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Dordrecht : Springer, [2012]

xx, 432 stran ; 25 cm

ISBN 978-94-007-4752-4 (vázáno)

Graduate texts in mathematics, ISSN 0072-5285 ; 265

Obsahuje bibliografii na stranách 417-423 a rejstříky

001460253

Konrad Schmüdgen // Unbounded Self-adjoint Operators on Hilbert Space // 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 extensions. // Mathematics // ISBN 97 // 94-007-4752-4 // 789400 // 747524 // ? springer.com // Part I Basics of Closed Operators // 1 Closed and Adjoint Operators... 3 // 1.1 Closed and Closable Operators and Their Graphs... 3 // 1.1.1 General Notions on Linear Operators... 3 // 1.1.2 Closed and Closable Operators... 5 // 1.2 Adjoint Operators... 8 // 1.3 Examples: Differential Operators... 13 // 1.3.1 Differentiation Operators on Intervals I... 13 // 1.3.2 Linear Partial Differential Operators ... 18 // 1.4 Invariant Subspaces and Reducing Subspaces... 20 // 1.5 Exercises... 22 // 2 The Spectrum of a Closed Operator

... 25 // 2.1 Regular Points and Defect Numbers of Operators... 25 // 2.2 Spectrum and Resolvent of a Closed Operator... 28 // 2.3 Examples: Differentiation Operators II... 33 // 2.4 Exercises... 35 // 3 Some Classes of Unbounded Operators... 37 // 3.1 Symmetric Operators ... 37 // 3.2 Self-adjoint Operators... 42 // 3.3 Accretive and Sectorial Operators... 48 // 3.4 Normal Operators... 51 // 3.5 Exercises... 54 // 3.6 Notes to Part I... 57 // Part II Spectral Theory // 4 Spectral Measures and Spectral Integrals... 61 // 4.1 Resolutions of the Identity and Operator Stieltjes Integrals... 61 // 4.2 Spectral Measures... 66 // xin // XIV // Contents // 4.2.1 Definitions and Basic Properties... 66 // 4.2.2 Three Technical Results on Spectral Measures... 70 // 4.3 Spectral Integrals ... 73 // 4.3.1 Spectral Integrals of Bounded Measurable Functions . ... 74 // 4.3.2 Integrals of Unbounded Measurable Functions... 75 // 4.3.3 Properties of Spectral Integrals... 79 // 4.4 Exercises... 84 // 5 Spectral Decompositions of Self-adjoint and Normal Operators ... 85 // 5.1 Spectral Theorem for Bounded Self-adjoint Operators... 85 // 5.2 Spectral Theorem for Unbounded Self-adjoint Operators... 89 // 5.3 Functional Calculus and Various Applications... 91 // 5.4 Self-adjoint Operators with Simple Spectra... 99 // 5.5 Spectral Theorem for Finitely Many Strongly Commuting Normal // Operators...101 // 5.5.1 The Spectral Theorem for n-Tuples...101 // 5.5.2 Joint Spectrum for n-Tuples...104 // 5.6 Strong

Commutativity of Unbounded Normal Operators...105 // 5.7 Exercises...Ill // 5.8 Notes to Part II...113 // Part HI Special Topics // 6 One-Parameter Groups and Semigroups of Operators...117 // 6.1 Groups of Unitaries...117 // 6.2 Differential Equations on Hilbert Spaces...123 // 6.3 Semigroups of Contractions on Banach Spaces...126 // 6.4 Semigroups of Contractions on Hilbert Spaces...131 // 6.5 Exercises...133 // 7 Miscellanea...137 // 7.1 The Polar Decomposition of a Closed Operator...137 // 7.2 Application to the Operator Relation A A* = A* A + /...139 // 7.3 The Bounded Transform of a Closed Operator...142 // 7.4 Analytic Vectors, Quasi-analytic Vectors, Stieltjes Vectors, and // Self-adjointness of Symmetric Operators...144 // 7.5 Tensor Products of Operators...154 // 7.5.1 Tensor Product of Hilbert Spaces...154 // 7.5.2 Tensor Product of Operators...156 // 7.6 Exercises...162 // 7.7 Notes to Part HI...164 // Part IV Perturbations of Self-adjointness and Spectra // 8 Perturbations of Self-adjoint Operators...167 // 8.1 Differential Operators with Constant Coefficients on L2(E ) . . . 167 // 8.2 Relatively Bounded Perturbations of Self-adjoint Operators ... 170 // 8.3 Applications to Schrödinger Operators: Self-adjointness...173 // 8.4 Essential Spectrum of a Self-adjoint Operator...178 // 8.5 Relatively Compact Perturbations of Self-adjoint Operators ... 179 // 8.6 Applications to Schrödinger Operators: Essential Spectrum ... 182 // 8.7 Exercises...185 // 9 Trace Class Perturbations

of Spectra of Self-adjoint Operators ... 189 // 9.1 Parts of the Spectrum of a Self-adjoint Operator...189 // 9.2 Aronszajn-Donoghue Theory of Rank One Perturbations...194 // 9.3 Krein’s Spectral Shift for Rank One Perturbations...200 // 9.4 Infinite Determinants ...203 // 9.5 Perturbation Determinants...206 // 9.6 Krein’s Spectral Shift for Trace Class Perturbations...209 // 9.7 Krein’s Trace Formula...213 // 9.8 Exercises...217 // 9.9 Notes to Part IV...218 // Part V Forms and Operators // 10 Semibounded Forms and Self-adjoint Operators ...221 // 10.1 Closed and Closable Lower Semibounded Forms...221 // 10.2 Semibounded Closed Forms and Self-adjoint Operators...225 // 10.3 Order Relations for Self-adjoint Operators...230 // 10.4 The Friedrichs Extension of a Semibounded Symmetric Operator . 233 // 10.5 Examples of Semibounded Forms and Operators...235 // 10.6 Dirichlet and Neumann Laplacians on Domains of ...238 // 10.6.1 Laplacian with Dirichlet Boundary Condition ...239 // 10.6.2 Laplacian with Neumann Boundary Condition...241 // 10.7 Perturbations of Forms and Form Sums...243 // 10.8 Exercises...248 // 11 Sectorial Forms and m-Sectorial Operators...251 // 11.1 Bounded Coercive Forms on Embedded Hilbert Spaces...251 // 11.2 Sectorial Forms...255 // 11.3 Application to Second-Order Elliptic Differential Operators ... 258 // 11.4 Exercises...262 // 12 Discrete Spectra of Self-adjoint Operators...265 // 12.1 The Min-Max Principle...265 // 12.2 Negative or Positive Eigenvalues

of Schrödinger Operators ... 270 // 12.3 Asymptotic Distribution of Eigenvalues of the Dirichlet Laplacian . 273 // 12.4 Exercises...278 // 12.5 Notes to Part V...280 // xyi Contents // Part VI Self-adjoint Extension Theory of Symmetric Operators // 13 Self-adjoint Extensions: Cayley Transform and Krein Transform . . 283 // 13.1 The Cayley Transform of a Symmetric Operator...283 // 13.2 Von Neumann’s Extension Theory of Symmetric Operators ... 287 // 13.3 Existence of Positive Self-adjoint Extensions and Krein-von // Neumann Extensions ...290 // 13.4 Positive Self-adjoint Extensions and Krein Transform...296 // 13.4.1 Self-adjoint Extensions of Bounded Symmetric Operators . 296 // 13.4.2 The Krein Transform of a Positive Symmetric Operator . . 297 // 13.5 Self-adjoint Extensions Commuting with a Conjugation or // Anticommuting with a Symmetry...299 // 13.6 Exercises...302 // 14 Self-adjoint Extensions: Boundary Triplets ...307 // 14.1 Linear Relations...307 // 14.2 Boundary Triplets of Adjoints of Symmetric Operators...311 // 14.3 Operator-Theoretic Examples...315 // 14.4 Examples: Differentiation Operators HI...318 // 14.5 Gamma Fields and Weyl Functions...322 // 14.6 The Krein-Naimark Resolvent Formula...325 // 14.7 Boundary Triplets and Semibounded Self-adjoint Operators ... 329 // 14.8 Positive Self-adjoint Extensions...333 // 14.9 Exercises...339 // 15 Sturm-Liouville Operators...343 // 15.1 Sturm-Liouville Operators with Regular End Points...343 // 15.2 Limit Circle Case

and Limit Point Case...347 // 15.3 Boundary Triplets for Sturm-Liouville Operators...352 // 15.4 Resolvents of Self-adjoint Extensions...356 // 15.5 Exercises...361 // 16 The One-Dimensional Hamburger Moment Problem...363 // 16.1 Moment Problem and Jacobi Operators...363 // 16.2 Polynomials of Second Kind ...374 // 16.3 The Indeterminate Hamburger Moment Problem...378 // 16.4 Nevanlinna Parameterization of Solutions...386 // 16.5 Exercises...389 // 16.6 Notes to Part VI...391 // Appendix A Bounded Operators and Classes of Compact Operators // on Hilbert Spaces...393 // Appendix ? Measure Theory...397 // Appendix C The Fourier Transform...403 // Appendix D Distributions and Sobolev Spaces ...405 // Appendix E Absolutely Continuous Functions ...409 // Contents xvii // Appendix F Nevanlinna Functions and Stieltjes Transforms...411 // References...417 // Classical Articles ...417 // Books ...418 // Articles...420 // Subject Index...425 // Symbol Index...431