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: