250422 VO Topics in Spectral Theory (2008S)

The principal motivation for this course stems from solving heat conduction problems (and certain problems in quantum theory, acoustics, and electromagnetism). In particular, we will show that the solution of such problems most naturally leads to the spectral analysis of general Sturm-Liouville operators (i.e., second-order differential operators involving three coefficients). In this context it will become clear that eigenfunction expansion methods going back to Joseph Fourier lie at the heart of spectral analysis.

After some ODE preliminaries, we will study the regular Sturm-Liouville problem in detail, including a discussion of all self-adjoint boundary conditions and associated eigenfunction expansions. This will be followed by the singular Sturm-Liouville problem on arbitrary intervals and the basics of Weyl-Titchmarsh theory. Subsequently, we will turn to the computation of the (matrix-valued) spectral function in the regular and singular cases.

Useful Background:

Basic knowledge of functional analysis in Hilbert space, especially, the notion of closed, symmetric, and self-adjoint operators, and elements of
spectral theory. We will summarize some of these concepts and that of the von Neumann theory of self-adjoint extensions of closed symmetric operators in the course of these lectures.