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Spectral and Scattering Theory
(dt. Spektral- und Streutheorie)

Level, degree of commitment Specialization module, depends on importing study program
Forms of teaching and learning,
workload		 Lecture (4 SWS), recitation class (2 SWS),
270 hours (90 h attendance, 180 h private study)
Credit points,
formal requirements		 9 CP
Course requirement(s): Successful completion of at least 50 percent of the points from the weekly exercises.
Examination type: Written or oral examination (individual examination)
Language,
Grading		 English,
The grading is done with 0 to 15 points according to the examination regulations for the degree program M.Sc. Mathematics.
Subject, Origin Mathematics, M.Sc. Mathematics
Duration,
frequency		 One semester,
irregular
Person in charge of the module's outline Prof. Dr. Pablo Ramacher

Contents

Subject of this lecture are the spectral theory of bounded and unbounded operators on Hilbert spaces, as well as elements of scattering theory. Specifically, the following contents will be discussed:

  • The functional calculus for bounded and unbounded operators (elementary theory of C^* algebras, Gelfand-Naimark duality)
  • The Spectral Theorem for bounded and unbounded Operators
  • Existence and completeness of wave operators for trace class perturbations and the invariance principle (theorem of Kato-Rosenblum)

Qualification Goals

Students will

  • correctly recognize and appreciate the relevance of spectral analytical methods to concrete problems, such as those from the theory of partial differential equations, and possess the appropriate tools for solving these problems,
  • understand how methods of algebra, analysis, geometry and topology interact,
  • re-evaluate their knowledge from the basic modules and some advanced modules (e.g., function theory and vector analysis and functional analysis),
  • recognize the relationships of spectral theory to other areas of mathematics and to other sciences,
  • have deepened mathematical working methods (developing mathematical intuition and its formal justification, abstraction, proof),
  • have improved their oral communication skills in exercises by practicing free speech in front of an audience and in discussion.

Prerequisites

None. The competences taught in the following modules are recommended: either Foundations of Mathematics and Linear Algebra I and Linear Algebra II or Basic Linear Algebra, either Analysis I and Analysis II or Basic Real Analysis. Knowledge of general Measure and Integration Theory as well as Complex Analysis and Functional Analysis is helpful.


Recommended Reading

  • Literatur: Yosida, Functional Analysis; Reed-Simon, Methods of Modern Mathematical Physics, vol. I und III.



Please note:

This page describes a module according to the latest valid module guide in Winter semester 2023/24. Most rules valid for a module are not covered by the examination regulations and can therefore be updated on a semesterly basis. The following versions are available in the online module guide:

The module guide contains all modules, independent of the current event offer. Please compare the current course catalogue in Marvin.

The information in this online module guide was created automatically. Legally binding is only the information in the examination regulations (Prüfungsordnung). If you notice any discrepancies or errors, we would be grateful for any advice.