Spectral theory

Lecturer: Erik Skibsted

Content:
The spectral theory for unbounded self-adjoint operators on a Hilbert space is an indispensable tool for a large number of problems in mathematical physics. The main point of spectral theory is to diagonalize operators.

We start out by defining the notion of unbounded self-adjoint operators and settling basic properties like the fact that the spectrum of such operator is a subset of the real axis. Then we study two versions of the spectral theorem, one on the functional calculus form and the other on multiplication operator form. Here we shall rely on the spectral theory for bounded self-adjoint operators as developed in the course Analysis 2, in particular the notion of integration with respect to a family of spectral projections.

Next we shall study the relationship between self-adjoint operators and closed quadratic forms which is more subtle than for the case of bounded operators/forms. The Sturm-Liouville operators are particularly illuminating examples. The Fourier transform will be introduced as a tool for studying differential operators with constant coefficients on .

The next topic will be theory for locating the spectrum and determining its nature. Here the compact operators encountered in Analysis 2 are important. (The Hilbert-Schmidt criterion will be shown to be applicable for the resolvent of Sturm-Liouville operators.) Another tool is the Rayleigh-Ritz variational formula.

In time permits we may embark on a more systematic study of the theory of self-adjoint extensions (decifiency indices, Nelson's commutator theorem, etc.) and Stone's theorem or even more general semigroup theory on a Hilbert space.

Students are expected to do exercises. The course may serve as a background for further study of partial differential equations and/or functional analysis.

Prerequisites: Analysis 2.

Literature: E.B. Davies, Spectral theory and differential operators, Cambridge University Press, 1995, 1. edition; notes.