Spectral Theory of Self-adjoint Operators in Hilbert Spaces
Spektraltheorie selbstadjungierter Operatoren im Hilbertraum
8 CPirregularlyEN
Prerequisites
Recommended, as written
Knowledge of the contents of modules • 11103: Analysis I • 11104: Analysis II • 11201: Analysis III • 11303: Funktionalanalysis or 13844: Functional Analysis
Contents
• Theory of unbounded linear operators • Hermitian and self-adjoint operators • Resolvents and the spectral set of unconstrained operators • Spectral theorem of self-adjoint operators • Theorems of E. Helly • Spectral theorem for unitary operators • Time-dependent Schrödinger equation • The Friedrichs continuation for semi-constrained Hermitian operators • Elliptic differential operators and Schrödinger operators with their spectra
Learning outcomes
After successfully completing the module, students have mastered the mathematical fundamentals of linear spectral theory. They are able to determine independently when linear operators have a discrete or continuous spectrum. The students know in detail special classes (e.g. the class of Schrödinger operators) of linear operators, which play an important role in other areas of mathematics and physics. In addition, they have expanded their knowledge of analysis.
Assessment
• oral examination, 30 minutes
Literature (2)
- Friedrich Sauvigny: Spektraltheorie selbstadjungierter Operatoren im Hilbertraum und elliptischer Differentialoperatoren. Springer Spektrum, Berlin, 2019.
- Reed Simon: Methods of Mathematical Physics Vol I-IV, Academic Press, 1978.
Remarks
• Study programme Angewandte Mathematik M.Sc.: Compulsory elective module in the complex „Analysis/Algebra/Kombinatorik“ • Study programme Mathematics M.Sc.: Compulsory elective module in the complex „Analysis/Algebra/Combinatorics“
Source: BTU's module description · retrieved 29 Sep 2026Original