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Spectral Theory and Partial Differential Equations

17th July 2006 to 11th August 2006

Organisers: Professor M van den Berg (Bristol), Professor B Helffer (Orsay), Professor A Laptev (Stockholm), Professor AV Sobolev (Birmingham)

Programme Theme

Spectral theory and partial differential equations stand at a meeting point of several different parts of mathematics and physics. Within mathematics it links spectral properties of elliptic and parabolic operators to the geometry and topology of the underlying manifold. Within physics it links, for example, the stability of matter to the properties of the potentials in the Schrödinger operators.

Some of the fundamental problems of spectral theory have been quite well understood. These include, for instance, the relation between the asymptotic properties of various spectral quantities (spectral counting function, heat content and heat trace functions) and the geometry of the underlying manifold, the general properties of periodic and magnetic operators etc. On the other hand certain questions of spectral geometry (e.g. connection between the bottom of the spectrum, nodal lines, multiplicities of eigenvalues and the geometric properties of the region or manifold), and the theory of periodic operators (e.g. detailed properties of the band-gap spectrum, absolute continuity, the number of gaps) remain unanswered.

Some fundamental questions in the theory of multi-dimensional Schrödinger operators, such as absolute continuity, are open and at present substantial efforts are being made towards understanding these through the so-called trace formulae. Many of these questions have important applications in physics (solid state physics, statistical physics, large particle systems, quantum mechanics, photonic crystals).

The aim of the programme is to focus the expertise in Spectral Theory on the issues mentioned above and incite useful collaborations involving mathematicians from the UK and other countries.

The following is the list of people who have so far agreed to be participants on the Programme; M. Ashbaugh, R. Banuelos, M. Birman, E.B. Davies, A. Grigoryan, P.B.Gilkey, V. Guillemin, T. Hoffmann-Ostenhof, P. Kuchment, E. Lieb, M. Loss, V. Maz’ya, R. Melrose, N. Nadirashvili, P. Sarnak, B. Simon, J. Sjöstrand, U. Smilansky, M. Solomyak, S. Zelditch, M. Zworski.

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