Address:Institut für Mathematik
Universität Wien
Nordbergstr. 15/C717
1090 Wien, Austria
Telephone: +43 1 4277 507 17
Fax: +43 1 4277 9 506
Email: rada.weishaeupl@univie.ac.at
Advisor: P. A. Markowich
Research: My area of research focuses on the Bose-Einstein Condensate. The properties of a BEC at temperatures much smaller than the critical condensation temperature are modelled by the Gross-Pitaevskii equation, which is a nonlinear Schroedinger equation with cubic nonlinearity and a harmonic potential. By changing the shape of the trapping potential it is possible to produce dilute and cold gases in highly anisotropic configurations, where the motion of particles is quenched in one or two directions. Thus, my work concetrates on the dimension reduction of the GPE, for both disk shaped and cigar shaped condensates.
We can prove rigorously the limit for the case in which the initial wavefunction is concentrated on the first energy subband. This is performed by projecting the three dimensional wavefunction on the ground state corresponding to the potential in the strongly confined direction. This leads to a two-dimensional GPE with an effective scattering length.
In the general case (initial state lies in more than one subband) we have formal results. By Fourier expansion of the wavefunction with respect to the eigenstates corresponding to the potential in the strongly confined direction and then averaging, we obtain in the limit a coupled system of two dimensional Schrödinger equations representing the coupling of all energy subbands.
Furthermore we also treat the numerics of dimension reduction for the GPE. In the case of a single subband initial datum we use a
time-splitting spectral method (TSSP), for the much more complicated case of general initial data we designed two numerical discretization techniques which take account of the complicated coupling of the (band) Schrödinger equations. The first method is based on a multistep time splitting-spectral scheme, whereas the second discretization is based on a Hermite pseudo-spectral method combined with a Crank-Nicolson type method.
Publications:
The Nonlinear Schrödinger equation with a strongly anisotropic harmonic
potential, with N. Ben Abdallah, F. Mèhats, and Ch. Schmeiser, SIAM J. Math. Anal.,
Vol 37, No.1 (2005) 189-199
On the Gross-Pitaevskii equation with strongly anisotropic confinement:
formal asymptotics and numerical experiments, with W. Bao, P. A. Markowich,
and Ch. Schmeiser, M3AS, Vol 15, No. 5 (2005) 767-782
A Hermite Pseudo-spectral method for solving systems of Gross-Pitaevskii Equations, with Ch. Schmeiser, P. A. Markowich, and J. P. Borgna, preprint
Convergence rate of dimension reduction on Bose-Einstein condensates, with W. Bao, Y. Ge, D. Jaksch, and P.A. Markowich, preprint
Activities:
2005: Talk at the 3rd ALPHA-Meeting in Strobl, Austria
Talk at the 3rd A-HYKE Meeting in Rome, Italy
Participation at
Nanoscale Material Interfaces: Experiment, Theory and Simulation at the
National University of Singapore
2004: Participation at Modeles Mathematiques non lineares pour les
suprafluides et la condensation dans un gaz at University of
Picardie Amiens, France
Three months research stay at Universidad de Chile, Santiago de Chile
2003: Reserch visit
at the University Paul Sabatier Toulouse, France
Talk at the meeting of
the Austrian Mathematical Society in Bolzano, Italy
Reserch visit at
the University of Technology Berlin, Germany
Talk at ICIAM in Sydney, Australia
Participation at Emerging
Applications of the Nonlinear Schrödinger Equations at IPAM (UCLA), USA