C﹡-algebraic methods in spectral theory

Serge RICHARD Professor

Department: Graduate School of Mathematics

Course Overview

Course Overview

This course will provide an overview on some of the most recent tools introduced in functional analysis for the study of operators related to quantum mechanics. During the first lectures, we shall review some basics properties of bounded and unbounded operators on Hilbert spaces, and introduce the spectral theorem for self-adjoint operators. After reviewing some definitions and properties related to C^*-algebras, we shall show how crossed product C^*-algebras are naturally linked to generalized Schroedinger operators, and how information on these operators can be deduced from representations of these algebras. A related construction involving twisted crossed product algebras and its application for magnetic systems will then be discussed.

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Syllabus

Keywords

Self-adjoint operators, spectrum, C*-algebras, crossed product, magnetic systems.

Plan of the course

Tentative program:

1. Linear operators on a Hilbert space,
2. C*-algebras,
3. Dynamical systems and crossed product C*-algebras,
4. Schrodinger operators and essential spectrum,
5. Twisted crossed product algebras,
6. Pseudodifferential calculus,
7. Magnetic systems.

Textbooks

There is no specific book related to this course. References and additional material will be provided during the lectures.

Course Requirements

This course is open for any students at Nagoya University as one of the "open subjects" of general education.

Required Knowledge

Knowledge on standard undergraduate functional analysis.

Additional advice

This course will not be very technical but rather interdisciplinary.

Grading

Grades based on attendance, voluntary works and written reports.

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Class Materials

Lecture Handouts

Title

Title (PDF, 47KB)

Contents

Contents (PDF, 112KB)

Chapter 1

Linear operators on a Hilbert space (PDF, 345KB)

Chapter 2

C*-algebras (PDF, 368KB)

Chapter 3

Crossed product C*-algebras (PDF, 351KB)

Chapter 4

Schroedinger operators and essential spectrum (PDF, 328KB)

Chapter 5

Twisted crossed product C*-algebras (PDF, 347KB)

Chapter 6

Pseudodifferential calculus (PDF, 262KB)

Chapter 7

Magnetic systems (PDF, 309KB)

Bibliography

Bibliography (PDF, 142KB)

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