This textbook provides a modern account of the theory of unbounded self‑adjoint operators on Hilbert spaces and their spectral theory, with emphasis on applications in mathematical physics (Schrödinger operators) and analysis (Dirichlet and Neumann Laplacians, Sturm–Liouville operators, and the Hamburger moment problem). The new edition has been thoroughly revised and updated and features an extensive treatment of the self‑adjoint extension theory of symmetric operators. In addition, a number of advanced special topics are presented at the textbook level, accompanied by numerous illustrative examples and exercises.

The main themes of the book are:

• Basics of unbounded operators and fundamental self‑adjointness criteria
• Spectral integrals, spectral theorems, and functional calculus for self‑adjoint and normal operators
• Perturbations of self‑adjointness and of the spectra of self‑adjoint operators
• Forms and operators
• Boundary triplets
• The Krein transform and the Krein–Vishik–Birman theory of positive self‑adjoint extensions

With several updates and additions, this edition further enhances an established standard reference and accessible entry point to the subject.

Konrad Schmüdgen is an emeritus professor at the Mathematical Institute of Leipzig University. His expertise is in operator theory, unbounded operator algebras, quantum groups and real algebraic geometry. He has authored several books, including the monograph with A. Klimyk on Quantum Groups and their Representations (1997) and the Graduate Texts in Mathematics The Moment Problem (2017) and An Invitation to Unbounded Representations of ∗-Algebras on Hilbert Space (2020).