WK Differential Equations - Student Member:
Christof Sparber

Address:
Institut für Mathematik
Universität Wien
Nordbergstraße 15
(Office no. C 715)
1090 Wien, Austria

Telephone: +43 1 4277 507 14
Fax: +43 1 4277 506 50
Email: christof.sparber@univie.ac.at

Advisor: P. Markowich
Co-Advisor: N. Mauser



RESEARCH:

Analysis of partial differential equations arising from mathematical Physics. In particular, my current focus of research mainly is on the following two topics:

• Geometrical optics asymptotics:
  
  We study the high frequency limit for linear and/or nonlinear dispersive PDEs, in particular those which arise in the description of quantum mechanical systems, i.e. (time-dependent) Schrödinger type and Dirac type equations: In this quantum mechanical context high frequency asymptotics are usually considered to be equivalent to (semi-) classical limits. One research direction is the investigation of Wigner transformation methods in comparison with more traditional WKB-type asymptotics. Another point of interest is the extension of WKB-approximations to (so far only weakly) nonlinear PDEs. In particular, we studied in this way the semi-classical asymptotics for the (weakly) coupled Maxwell-Dirac system. Recently we investigate the simultaneous high-frequency and homogenization limit for (weakly) nonlinear PDEs with coefficients that are periodic w.r.t. to some lattice.
  

• Quantum Fokker-Planck type models:
  
  The inclusion of dissipative and/or diffusive phenomena into quantum mechanics appears in the modeling of so called open quantum systems, i.e. quantum systems interacting with their environment. In order to be physically acceptable (i.e. conservation of mass etc.) the considered evolution equations have to be in Lindblad form. We mainly focus on a particular class of such Lindblad models, namely the so called Quantum Fokker-Planck equation. To take into account the Coulomb interaction between particles we moreover include a self-consistent coupling to a Poisson equation for the electric potential. The considered equation simplifies to the classical kinetic Vlasov-Fokker-Planck equation (or Kramers equation) in the classical limit. We recently proved well posedness of the nonlinear system as an evolution equation on the Banach space of trace-class operators. The long time behavior of the linear model (i.e. without Poisson-coupling) in a harmonic confinement potential has also been investigated. We hope to extend these latter results to more general models, in particular we try to carryover entropy-entropy-dissipation techniques, already successfully used in classical kinetic theory, into this quantum mechanical context.
  
  

PUBLICATIONS:

• Refereed publications:
  
  
  P1. Wigner functions vs. WKB-techniques in multivalued geometrical optics, Asympt. Anal. 33 (2003), no. 2, 153-187;
  joint work with N. Mauser, P. Markowich: preprint ps.gz.
  
  
  P2. Semiclassical asymptotics for the Maxwell-Dirac system, J. Math. Phys. 44 (2003), issue 10, 4555-4572;
  joint work with P. Markowich: preprint ps.gz.
  
  
  P3. On the long time behavior of the quantum Fokker-Planck equation, Monatsh. f. Math. 141 (2004), no. 3, 237-257;
  joint work with J. A. Carrillo, J. Dolbeault, P. Markowich: preprint ps.gz.
  
  
  P4. Semiclassical asymptotics for weakly nonlinear Bloch waves, to appear in J. Stat. Phys. 2004;
  joint work with R. Carles, P. Markowich: preprint ps.gz.
  
  
  P5. Quantum dynamical semigroups for quantum diffusion models with Hartree interaction, to appear in Comm. Math. Phys. 2004;
  joint work with A. Arnold: preprint ps.gz.
  

• Proceedings:
  
  
  Pr1. Highly oscillatory partial differential equations, to appear in the Proceedings of the ICIAM, Sydney 2003;
  joint work with P. Markowich: preprint ps.gz.
  

• Master Thesis:
  
  A Wignerfunction Approach to (Multivalued) Geometrical Optics; advisors: N. Mauser, P. Markowich
  

• Ph.D. Thesis:
  
  Rigorous Results in Nonlinear Quantum Dynamics; advisors: P. Markowich, N. Mauser: thesis ps.gz.
  
  
  
  
  
  

SCIENTIFIC ACTIVITIES:

• August 2001: Participant in the summer school: Relativistic quantum models.
• November 2001: Talk given at the Workshop on Semiclassical limits: WKB methods vs. Wigner transform methods.
• May 2002: Talk given at the Vienna Workshop on Nonlinear Models and Analysis.
  
• January - February 2003: HYKE pre-doc employment at the Institute for Applied Mathematics, University of Saarbrücken: Research group of A. Arnold.
  
• February 2003: Participant in the frist annual meeting of the HYKE-network: A HYKE.
• March, September 2003: Short term research visits (together with P. Markowich) at the Institute for Computational Engineering and Sciences (formerly known as TICAM).
• April 2003: Participant in the workshop on Semi-classical Methods in Physics and Chemistry at the Mathematical Science and Research Institute, Berkeley.
• June - July 2003: HYKE pre-doc employment at the Unité de Mathématiques Pures et Appliquées, Ecole Normale Supérieure de Lyon: Research group of C. Villani.
  
• November 2003: Talk given at the mathematical research institute Oberwolfach: Workshop on Classical and Quantum Mechanical Models of Many-Particle Systems.