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 (lecturer: Davide Lonigro): Wednesdays 10:15–11:45, Seminarraum 307 im Tandemlabor, old ECAP building.

• Exercise classes (lecturer: Leonhard Richter): 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 (24.06.26).

Solutions of exercises: Here is the latest version (12.06.26).

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.

• 13.05.26: Properties of the adjoint; Adjoint of the sum of operators; Symmetric and self-adjoint operators; Multiplication operators: denseness of domain and adjoint.

• 20.05.26: Extensions of operators and adjoint; Closure of an operator; Essential self-adjointness; The momentum operator: boundary conditions and symmetry; Absolutely continuous functions and Sobolev spaces; Adjoint and closure of momentum operator with Dirichlet boundary conditions.

• 27.05.26: Direct sum of Hilbert spaces; Graph of an operator; Topological definition of closure of an operator; Equivalence between definitions of operator closure; Closure of multiplication operator on compactly supported functions.

• 03.06.26: Closure of the momentum operator with Dirichlet boundary conditions; Inverse operator; Adjoint and closure of inverse operator; Necessary and sufficient criterion of self-adjointness; necessary and sufficient criterion of essential self-adjointness.

• 10.06.26: Applications of the criterion of (essential) self-adjointness: multiplication operator on L2 and l2 spaces, momentum operators with Dirichlet and periodic boundary conditions; Closed graph and Hellinger-Toeplitz theorems; Resolvent and spectrum: basic definitions; Resolvent of the adjoint.

• 17.06.26: First resolvent formula; Recap on complex analysis; Resolvent set is open, resolvent map is holomorphic; Resolvent and spectrum of bounded operators and Neumann series; Resolvent and spectrum of symmetric and self-adjoint operators.

• 24.06.26: Operator norm of multiplication operators; Resolvent and spectrum of multiplication operators; Resolvent and spectrum of the periodic momentum; Resolvent and spectrum of the inverse; Resolvent and spectrum of unitary operators; Weyl sequences and generalized eigenvectors; Weyl sequences for the position operator.

Diary of the exercise classes

• 24.04.26: Polarization identity; Scalar product on the space of continuous functions; The space of continuous functions is not complete.

• 08.05.26: Equivalent expressions for the operator norm; Linear operators are bounded iff they are continuous.

• 15.05.26: Properties of the adjoint of bounded operators; Equivalence between symmetry and real-valuedness of expectation values; Continuous linear extension theorem.

• 22.05.26: Adjoint of sum vs sum of adjoints; Adjoint of product vs product of adjoints; Kernel of the adjoint.

• 29.05.26: Double adjoint of symmetric operators; Integration by parts for absolutely continuous functions.

• 12.06.26: Bounded operators are closed; Product of closed and bounded is closed; The number operator on sequences is closed; The kernel of closed operators is closed.

• 19.06.26: Normal operators are closed; The Poincaré inequality; Eigenvalues of symmetric operators.

• 26.06.26: Subspace of finite sequences; Multiplication operators on finite sequences; Hellinger–Toeplitz theorem.