Spectral Theory
Tid: Må 2012-09-03 kl 13.15 - 15.00
Plats: Room 306, building 6, Kräftriket, Department of mathematics, Stockholm university
The course will be devoted mainly to the spectral theory of differential operators with focus on applications in quantum mechanics. Differential operators can be seen as linear transformations in infinite dimensional spaces and spectral theory can be considered as an attempt to understand diagonalization of infinite size matrices corresponding to such transformations. As a byproduct one obtains expansions in terms of eigenfunctions generalizing classical Fourier analysis. The role of spectral theory is not limited to proving the spectral theorem for self-adjoint operators, but forms the foundation of quantum mechanics, where physical systems are described precisely by self-adjoint operators acting in infinite dimensional Hilbert spaces. Such operators similar to Hermitian matrices have real spectrum ensuring that physical observables for different states give real numbers, which can be measured in experiments. Development of spectral theory can be successfully carried out only if ideas from both mathematics and physics are used simultaneously.