this subclass the existence of finite dimensional invariant tori in a neighborhood of each stationary solution. We conclude with some comments on invariant measures.



Hans Christoph Kaiser (Berlin),


«An open quantum system driven by an external flow»


We regard an open quantum system which is embedded into a potential flow. The system is driven by this flow acting on the boundary of the bounded spatial domain designated to quantum mechanics. We investigate the spectral properties of the corresponding essentially non-selfadjoint Schroedinger-type operator.



Laurent Michel (Paris 13)


« Scattering amplitude for the Schrodinger equation with strong magnetic field »



Yassine Patel (Rennes),


«Asymptotical model for far-from-equlibrium systems : non-linear-1D-Schrodinger-Poisson-systems with quantum wells.»



Galina Perelman (Ecole Polytechnique Paris)


 «Absolutely continuous spectrum of multi-dimensional Schrodinger operators with slowly decaying potentials»


We consider three-dimensional Schrodinger operators with potentials decaying as $|x|^{-1/2-\epsilon}, \epsilon >0$. We show that the a.c. spectrum of these operators is essentially supported by $[0,\infty)$ provided the gradient angular component of the potential decays as  $|x|^{-3/2-\epsilon}$.



Carlo Presilla (Rome),


«Analytical probabilistic approach to the spectral properties of many-body lattice quantum systems»


I review a recently proposed probabilistic approach to the study of the spectral properties of many-body lattice quantum systems. These properties, in particular the ground-state energy, are determined as an exact series expansion in the cumulants of the long-time multiplicities of two macroscopic variables, namely the potential and hopping energies of the system. Once the cumulants are known, even at a finite order, this approach provides analytical results as a function of the Hamiltonian parameters



Joachim Rehberg (Berlin),


«Some analytical ideas concerning the Quantum Drift Diffusion system»



Andrea Sacchetti (Modena),


«Nonlinear double well Schroedinger equations in the semiclassical limit»


We consider time-dependent Schroedinger equations with a double well potential and an external nonlinear, both local and non-local, perturbation. In the semiclassical limit, the finite dimensional eigenspace associated to the lowest eigenvalues of the linear operator is almost invariant for times of the order of the beating period and the dominant term of the wavefunction is given by means of the solutions of a finite dimensional dynamical system. In the case of local nonlinear perturbation we assume the spatial dimension d=1 or d=2.



Alessandro Teta (l'Aquila),


«On the asymptotic dynamics of a quantum system composed by heavy and light particles, with application to decoherence»


We consider a quantum system of K heavy plus N light particles with an initial state in a product form and we characterize the asymptotic dynamics for $m/M \rightarrow 0$, with a control of the error.

The result is then applied to the explicit computation of the decoherence effect on one heavy particle due to the scattering of the light ones.



Oliver Vanbésien (Lille),


«Modelling of metamaterials - from ab initio methods to homogeneization techniques»


In this communication, we will explore the abnormal electromagnetic and optical properties of periodically artificial materials named "metamaterials". One class of these new structures exhibits particular properties such as negative refraction or evanescent wave amplification (left-handed materials) which could be at the origin of original concepts over the whole wavelength spectrum, from microwave to infrared and optics, as for example the so-called "perfect lens". Part of the talk will be devoted to the simulation of these materials, generally structured at a sub-wavelength scale, starting from ab-initio simulations to homogeneization techniques in order to derive ad-hoc parameters needed when practical applications are envisaged.