Mathematical Foundations of Quantum Theory
Summer Semester 2025, 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 (25.07.25).

Solutions of exercises: Here is the latest version (19.07.25).

Diary of the lectures

• 29.04.25: The axioms of quantum mechanics; pre-Hilbert spaces; orthogonal vectors and the Cauchy–Schwarz inequality; Hilbert spaces; L2 and l2 spaces.

• 07.05.25: Complete orthonormal sets; Bounded linear operators; Adjoint of a bounded operator; Self-adjoint and unitary operators, projectors; Notions of convergence for bounded operators.

• 16.05.25: 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.

• 21.05.25: Adjoint of the sum of operators; Symmetric and self-adjoint operators; Multiplication operators: denseness of domain and adjoint; Adjoint of extensions and maximal symmetry of self-adjoint operators.

• 28.05.25: 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.

• 04.06.25: Adjoint and closure of momentum operator with periodic boundary conditions; 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.

• 18.06.25: 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.

• 25.06.25: 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.

• 02.07.25: 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; Operator norm of multiplication operators.

• 09.07.25: 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.

• 16.07.25: Crash course on measure theory I: sigma-algebras, measures, Borel sigma-algebras, regular Borel measure, distribution functions; Crash course on measure theory II: measurable functions, Lebesgue integral; Projection-valued measures; Properties of PVMs.

• 23.07.25: Spectral measures; Lebesgue integral of bounded functions with respect to a PVM; Lebesgue integral of unbounded functions with respect to a PVM; Representation of bounded sesquilinear forms; Statement of the spectral theorem for unbounded self-adjoint operators.

• 25.07.25: Proof of the spectral theorem; Spectrum as topological support of the PVM; Functional calculus on a self-adjoint operator; Unitary evolution generated by a self-adjoint operator; Stone's theorem; The axioms of quantum mechanics, revisited.

Diary of the exercise classes

• 12.05.25: Polarization identity and parallelogram law; L2 scalar product of continuous functions; Non-completeness of continuous functions; Equivalent definitions of operator norm; Equivalence between boundedness and continuity for linear operators.

• 23.05.25: Properties of the adjoint of bounded operators; Characterization of symmetry; Strong and weak convergence in a dense subset; Equivalent definitions of denseness; Continuous linear extension theorem.

• 06.06.25: A+(-A) is not always zero; Adjoint of the composition of unbounded operators; Relation between kernel of A* and range of A; Closure of a symmetric operator is symmetric; Integration by parts for absolutely continuous functions.

• 11.06.25: Characterization of derivatives of continuous functions; Closure and orthogonal complement; Eigenvalues of momentum operators; Orthogonal direct sum of Hilbert spaces; Bounded operators are closed.

• 13.06.25: Product of closed and bounded operator; Number operator on l2 is closed; Kernel of closed operator is closed; Normal operators are closed; Poincaré inequality.

• 27.06.25: Eigenvalues and eigenvectors of symmetric operators; Denseness of finite sequences in l2; Essential self-adjointness of multiplication operators on l2; Direct sum of self-adjoint operators; Resolvent operators and Herglotz–Nevanlinna functions; Friedrichs Hamiltonians.

• 04.07.25: The Hellinger–Toeplitz theorem; Spectrum in finite dimensions; Closing the proof of holomorphy of the resolvent; Isospectrality of similar operators; Pseudo-self-adjoint operators have real spectrum.

• 11.07.25: The second resolvent identity; Spectrum of unitary operators; Closing the proof of the Weyl criterion; Weyl sequences for the position operators; A differential operator with empty spectrum.

• 18.07.25: Sigma-algebras are closed under intersections; Discontinuities of distribution functions; Integrability of the Dirichlet function; Lebesgue integral w.r.t. Dirac measure; PVM in finite dimensions; Properties of PVMs.