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German original

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

Level, degree of commitment in original study programme Advanced 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: Successful completion of at least 50 percent of the points from the weekly exercises.
Examination type: Written or oral examination
Language,
Grading
German,
The grading is done with 0 to 15 points according to the examination regulations for study course M.Sc. Mathematics.
Original study programme M.Sc. Mathematik / Vertiefungsbereich Mathematik
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

The students shall

  • learn to recognize and assess the relevance of spectral analytical methods for concrete problems, e.g. from the theory of partial differential equations, and to acquire the appropriate instruments for solving these problems,
  • learn how methods of algebra, analysis, geometry and topology interact,
  • Re-evaluate contents from the basic modules and some advanced modules (e.g. function theory, Analysis III and functional analysis),
  • to understand the relations of spectral theory to other fields of mathematics and other sciences,
  • practice mathematical working methods (development of mathematical intuition and its formal justification, training of the ability to abstract, proof techniques),
  • improve their oral communication skills in the exercises by practicing free speech in front of an audience and during 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

(not specified)



Please note:

This page describes a module according to the latest valid module guide in Wintersemester 2020/21. 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.