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This entry is from Winter semester 2016/17 and might be obsolete. You can find a current equivalent here.

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

Level, degree of commitment Specialization module, compulsory elective module
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): Written or oral examination
Examination type: Successful completion of at least 50 percent of the points from the weekly exercises.
Language,
Grading		 German,
The grading is done with 0 to 15 points according to the examination regulations for the degree program 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

  • learn to recognize and appreciate the relevance of spectral analytical methods to concrete problems, such as those from the theory of partial differential equations, and acquire the appropriate tools for solving these problems,
  • learn how methods of algebra, analysis, geometry, and topology interact,
  • re-evaluate knowledge gained in the basic modules and some advanced modules (e.g., Theory of Functions, Analysis III, and Functional Analysis),
  • recognize the relationships of spectral theory to other areas of mathematics and to other sciences.
  • practice mathematical ways of working (developing mathematical intuition and its formal justification, training the ability to abstract, reasoning)
  • improve their oral communication skills in exercises by practicing free speech in front of an audience and in discussion.

Prerequisites

Translation is missing. Here is the German original:

Keine. Empfohlen werden die Kompetenzen, die in den Grundmodulen vermittelt werden. Kenntnisse aus der allgemeinen Maß- und Integrationstheorie sowie der Funktionentheorie und Funktionalanalysis sind hilfreich.


Applicability

Module imported from M.Sc. Mathematics.

It can be attended at FB12 in study program(s)

  • B.Sc. Mathematics
  • M.Sc. Mathematics
  • LAaG Mathematics

When studying LAaG Mathematics, this module can be attended in the study area Specialization Modules.

Die Wahlmöglichkeit des Moduls ist dadurch beschränkt, dass es der Reinen Mathematics zugeordnet ist.


Recommended Reading

(not specified)



Please note:

This page describes a module according to the latest valid module guide in Winter semester 2016/17. 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.