# Lecture: Functional Analysis in Physics

Andrzej Sitarz, spring semester 2021/22

Operators in Quantum Mechanics and Quantum Field Theory are basic objects that mathematically allow describing physical systems. Looking at their properties and spectra we learn about physical phenomena. The language and methods of functional analysis and operator algebras become important in gravity, particle physics, QFT, solid-state physics, quantum computers and quantum graphs. The lecture extends the short introduction to the topic from course lectures aiming to demonstrate the mathematical basis of fundamental properties that are used in physics.

*The background picture illustrates spectral properties of a
Hamiltonian related to electrons in 2D systems.*

## The main topics of the lecture:

- Basics: Banach and Hilbert spaces. Topologies and convergence.
- Linear operators: bounded and unbounded.
- Dual spaces. Adjoint operators.
- Operator spectrum, basic theorems on the spectra of operators.
- Functional calculus - allowed operations on operators.
- Operator algebras, C* algebras and their properties.
- More on properties and applications of C* algebras, von Neumann algebras.
- Compact operators, unbounded operators.
- Sobolev spaces. Differential operators and their spectra.

Full subject description sheet (in Polish).

Full subject description sheet (in English).

USOS page (in Polish)

USOS page (in English)