# Functional Analysis

## Spectral Theory in Quantum Mechanics

Winter Term 2016/2017

Graduate Seminar on Analysis (S4B1)

Prof. Dr. J.J.L. Velázquez, Dr. A. Nota

Seminar sessions Thursdays, 4-6 pm., room N0.007

**Preliminary Meeting ****Friday, 22.07.2016, 1.15 pm in room 2.****025**

**Second Preliminary Meeting 29.09.2016 at 14 c.t. in room 2.025**

** **

### Synopsis

In this seminar we will cover several mathematical issues of functional analysis that are useful for quantum mechanics as well as for many other branches of mathematical physics: from self-adjointness, the spectral theorem, Stone's and the Rage Theorem to perturbation theory for self-adjoint operators.

Moreover we will focus on the detailed study of the free Schrödinger operator respectively position, momentum and angular momentum operators; on the

theory for Sturm-Liouville operators and apply it to spherically symmetric problems, in particular to the hydrogen atom. We will illustrate some of the abstract results of spectral theory by presenting concrete examples investigating, for instance, the self-adjointness of atomic Schrödinger operators and their essential spectrum and the scattering theory in the case of short range potentials.

### Prerequisites

Basic knowledge of PDEs and Functional Analysis is essential.

### Organization

The seminar sessions are on Thursdays 4-6 pm in room N0.007.

Topics will be distributed in a preliminary meeting on 22.07.16 at 13.15 c.t. in room 2.025. There will be a second preliminary meeting on 29.09.2016 at 14 c.t. in room 2.025.

Interested students are asked to conatct us by email (A. Nota or J. Velázquez) in advance.

The complete list of the topics is available here.

The schedule of the talks is available here.

### Literature

Chapters extracted from

- Teschl G.,
*Mathematical Methods in Quantum Mechanics*, American Mathematical Society, 2009.

and, for further readings, chapters extracted from

- Reed M., Simon B.,
*Methods of Modern Mathematical Physics*, I: Functional Analysis, Academic Press, 1980.

- Reed M., Simon B.,
*Methods of Modern Mathematical Physics*, IV: Analysis of Operators, Academic Press, 1978.