Blyth 1997

The 1997 R. A. Blyth Lectures


 

Microlocal Methods in Partial Differential Equations


Professor Johannes Sjöstrand, Ecole Polytechnique, Paris

 


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 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 have 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 Schrödinger operator.

These lectures are closely connected with the program Microlocal Methods in Geometric Analysis and Mathematical Physics at the Fields Institute (Fall 97).

 


  1. Microlocal Methods in Partial Differential Equations, Monday September 15, 4:20 p.m., Fields Institute, 222 College St., Room 230.
  2. Resonances: Introduction and Overview of Some Recent Results, Wednesday September 17, 4:20 p.m., Fields Institute, 222 College St., Room 230.
  3. Trace Formulae for Resonance and Applications, Friday September 19, 4:20 p.m., Fields Institute, 222 College St., Room 230.

The Blyth Lecture Reception will follow the first lecture on Monday September 15, 1997, at the Fields Institute.