Address:Institut für Mathematik
Universität Wien
Botzmanngasse 9
1090 Wien, Austria
Telephone: +43/1 4277 506 79
Fax: +43/1 4277 506 65
Email: hans.peter.stimming@univie.ac.at
Advisor: N. Mauser
Co-advisor:
Research: My work concentrates on the analysis and numerics of Nonlinear Schroedinger equations.
A main interest is in the so-called semiclassical limit of Nonlinear Schroedinger equations, which is the limit of the equation if the (scaled) Planck constant is tending to zero.
At the TU Berlin I started working on the Linear Schroedinger eq. where Wigner measure techniques give the semiclassical limit and at the same time provide a stringent tool to verify numerical schemes of finite difference type in this limit. I worked on applying this theory to the Dufort-Frankel scheme for the linear Schroedinger eq.
For nonlinear Schroedinger equations this method generally fails. An alternative is inverse scattering theory which so far is available only for a particular case in one dimension. In a cooperation with Spyros Kamvissis we tried to generalize this result to an integrable two-dimensional case, the Davey-Stewartson system.
In cooperation with Remi Carles from the Univ. Bordeaux I we studied the semiclassical limit of the Hartree equation by a method based on geometric considerations. It consists of treating the equation with a particular scaling of the nonlinearity ( with respect to the "semiclassical parameter" ) and ensuring that the solution follows a particular geometric pattern. In this case it is possible to "extract" the action of the nonlinear part of the equation in the frame of a WKB analysis. A result of this work will be publishable shortly.
On the numeric side, I work on spectral methods which have proven to be more effective for the task of semiclassics. For making simulations in three space dimensions feasible I make use of a massive parallel computer, the University of Vienna's "Schroedinger 2". This means that the possibilities of adapting a given code for a distributed memory parallel architecture had to be studied. I applied parallel numerics to an equation known as "Schrödinger-Poisson-Xalpha" equation, which arises from a local density approximation of quantum particle dynamics. For this equation I also did a study on the existence theory in one dimension which is not covered by the standard results on NLS.
I also applied the numeric tools obtained for this equation to the above mentioned Davey-Stewartson system, investigating the Soliton type behaviour of the integrable versions of the equation and blowup phenomena. This work is done in cooperatoin with Christophe Besse from the U.P.S. Toulouse.
Publications:
P. A. Markowich, P. Pietra, C. Pohl, H. P. Stimming: ``A Wigner-measure analysis of the Dufort-Frankel scheme for the Schr\"odinger equation.'', SIAM J. Numer. Anal. 40, No.4, 1281-1310 (2002).
Weizhu Bao, N. J. Mauser, H. P. Stimming: ``Effective one particle quantum dynamics of electrons : a numerical study of the Schr\"odinger-Poisson-X$\alpha$ model", submitted to CMS (2003)
Conference talks:
Pauli seminar on applied Mathematics, Univ. of Vienna, June 2000. "Introduction to Nonlinear Schrödinger equations"
TMR Workshop on ADVANCES IN MATHEMATICAL SEMICONDUCTOR MODELING, Pavia (Italy); September 21 - 23, 2000 "Wigner-measure analysis of the Dufort-Frankel scheme for the Schroedinger equation.''
WPI workshop "Nonlinear Dispersive Equations" at ESI, Vienna, 19.07.2001, Title: "The Schrödinger-Poisson-Xalpha model."
SIAM 50th Anniversary Meeting, July 8-12 2002, Philadelphia PA, USA, Title: "The Schrödinger-Poisson-Xalpha model."
Nanolab Spring School QUANTUM TRANSPORT IN NANOSTRUCTURES , May 19-23, 2003, Universite Paul Sabatier Toulouse, France, Title: "Effective one particle quantum dynamics of electrons : The Schrödinger-Poisson-X-alpha model. Simulations on parallel architecture."
Travel activities:
Colloque "Analyse des Equations aux Dérivées Partielles", 3 - 7 June 2002, Forges-les-Eaux (France)
Colloque "Systèmes hyperboliques et oscillations", 18 - 20 September 2002. Institut de Mathématiques de Bordeaux, Université Bordeaux 1.
CEMRACS 2003, Numerical methods for hyperbolic and kinetic problems, 21 July - 29 August 2003, Marseille (France). Project "NLSNUM "