4 hours of lectures and exercises per week.
Lecturer
Erik Skibsted
Content
The spectral theory for unbounded self-adjoint operators on a Hilbert space is an indispensable tool for a variety 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 an operator is a subset of the real axis. We show how to construct self-adjoint operators by extension of a given symmetric operator using the Cayley transform and the concept of deficiency indices. Then we prove a version of the spectral theorem. Our proof is based on spectral theory for bounded normal 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 Rn and as a tool for constructing compact operators, cf. Sobolev embedding theorems.
The next topic will be theory for locating the spectrum and determining its nature. Here the compact operators encountered in Analysis 2 are important. Another tool is the Rayleigh-Ritz variational formula.
If time permits we may embark on a more systematic study of the theory of self-adjoint extensions (including for example Nelson's commutator theorem) 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.