Instructor: Fritz Gesztesy
Description: We intend to offer a basic course on spectral theory for self-adjoint second-order ordinary and partial differential and difference operators in a Hilbert space context.
More precisely, in the ordinary differential and difference operator context we will describe the Weyl-Titchmarsh-Kodaira approach to Sturm-Liouville-type operators, that is, their spectral functions and matrices, Weyl-Titchmarsh functions, multiplicity theory, eigenfunction expansions, etc. In the case of partial differential operators we will describe operator-valued spectral functions, operator-valued Weyl-Titchmarsh functions, eigenfunction expansions, and alike.
Typical concrete situations we have in mind are Schrodinger, Dirac, Jacobi, and CMV operators (the latter have recently gained enormous popularity in connection with the trigonometric moment problem).
Prerequisites: Basic complex and functional analysis and basic material in ordinary and partial differential equations. More advanced results in these fields will be developed as needed during the course. Additional reading materials will be provided whenever appropriate.