Blyth Lectures are delivered by Johannes Sjostrand!
September 15 (Mon), 17 (Wed), 19 (Fri) 1997 at 4:20 p.m. at Fields
Microlocal methods in partial differential equations
have been developed over the last 30 years. They permit to analyze
singularities of solutions, both their position and the responsible
oscillations within the limit of the Heisenberg uncertainty principle.
We discuss some applications to the propagation of singularities for
equations such as the wave equation and related results for eigenvalues.
Resonances. Introduction and overview of some recent results.
Resonances appear in a large number of problems as poles of the meromorphic
extension of the scattering matrix, and physically they correspond
to unstable states or time-decaying modes. They can also be viewed
as complex eigenvalues when the original equation is considered in
a suitable space. We are particularly interested in the resonances
near the real axis, and we shall discuss some results on the distribution
of resonances, sometimes in relation with properties of the corresponding
classical dynamical system. Suitable forms of microlocal analysis
are essential tools.
Trace formulae for resonances and applications.
Such a formula has been developed for compactly supported perturbations
of the Laplacian in odd dimensions, starting with Lax-Phillips. (We
will here only discuss the Euclidean case). It has recently been possible
to consider long range perturbations in all dimensions. Applications
concern the existence of resonances and lower bounds on their density,
including the case of the semi-classical Schroedinger operator.