Microlocal methods in partial differential equations.
  (For the general audience)

  Microlocal methods in partial differential equations.
  (For the general audience)

  Trace formulae for resonances and applications.
  (For the expert)

Abstract

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.