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Spectral Theory of Relativistic Operators

Participation in INI programmes is by invitation only. Anyone wishing to apply to participate in the associated workshop(s) should use the relevant workshop application form.

23rd July 2012 to 17th August 2012
Malcolm Brown [Cardiff], Cardiff University
Maria Esteban [CEREMADE], CNRS & Université Paris-Dauphine
Karl Schmidt [Cardiff], Cardiff University
Heinz Siedentop [Munich], Ludwig-Maximilians-Universität München


Programme Theme

Relativistic operators are used to model important physical systems which include transport properties of graphene, and relativistic quantum field theory. This meeting will focus on the following areas of current research interest in such operators applied to mathematical physics.

1. For classical (one-particle) Dirac operators, current topics of interest include the Weyl-type theory, dissolution of eigenvalues of corresponding relativistic systems into resonances, asymptotics of the spectral function and spectral concentration as well as the role of the mass term of Dirac operators. We shall concentrate on the structure of the spectrum, quadratic form methods, singularly perturbed problems, and selfadjoint extensions as well as applications of the Dirac operator in mathematical physics and chemistry.

2. A topic of central importance to be covered by the workshop is the stability of matter and asymptotic behaviour of the ground state energy for relativistic many-particle systems. This problem concerns the question whether the energy of the system of particles is bounded from below and whether it is bounded below by a constant multiple of the number of particles. Although the problems were settled for non-relativistic quantum mechanical electrons and nuclei, many challenges remain when relativistic considerations are included.

3. Another major topic of the workshop concerns the multi-particle operators and connections to quantum electrodynamics. A new chapter of mathematical analysis has been opened by studying the interaction of photons with fast moving (relativising) electrons, positrons, and photons. The central problem is to investigate the physically relevant minimisers of an energy functional. The mathematical challenge is to formulate and to investigate such models analytically and thereby to provide a theoretical basis for their numerical treatment.

Final Scientific Report: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons