We shall consider spectral theory of bounded and unbounded linear operators in Hilbert spaces. In this setting, spectrum is a generalization of the concept of eigenvalues of matrices. In addition to a treatment of the general theory, the goal of the course is to access the plentiful applications of spectral theory to partial differential equations and mathematical physics. Spectrum of the Schrödinger equation descibes the structure of atoms and molecules; spectrum of the Laplace-Dirichlet/Neumann problem in a bounded planar domain describes the vibrations of a drum membrane (see for example http://www.acs.psu.edu/drussell/Demos/MembraneCircle/Circle.html), or in a 3-d domain, propagation of electromagnetic waves; spectrum of the linearized elasticity system describes the mechanical vibrations in a solid and so on.
Topics include: unbounded operators in Hilbert spaces, symmetric and self-adjoint operators, self-adjoint extensions, definition of spectrum, eigenvalues, continuous, essential and residual spectrum, spectral theorem, spectrum of compact operators, max-min principle, applications to differential equations. The course material consists of lecture notes + additional reading.

Exercise problems can be found via the link below. Solutions can be found under the course material link.

The second exam will take place Thursday May 3rd at 14.15-16.00 in the room C122. The material for this exam is the rest of the lecture note (starting from Section 6.4.) and exercises 6-10.