Analysis und Mathematische Physik

(freitags 14 Uhr c.t., Hörsaal E05)

tragen Mitglieder unserer Arbeitsgruppen und auswärtige Gäste über

ihre Forschungsarbeiten vor.

Am 19. April 2002 sprach ** Dr. Doris Jakubaßa (LMU München)**
zum Thema

**Pseudodifferential Operators Applied to Relativistic
Hydrogen Like Ions**.

Am 10. Mai 2002 sprach ** Prof. Dr. Timo Weidl (Universität Stuttgart)**
über

**Spectral Estimates in Quantum Wave Guides**.

Am 17. Mai 2002 sprach **Laszlo Erdös (Georgia Institute of Technology, Atlanta)**
über eine

**Rayleigh-Type Isoperimetric Inequality with a Homogeneous Magnetic
Field and Its Application**

Abstract: We prove that the two dimensional free magnetic Schrödinger
operator, with a fixed constant magnetic field and Dirichlet
boundary conditions on a planar domain with a given area, attains its
smallest possible eigenvalue if the domain is a disk.
This
generalizes the classical Faber-Krahn inequality for magnetic fields.
The result is used to determine the low energy
asymptotic behaviour of the integrated density of states of the magnetic
Schrödinger operator with Poissonian random
potential.

Am 24. Mai 2002 sprach **Adrian Tip** über

** Absorptive Photonic Crystals: Recovery of a Unitary
Time Evolution and Complex Band Structure**

Abstract: In the presence of absorption Maxwell's equations for a macroscopic
dielectric show a time convolution. In particular, the
time evolution is unidirectoral. We show how Maxwell's equations can be
extended to a larger set of equations with a unitary
time evolution exp[iKt] in a suitable Hilbert space. We then consider
the situation where spatial periodicity is present. Such
systems are called "photonic crystals" and are at present intensely
studied in view of technological applications. After making
a Bloch-Floquet decomposition and a complex dilatation we find,
using analyticity and compactness argument, that the
complex dilated version of K has spectrum consisting of areas in the
lower complex half plane. We shall present a few
numerical results for a two-dimensional case. Some physical implications
are also discussed.

Am 31. Mai 2002 sprach **Tanja Suslina (St. Petersburg)** über

**Absolute Continuity of the Spectrum of Periodic Schrödinger Operators**.

Am 7. Juni 2002 sprach **Michail Solomyak (Weizmann Institute of Science)**
über

**Schrödinger Operators on Homogeneous Metric Trees: Spectrum in Gaps**

Abstract: This is a joint paper with Alex Sobolev from Brighton, accepted by
Rev. Math. Phys.
We consider a very special metric tree for which the spectrum of the free
Laplacian has the band-gap structure, and in each
band the spectrum has infinite multiplicity. We study the spectrum in gaps
appearing when we perturb the Laplacian by a
decaying potential (positive or negative but always of fixed sign).
The character of the results is close to the ones of Alama-Deift-Hempel and
of Sobolev concerning the spectrum in gaps of the Hill operator.

Am 14. Juni 2002 sprach **Christian Hainzl (LMU München)** über

**Mass Renormalization and Energy Level Shift in Non-Relativistic QED**.

Am 21. Juni 2002 sprach **Michail Birman (St. Petersburg, Rußland)** über

**Threshold Effects Near the Lower Edge of the Spectrum for Periodic Differential Operators
of Mathematical Physics**.

Abstract: We consider vector periodic differential operator A admitting a factorization
A=X*X, where X is a homogeneous differential operator of first order.
Many operators of mathematical physics have this form. The effects that
depend only on a rough behavior of the spectral decomposition of A in a
small neighborhood of zero are called threshold effects at the lower edge of
the spectrum. An example of a threshold effect is the behavior of a
differential operator in the small period limit. The "effective
characteristics", namely, the homogenized medium, the effective mass,
the effective Hamiltonian, etc. arise in these problems. We propose a
general approach to these problems based on the spectral perturbation
theory for operator-valued functions admitting analytic factorization.

Am 28. Juni 2002 sprach **Alexander Wolf (Erlangen)** über

**The Generalized Douglas-Kroll Transformation**.

Am 5. Juli 2002 sprachen **Bernard Helffer (Paris, Frankreich)** über

**Semi-classical Analysis of the Ground State of the Neumann Realization of a
Magnetic Schrödinger Operator**

und **Robert Seiringer (Princeton, U.S.A.)** über

** Poincaré inequalities**.

Am 12. Juli 2002 sprechen **M. Hirokawa** über

** Two Charges Interacting through a Massless Scalar Field:
Removal of Infrared and Ultraviolette Cutoffs **

und **Jan Derezinski (Warschau, Polen)** über

**Simple Models of the Infrared Problem**

Abstact: The infrared problem consists in the divergence of certain integrals
for small momenta in quantum field theory, notably in QED. It is believed that one of the
reasons for this problem is the appearance of non-Fock representations of
canonical commutation relations. I will describe a class of simple but non-trivial
models, studied recently by C. Gerard and myself, for which one can show rigorously
the existence of non-Fock asymptotic fields. I will describe a number of
open problems concerning these models.

Am 19. Juli 2002 spricht ** Semjon Wugalter (LMU)** über

** Enhanced Binding and Energy Shift in Nonrelativistic QED: New Results **.

A. M. Hinz, andreas.hinz@mathematik.uni-muenchen.de, 2002-07-08