Davide Lonigro
Researcher in quantum mechanics
Davide Lonigro
Researcher in quantum mechanics
⚠️ No exercise class on May 1 and no lecture on May 6. The exercise class on May 8 will take place as usual.
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: Here is the latest version (29.04.26).
Solutions of exercises: TBA
Diary of the lectures
15.04.26: The axioms of quantum mechanics; pre-Hilbert spaces; orthogonal vectors and the Cauchy–Schwarz inequality; Hilbert spaces; L² spaces.
22.04.26: l² spaces; Complete orthonormal sets; Bounded linear operators; Adjoint of a bounded operator.
29.04.26: Self-adjoint and unitary operators, projectors; Notions of convergence for bounded operators; The necessity for unbounded operators; Definition of (possibly) unbounded operators; Extensions and restrictions; Sum and composition of unbounded operators; Adjoint of an unbounded operator.
Diary of the exercise classes
TBA