C*-algebraic methods in spectral theory

Goals of the Course

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 quickly 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.

Objectives of the Course

Understand the constructions related to crossed product C*-algebras, and observe how this algebraic framework can be applied successfully to the spectral theory of operators.

Course Content

1. Linear operators on a Hilbert space

2. C*-algebras

3. Crossed product C*-algebras

4. Schroedinger operators and essential spectrum

5. Twisted crossed product C*-algebras

6. Pseudodifferential calculus

7. Magnetic systems

Course Prerequisites

Knowledge on standard undergraduate linear algebra, calculus and advanced calculus.

Related Courses

Any courses on function spaces or on operator algebras.

Course Evaluation Method and Criteria

Grades based on attendance and on written reports. An active participation of the students is expected.

Textbook

Lecture Notes

Additional information

Recommended Reading

Conditions of Other department student's attendance

This course is open for any students at Nagoya University. Motivated undergraduate students are also welcome.

This lecture is provided under Creative Commons Attribution-Non Commercial-ShareAlike 4.0 International.

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