Schmüdgen Konrad. Unbounded Self-adjoint Operators on Hilbert Space

Springer. — 2012. — 434 p. — (Graduate Texts in Mathematics 265). — ISBN 978-94-007-4752-4.Неограниченные самосопряженные операторы в гильбертовом пространстве
The book is a graduate text on unbounded self-adjoint operators on Hilbert space and their spectral theory with the emphasis on applications in mathematical physics (especially, Schrödinger operators) and analysis (Dirichlet and Neumann Laplacians, Sturm-Liouville operators, Hamburger moment problem) . Among others, a number of advanced special topics are treated on a text book level accompanied by numerous illustrating examples and exercises. The main themes of the book are the following:
- Spectral integrals and spectral decompositions of self-adjoint and normal operators
- Perturbations of self-adjointness and of spectra of self-adjoint operators
- Forms and operators
- Self-adjoint extension theory :boundary triplets, Krein-Birman-Vishik theory of positive self-adjoint extensionPart I Basics of Closed Operators
1 Closed and Adjoint Operators
2 The Spectrum of a Closed Operator
3 Some Classes of Unbounded Operators
Part II Spectral Theory
4 Spectral Measures and Spectral Integrals
5 Spectral Decompositions of Self-adjoint and Normal Operators
Part III Special Topics
6 One-Parameter Groups and Semigroups of Operators
7 Miscellanea
Part IV Perturbations of Self-adjointness and Spectra
8 Perturbations of Self-adjoint Operators
9 Trace Class Perturbations of Spectra of Self-adjoint Operators
Part V Forms and Operators
10 Semibounded Forms and Self-adjoint Operators
11 Sectorial Forms and m-Sectorial Operators
12 Discrete Spectra of Self-adjoint Operators
Part VI Self-adjoint Extension Theory of Symmetric Operators
13 Self-adjoint Extensions: Cayley Transform and Krein Transform
14 Self-adjoint Extensions: Boundary Triplets
15 Sturm–Liouville Operators
16 The One-Dimensional Hamburger Moment Problem