250422 VO Topics in Spectral Theory (2008S)

6.00 ECTS (4.00 SWS), SPL 25 - Mathematik

Details

Sprache: Englisch

Lehrende

Friedrich Gesztesy

Termine (iCal) - nächster Termin ist mit N markiert

Dienstag 04.03. 11:00 - 13:00 Hörsaal 1 2A120 1.OG UZA II Geo-Zentrum
Montag 10.03. 11:00 - 13:00 Hörsaal 1 2A120 1.OG UZA II Geo-Zentrum
Dienstag 11.03. 11:00 - 13:00 Hörsaal 1 2A120 1.OG UZA II Geo-Zentrum
Montag 31.03. 11:00 - 13:00 Hörsaal 1 2A120 1.OG UZA II Geo-Zentrum
Dienstag 01.04. 11:00 - 13:00 Hörsaal 1 2A120 1.OG UZA II Geo-Zentrum
Montag 07.04. 11:00 - 13:00 Hörsaal 1 2A120 1.OG UZA II Geo-Zentrum
Dienstag 08.04. 11:00 - 13:00 Hörsaal 1 2A120 1.OG UZA II Geo-Zentrum
Montag 14.04. 11:00 - 13:00 Hörsaal 1 2A120 1.OG UZA II Geo-Zentrum
Dienstag 15.04. 11:00 - 13:00 Hörsaal 1 2A120 1.OG UZA II Geo-Zentrum
Montag 21.04. 11:00 - 13:00 Hörsaal 1 2A120 1.OG UZA II Geo-Zentrum
Dienstag 22.04. 11:00 - 13:00 Hörsaal 1 2A120 1.OG UZA II Geo-Zentrum
Montag 28.04. 11:00 - 13:00 Hörsaal 1 2A120 1.OG UZA II Geo-Zentrum
Dienstag 29.04. 11:00 - 13:00 Hörsaal 1 2A120 1.OG UZA II Geo-Zentrum
Montag 05.05. 11:00 - 13:00 Hörsaal 1 2A120 1.OG UZA II Geo-Zentrum
Dienstag 06.05. 11:00 - 13:00 Hörsaal 1 2A120 1.OG UZA II Geo-Zentrum
Montag 19.05. 11:00 - 13:00 Hörsaal 1 2A120 1.OG UZA II Geo-Zentrum
Dienstag 20.05. 11:00 - 13:00 Hörsaal 1 2A120 1.OG UZA II Geo-Zentrum
Montag 26.05. 11:00 - 13:00 Hörsaal 1 2A120 1.OG UZA II Geo-Zentrum
Dienstag 27.05. 11:00 - 13:00 Hörsaal 1 2A120 1.OG UZA II Geo-Zentrum
Montag 02.06. 11:00 - 13:00 Hörsaal 1 2A120 1.OG UZA II Geo-Zentrum
Dienstag 03.06. 11:00 - 13:00 Hörsaal 1 2A120 1.OG UZA II Geo-Zentrum

Information

Ziele, Inhalte und Methode der Lehrveranstaltung

The principal motivation for this course stems from solving heat conduction problems (and certain problems in quantum theory, acoustics, and electromagnetism). In particular, we will show that the solution of such problems most naturally leads to the spectral analysis of general Sturm-Liouville operators (i.e., second-order differential operators involving three coefficients). In this context it will become clear that eigenfunction expansion methods going back to Joseph Fourier lie at the heart of spectral analysis.

After some ODE preliminaries, we will study the regular Sturm-Liouville problem in detail, including a discussion of all self-adjoint boundary conditions and associated eigenfunction expansions. This will be followed by the singular Sturm-Liouville problem on arbitrary intervals and the basics of Weyl-Titchmarsh theory. Subsequently, we will turn to the computation of the (matrix-valued) spectral function in the regular and singular cases.

Useful Background:

Basic knowledge of functional analysis in Hilbert space, especially, the notion of closed, symmetric, and self-adjoint operators, and elements of
spectral theory. We will summarize some of these concepts and that of the von Neumann theory of self-adjoint extensions of closed symmetric operators in the course of these lectures.

Art der Leistungskontrolle und erlaubte Hilfsmittel

Mindestanforderungen und Beurteilungsmaßstab

Prüfungsstoff

Literatur

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Krieger, 1985.

K. Jörgens and F. Rellich, Eigenwerttheorie gewöhnlicher Differentialgleichungen, Springer, 1976.

M. Reed and B. Simon, Methods of Modern Mathematical Physics II. Fourier Analysis, Self-Adjointness, Academic Press, 1975.

J. Weidmann, Linear Operators in Hilbert Spaces, Springer, 1980.

J. Weidmann, Lineare Operatoren in Hilbert Räumen. Teil II. Anwendungen, Teubner, 2003.

Zuordnung im Vorlesungsverzeichnis

MANV

Letzte Änderung: Do 31.10.2024 00:15