Mathematical Foundations of Quantum Theory
Summer Semester 2026, FAU

Abstract. In the mathematical formalism of quantum mechanics, unbounded operators play a central role. Yet, their rigorous treatment requires a level of mathematical care that goes well beyond the finite-dimensional linear algebra often used in introductory course. This lecture series offers a physicist-friendly introduction to the mathematical formalism of quantum theory, with a special focus on unbounded operators in Hilbert spaces. We will progressively build up the necessary tools to understand the spectral theorem for self-adjoint operators and Stone's theorem on one-parameter unitary groups. Along the way, we will emphasize intuitive motivations, physically meaningful examples, and the connection to real-world models in quantum mechanics. The course is designed to be accessible without requiring a background in advanced mathematics, while still treating the subject with full mathematical rigor.

Schedule (unless otherwise specified):

• Main lectures: Wednesdays 10:15–11:45, Seminarraum 307 im Tandemlabor, old ECAP building.

• Exercise classes: Fridays 10:15–11:45, Raum 308 im Tandemlabor, old ECAP building.

Main reference: Chapters 1–3 and Appendix A from G. Teschl, Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators. American Mathematical Society, Graduate Studies in Mathematics, Volume 99, 2009. The book can be freely downloaded here.

Slides: TBA

Solutions of exercises: TBA

Diary of the lectures

• TBA

Diary of the exercise classes

• TBA