Analysis und Mathematische Physik

apl. Prof. Dr. Andreas M. Hinz

Universität München, Sommersemester 2002

In meinem gemeinsam mit Prof. Dr. R. Denk und Prof. Dr. H. Siedentop , durchgeführten Oberseminar
(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