Course Description

This course was first developed in the Fall of 2018 to provide an introduction to Functional Analysis and Quantum Mechanics. Since then, it has been offered in alternate years. The course is aimed at 3rd and 4th year MATH majors, and at Masters/PhD students who have not taken a Functional Analysis course before. Students in related disciplines (such as Physics, Engineering, Computing, etc.) are also encouraged to attend. Undergraduate students will take this course as MATH 429, while graduate students will take it as MATH 829.

The course will focus on the extensions of Linear Algebra and Calculus to the infinite dimensional setting. As motivating topic, we will address several problems from Mathematical Physics and in particular the mathematical foundations of Quantum Mechanics.

I plan to cover the following topics: metric spaces, Banach spaces, inner product spaces, Hilbert spaces, operators and their spectrum, mathematical foundations of Quantum Mechanics, examples of Schrödinger equations, and the harmonic oscillator. The course will move quickly through the material: we will start with a review of Cauchy sequences and in about 10 weeks we will arrive at a full proof of the spectral theorem for unbounded self-adjoint operators, which we will then use in Quantum Mechanics.

Prerequisites

The only strict prerequisites for this course are MATH/MTHE 281 and a course in linear algebra (MATH 110 or 111 OR 112). Students must be familiar with the following topics:

• sequences and limits in R^d, convergence for sequences and series of functions (extensively covered in MATH/MTHE 281),

• linear algebra in general vector spaces (covered, e.g., in the second semester of MATH 110),

• complex numbers and their properties.

Lectures

There will be two in-person lectures per week. Each lecture will be 1.5 hours long. Here's the schedule

• Monday 6:30pm - 8:00pm in Jeffery 102

• Tuesday 6:30pm - 8:00pm in Jeffery 102

Teaching Team

Instructor: Francesco Cellarosi (fc19@queensu.ca).

Teaching Assistant: Ahmed Shaltut (aghg@queensu.ca)

Textbooks

While preparing this course, I have consulted a list of textbooks at various levels. The course will be based on a combination of materials from many different sources, and I will share my notes with the students.

Three main reference texts for the Functional Analysis part of the course are:

• Functional Analysis: An Introduction by Yuli Eidelman, Vitali Milman, and Antonis Tsolomitis. Graduate Studies in Mathematics, Volume 66, 2004. ISBN: 978-0-8218-3646-0. The prerequisites for the first half of the book are minimal amounts of linear algebra and calculus.

• Functional Analysis by George Bachman and Lawrence Narici. Dover Publications Inc. QA320.B24 2000. This book targets 3rd and 4th year students specifically. It covers a lot of material, and has the advantage of being self-contained.

• Applied functional analysis. Main principles and their applications. by Eberhard Zeidler. Volume 109 of Applied Mathematical Sciences. Springer-Verlag, New York, 1995.

• Functional Analysis, Spectral Theory and Applications by Manfred Einsiedler and Thomas Ward. Graduate Texts in Mathematics. Springer International Publishing AG 2017.

For the Quantum Mechanics part of the course, I will follow:

• Mathematical Methods in Quantum Mechanics. With Applications to Schrödinger Operators by Gerald Teschl. Graduate Studies in Mathematics, Volume 157, 2014. ISBN: 978-1-4704-1704-8. We will cover no more than the first three chapters of this book.

• Quantum Mechanics for Mathematicians by Leon A. Takhtajan. Graduate Studies in Mathematics, Volume 95. American Mathematical Society. QC174.12.T343 2008. We will only cover parts of chapters 2 and 3 of this book.

• Quantum Mechanics: Foundations and Applications by Arno Bohm. Springer-Verlag. QC174.12.B63 1993. We will cover no more than the first two chapters of this book.