Spectral theory and partial differential equations

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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. Mazya, R. Melrose, N. Nadirashvili, P. Sarnak, B. Simon, J. Sjöstrand, U. Smilansky, M. Solomyak, S. Zelditch, M. Zworski.

Click here to download the programme's final scientific report

Preprints

Title	Year	Programme
Strong diamagnetism for general domains and applications Authors: Bernard Helffer, Soeren Fournais Download PDF	2006	STP
Four lectures in semiclassical analysis for non self-adjoint problems with applications to hydrodynamic instability Authors: Bernard Helffer Download PDF	2006	STP
Complex Osserman algebraic curvature tensors and Clifford families Authors: Peter B Gilkey, M Brozos-vazquez Download PDF	2006	STP
The periodic magnetic Schrödinger operators: spectral gaps and tunneling effect Authors: Yuri Kordyukov, Bernard Helffer Download PDF	2006	STP

Workshops

STPW01

Spectral theory and its applications

24 July 2006 to 28 July 2006

The Organisers would like to thank the following sponsors for their generous support of the event:

Seminars

Thursday 20th July 2006
14:00 to 15:00	Peter B Gilkey University of Oregon The spectral geometry of the canonical Riemannian submersion of a compact Lie group	Room 1
15:30 to 16:30	Leonid Polterovich Tel Aviv University Nodal inequalities on surfaces	Room 1

Wednesday 2nd August 2006
14:00 to 15:00	Symmetry tricks in spectral geometry	Room 2

Thursday 3rd August 2006
14:00 to 15:00	Dirk Hundertmark University of Illinois at Urbana-Champaign Decay estimates for dispersion management solitons	Room 2
15:30 to 16:30	Uzy Smilansky University of California, Irvine and Weizmann Institute of Science Geometrical and topological characterization of nodal domains of wave functions and of random waves	Room 2

Thursday 10th August 2006
14:00 to 15:00	Ira Herbst University of Virginia Absence of ground states for massless Nelson models	Room 1
15:30 to 16:30	Yuri Kordyukov Pereslavl-Zalessky of the Russian Academy of Sciences Spectral gaps for periodic Schroedinger operators with magnetic wells	Room 1

Spectral theory and partial differential equations

Organisers

Bernard Helffer
Ari Laptev
Alex Sobolev University of Birmingham
Michiel van den Berg University of Bristol

Speakers

Peter B Gilkey University of Oregon
Ira Herbst University of Virginia
Dirk Hundertmark University of Illinois at Urbana-Champaign
Yuri Kordyukov Pereslavl-Zalessky of the Russian Academy of Sciences
Leonid Polterovich Tel Aviv University
Uzy Smilansky University of California, Irvine and Weizmann Institute of Science

Spectral theory and partial differential equations

Programme theme

Preprints

Strong diamagnetism for general domains and applications

Four lectures in semiclassical analysis for non self-adjoint problems with applications to hydrodynamic instability

Complex Osserman algebraic curvature tensors and Clifford families

The periodic magnetic Schrödinger operators: spectral gaps and tunneling effect

Workshops

Spectral theory and its applications

Seminars

Organisers

Speakers

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