Self-adjoint operators in Hilbert spaces

Math 523 - Spring 2016

Section information: Classes meet on Mondays, Wednesdays and Fridays from 3:00PM to 4:00PM at Cupples I 207. Office hours are on Tu-Th from 12:00PM to 2:00PM.

Subject: This is an introduction to the spectral theory of self-adjoint operators, in particular differential operators and operators of relevance to quantum theory, although no knowledge of physics will be assumed. (Physics students are most welcome to join the class.)

Text: Intermediate spectral theory and quantum dynamics by Cesar R. de Oliveira, Birkhauser, 2009.
The following texts may be used occasionally for background or supplementary material:

• Functional Analysis I Michael Reed / Barry Simon, Academic Press, 1980;
• Quantum theory for mathematicians by Brian Hall, Springer, 2013. (Olin Library has the electronic file for download.)

Tentative list of topics:

• Basic facts about Banach and Hilbert spaces, and spectra.
• The adjoint of an operator and self-adjoint operators
• Fourier transform
• Unitary evolution groups and Stone's theorem
• The spectral theorem for unbounded self-adjoint operators
• The Kato-Rellich theorem
• Applications and further topics as time permits

Coursework: Your work will consist entirely of weekly homework assignments. I expect we'll have approximately 10 assignments. You are welcome to collaborate on them, discuss them with me or others, and bring up homework related questions for discussion in class before turning them in. The method for computing grades will be discussed in class.

Grades: Your final grade will be based on your homework assignments mean score.

Homework assignments:

• Homework 1: 1/29
  Solutions