Mathematisches Seminar: Spectral Theory

Block course (Tue, Wed, Thu), ending 13/11/2012 – times and location t.b.a.

First meeting: Mon 15/10/2012 at 10:00 in B 448

(Discussion of topics and assignment of talks)

If you plan to attend, please register by email until 14/10/2012.
Talks can be given in English or German!

Synopsis

Spectral theory is concerned with the generalisation of the eigenvalue decomposition of finite-dimensional matrices to the case of linear operators on a Hilbert space. As such it lies at the core of the mathematical foundations of the theory of linear partial differential equations with abundant applications, for example, in quantum mechanics.

We will mostly follow the approach of E. Brian Davies, who uses the Helffer-Sjöstrand formula to define the functional calculus. This approach is particularly straightforward and direct, and has the further advantage of using modern and powerful techniques which are of value elsewhere in the theory of differential equations. For example, the Helffer-Sjöstrand formula has had many important applications to spectral and scattering theory in recent years.

Suitable for

3rd year Bachelor students and Master students of Mathematics and Physics, TMP students
(the scope of the block seminar amounts to a course of 2 hours/week during the whole semester)

Prerequisites

basic knowledge of functional analysis

Literature

• E. B. Davies, Spectral theory and differential operators, Cambridge University Press, 1995
• M. Reed, B. Simon, Methods of modern mathematical physics, vol. 1, Academic Press, 1980