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Module Number:
| 14300
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| Module Title: | Spectral Theory of Self-adjoint Operators in Hilbert Spaces |
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Spektraltheorie selbstadjungierter Operatoren im Hilbertraum
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| Department: |
Faculty 1 - Mathematics, Computer Science, Physics, Electrical Engineering and Information Technology
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| Responsible Staff Member: | -
Prof. Dr. rer. nat. habil. Hauer, Daniel
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| Language of Teaching / Examination: | English |
| Duration: | 1 semester |
| Frequency of Offer: |
On special announcement
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| Credits: |
8
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| Learning Outcome: | 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. |
| 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
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| Recommended Prerequisites: | Knowledge of the contents of modules
- 11103: Analysis I
- 11104: Analysis II
- 11201: Analysis III
- 11303: Funktionalanalysis or 13844: Functional Analysis
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| Mandatory Prerequisites: | None |
| Forms of Teaching and Proportion: | -
Lecture
/ 4 Hours per Week per Semester
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Exercise
/ 2 Hours per Week per Semester
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Self organised studies
/ 150 Hours
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| Teaching Materials and Literature: | - 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.
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| Module Examination: | Final Module Examination (MAP) |
| Assessment Mode for Module Examination: | - oral examination, 30 minutes
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| Evaluation of Module Examination: | Performance Verification – graded |
| Limited Number of Participants: | None |
| Part of the Study Programme: | -
Master (research-oriented) /
Angewandte Mathematik /
PO 2019
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| Remarks: | - Study programme Angewandte Mathematik M.Sc.: Compulsory elective module in the complex „Analysis/Algebra/Kombinatorik“
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| Module Components: | - Lecture: Spectral Theory of Self-adjoint Operators in Hilbert Spaces
- Accompanying exercises
- Related Examination
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| Components to be offered in the Current Semester: | |
