TITLES AND ABSTRACTS

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**« Non linear spectral
problems and mean field models»**

**SpecNL**

**Paris – IHP - April 4 to April 8,
2005**

**Amandine
Aftalion (Paris 6),**

«Vortex
patterns for rotating Bose Einstein condensates» |

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**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),**

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« 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.

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**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.

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