Microlocal Analysis
This is the webpage for the 2011 TCC course on microlocal analysis.
Introduction
Microlocal analysis is part of the general theory of partial
differential operators. It is used to
study questions like solvability of PDE's, qualitative
properties of solutions of PDE's, spectral
asymptotics like Weyl's law and the pseudo-spectrum of non-normal
operators, to name just a
few applications. Microlocalisation is a process which
combines the standard
techniques of localisation and Fourier transform: one localises not
only in the space variable x, but as well in the
Fourier transform variable p. The resulting space of the variables
(p,x) is called phase space and is
a symplectic manifold. The symplectic geometry in this space and the
corresponding Hamiltonian
dynamical systems are then used to study the original PDE problems.
The beauty of the
field lies in this interaction between analysis and geometry.
We will develop the semiclassical version of this theory in which a
small parameter
is present, and we consider asymptotic expansions in this parameter.
In this
context microlocal analysis provides the mathematical framework
for the semiclassical limit in quantum mechanics and in a more
general setting
the relation between ray and wave dynamics in hyperbolic equations.
This course will develop the basic setup of the theory and then
give
a guided tour through some of the applications in spectral
asymptotics,
quantum ergodicity and normal form theory.
Syllabus
We plan to cover the following topics:
- Fourier transform on Schwartz space, Weyl-quantisation and
Wigner functions
- symbols, symbol-calculus, parametrix construction and
functional calculus
- Essential Support and Frequency Set, Egorov's theorem and
transport of singularities
- Weyl's law, local Weyl law and quantum ergodicity
- local symplectic geometry, Lagrangian states and time
dependent WKB theory
- Fourier Integral Operators and normal forms.
Prerequesites
Somebody once said that microlocal analysis is basically just a
combination of Fourier-transformations and partial
integration,
with a sprinkle of symplectic geometry to finish it of. We
will follow this philosophy and
try to get away with as few prerequisites as possible. But some
familiarity with bits of functional analysis
and spectral theory, theory of distributions and basic elements of
differential forms will be helpful. Much of this is actually
developed in Chapter 3 and the Appendix of the lecture notes
by Evans and Zworski.
Literature
The recommended text are the Lectures
on Semiclassical Analysis by Evans and Zworski
which are available on
http://math.berkeley.edu/~zworski/
Besides this there are a number of other good texts:
- M. Dimassi and J. Sjostrand, Spectral Asymptotics in the Semi-Classical Limit,
LMS Lecture Note Series 268, Cambridge University Press,
1999. A standard textbook we will use alongside Evans and
Zworski.
- V. Guillemin and S. Sternberg,
Lecture notes on Semi-Classical
Analysis, available on V. Guillemin's homepage. These provide
a more geometrical approach then the text by Evans and Zworski.
- V. Guillemin and S. Sternberg, Geometric Asymptotics,
Mathematical Surveys, No. 14, AMS, 1977. A classic! Still
beautiful to read.
- L. Hormander, The Analysis
of Linear Partial Differential Operators I-IV,
Springer, 1983-1985. The definite text (on the non-semiclassical
version of the theory) by one of the masters. For us the first
volume will be useful for background in Fourier
transformations, stationary phase and the theory of
distributions.
- A. Martinez, An
Introduction to Semiclassical and Microlocal Analysis,
Springer, 2002. An introductary text which focuses more on
complex analytic methods.
Assessment
Assessment will be by solving exercises on problem sheets which wil
be posted here.