§ 1. Vector Spaces And Normed Vector Spaces Page 1 1. Basic Notions Page 1 2. Bases Page 2 3. Linear Manifolds Page 3 4. Convergence And Norms Page 4 5. Topological Notions In A Normed Space Page 6 6. Infinite Series Of Vectors Page 7 7. Vector-Valued Functions Page 8 § 2. Linear Forms And The Adjoint Space Page 10 1. Linear Forms Page 10 2. The Adjoint Space Page 11 3. The Adjoint Basis Page 12 4. The Adjoint Space Of A Normed Space Page 13 5. The Convexity Of Balls Page 14 6. The Second Adjoint Space Page 15 § 3. Linear Operators Page 16 1. Definitions. Matrix Representations Page 16 2. Linear Operations On Operators Page 18 3. The Algebra Of Linear Operators Page 19 4. Projections. Nilpotents Page 20 5. Invariance. Decomposition Page 22 6. The Adjoint Operator Page 23 § 4. Analysis With Operators Page 25 1. Convergence And Norms For Operators Page 25 2. The Norm Of T^n Page 27 3. Examples Of Norms Page 28 4. Infinite Series Of Operators Page 29 5. Operator-Valued Functions Page 31 6. Pairs Of Projections Page 32 § 5. The Eigenvalue Problem Page 34 1. Definitions Page 34 2. The Resolvent Page 36 3. Singularities Of The Resolvent Page 38 4. The Canonical Form Of An Operator Page 40 5. The Adjoint Problem Page 43 6. Functions Of An Operator Page 44 7. Similarity Transformations Page 46 § 6. Operators In Unitary Spaces Page 47 1. Unitary Spaces Page 47 2. The Adjoint Space Page 48 3. Orthonormal Families Page 49 4. Linear Operators Page 51 5. Symmetric Forms And Symmetric Operators Page 52 6. Unitary, Isometric And Normal Operators Page 54 7. Projections Page 55 8. Pairs Of Projections Page 56 9. The Eigenvalue Problem Page 58 10. The Minimax Principle Page 60 Chapter Two. Perturbation Theory In A Finite-Dimensional Space Page 62 § 1. Analytic Perturbation Of Eigenvalues Page 63 1. The Problem Page 63 2. Singularities Of The Eigenvalues Page 65 3. Perturbation Of The Resolvent Page 66 4. Perturbation Of The Eigenprojections Page 67 5. Singularities Of The Eigenprojections Page 69 6. Remarks And Examples Page 70 7. The Case Of T(x) Linear In x Page 72 8. Summary Page 73 § 2. Perturbation Series Page 74 1. The Total Projection For The λ-Group Page 74 2. The Weighted Mean Of Eigenvalues Page 77 3. The Reduction Process Page 81 4. Formulas For Higher Approximations Page 83 5. A Theorem Of Motzkin-Tausky Page 85 6. The Ranks Of The Coefficients Of The Perturbation Series Page 86 § 3. Convergence Radii And Error Estimates Page 88 1. Simple Estimates Page 88 2. The Method Of Majorizing Series Page 89 3. Estimates On Eigenvectors Page 91 4. Further Error Estimates Page 93 5. The Special Case Of A Normal Unperturbed Operator Page 94 6. The Enumerative Method Page 97 § 4. Similarity Transformations Of The Eigenspaces And Eigenvectors Page 98 1. Eigenvectors Page 98 2. Transformation Functions Page 99 3. Solution Of The Differential Equation Page 102 4. The Transformation Function And The Reduction Process Page 104 5. Simultaneous Transformation For Several Projections Page 104 6. Diagonalization Of A Holomorphic Matrix Function Page 106 § 5. Non-Analytic Perturbations Page 106 1. Continuity Of The Eigenvalues And The Total Projection Page 106 2. The Numbering Of The Eigenvalues Page 108 3. Continuity Of The Eigenspaces And Eigenvectors Page 110 4. Differentiability At A Point Page 111 5. Differentiability In An Interval Page 113 6. Asymptotic Expansion Of The Eigenvalues And Eigenvectors Page 115 7. Operators Depending On Several Parameters Page 116 8. The Eigenvalues As Functions Of The Operator Page 117 § 6. Perturbation Of Symmetric Operators Page 120 1. Analytic Perturbation Of Symmetric Operators Page 120 2. Orthonormal Families Of Eigenvectors Page 121 3. Continuity And Differentiability Page 122 4. The Eigenvalues As Functions Of The Symmetric Operator Page 124 5. Applications: A Theorem Of Lidskii Page 124 § 1. Banach Spaces Page 127 1. Normed Spaces Page 127 2. Banach Spaces Page 129 3. Linear Forms Page 132 4. The Adjoint Space Page 134 5. The Principle Of Uniform Boundedness Page 136 6. Weak Convergence Page 137 7. Weak* Convergence Page 140 8. The Quotient Space Page 140 § 2. Linear Operators In Banach Spaces Page 142 1. Linear Operators: The Domain And Range Page 142 2. Continuity And Boundedness Page 145 3. Ordinary Differential Operators Of Second Order Page 146 § 3. Bounded Operators Page 149 1. The Space Of Bounded Operators Page 149 2. The Operator Algebra B(X) Page 153 3. The Adjoint Operator Page 154 4. Projections Page 155 § 4. Compact Operators Page 157 1. Definition Page 157 2. The Space Of Compact Operators Page 158 3. Degenerate Operators: The Trace And Determinant Page 160 § 5. Closed Operators Page 163 1. Remarks On Unbounded Operators Page 163 2. Closed Operators Page 164 3. Closable Operators Page 165 4. The Closed Graph Theorem Page 166 5. The Adjoint Operator Page 167 6. Commutativity And Decomposition Page 171 § 6. Resolvents And Spectra Page 172 1. Definitions Page 172 2. The Spectra Of Bounded Operators Page 176 3. The Point At Infinity Page 176 4. Separation Of The Spectrum Page 178 5. Isolated Eigenvalues Page 180 6. The Resolvent of the Adjoint Page 183 7. The Spectra of Compact Operators Page 185 8. Operators with Compact Resolvent Page 187 § 1. Stability of Closedness and Bounded Invertibility Page 189 1. Stability of Closedness Under Relatively Bounded Perturbation Page 189 2. Examples of Relative Boundedness Page 191 3. Relative Compactness and a Stability Theorem Page 194 4. Stability of Bounded Invertibility Page 196 § 2. Generalized Convergence of Closed Operators Page 197 1. The Gap Between Subspaces Page 197 2. The Gap and the Dimension Page 199 3. Duality Page 200 4. The Gap Between Closed Operators Page 201 5. Further Results on the Stability of Bounded Invertibility Page 205 6. Generalized Convergence Page 206 § 3. Perturbation of the Spectrum Page 208 1. Upper Semicontinuity of the Spectrum Page 208 2. Lower Semi-Discontinuity of the Spectrum Page 209 3. Continuity and Analyticity of the Resolvent Page 210 4. Semicontinuity of Separated Parts of the Spectrum Page 212 5. Continuity of a Finite System of Eigenvalues Page 213 6. Change of the Spectrum Under Relatively Bounded Perturbation Page 214 7. Simultaneous Consideration of an Infinite Number of Eigenvalues Page 215 8. An Application to Banach Algebras: Wiener's Theorem Page 216 § 4. Pairs of Closed Linear Manifolds Page 218 1. Definitions Page 218 2. Duality Page 221 3. Regular Pairs of Closed Linear Manifolds Page 223 4. The Approximate Nullity and Deficiency Page 225 5. Stability Theorems Page 227 § 5. Stability Theorems for Semi-Fredholm Operators Page 229 1. The Nullity, Deficiency and Index of an Operator Page 229 2. The General Stability Theorem Page 232 3. Other Stability Theorems Page 236 4. Isolated Eigenvalues Page 239 5. Another Form of the Stability Theorem Page 241 6. Structure of the Spectrum of a Closed Operator Page 242 § 6. Degenerate Perturbations Page 244 1. The Weinstein-Aronszajn Determinants Page 244 2. The W-A Formulas Page 246 3. Proof of the W-A Formulas Page 248 4. Conditions Excluding the Singular Case Page 249 § 1. Hilbert Space Page 251 1. Basic Notions Page 251 2. Complete Orthonormal Families Page 254 § 2. Bounded Operators in Hilbert Spaces Page 256 1. Bounded Operators and Their Adjoints Page 256 2. Unitary and Isometric Operators Page 257 3. Compact Operators Page 260 4. The Schmidt Class Page 262 5. Perturbation of Orthonormal Families Page 264 § 3. Unbounded Operators in Hilbert Spaces Page 267 1. General Remarks Page 267 2. The Numerical Range Page 267 3. Symmetric Operators Page 269 4. The Spectra of Symmetric Operators Page 270 5. The Resolvents and Spectra of Selfadjoint Operators Page 272 6. Second-Order Ordinary Differential Operators Page 274 7. The Operators T*T Page 275 8. Normal Operators Page 276 9. Reduction of Symmetric Operators Page 277 10. Semibounded and Accretive Operators Page 278 11. The Square Root of an m-Accretive Operator Page 281 § 4. Perturbation of Selfadjoint Operators Page 287 1. Stability of Selfadjointness Page 287 2. The Case of Relative Bound 1 Page 289 3. Perturbation of the Spectrum Page 290 4. Semibounded Operators Page 291 5. Completeness of the Eigenprojections of Slightly Non-Selfadjoint Operators Page 293 § 5. The Schrödinger and Dirac Operators Page 297 1. Partial Differential Operators Page 297 2. The Laplacian in the Whole Space Page 299 3. The Schrödinger Operator with a Static Potential Page 302 4. The Dirac Operator Page 305 § 1. Sesquilinear and Quadratic Forms Page 308 1. Definitions Page 308 2. Semiboundedness Page 310 3. Closed Forms Page 313 4. Closable Forms Page 315 5. Forms Constructed from Sectorial Operators Page 318 6. Sums of Forms Page 319 7. Relative Boundedness for Forms and Operators Page 321 § 2. The Representation Theorems Page 322 1. The First Representation Theorem Page 322 2. Proof of the First Representation Theorem Page 323 3. The Friedrichs Extension Page 325 4. Other Examples for the Representation Theorem Page 326 5. Supplementary Remarks Page 328 6. The Second Representation Theorem Page 331 7. The Polar Decomposition of a Closed Operator Page 334 § 3. Perturbation of Sesquilinear Forms and the Associated Operators Page 336 1. The Real Part of an M-Sectorial Operator Page 336 2. Perturbation of an M-Sectorial Operator and Its Resolvent Page 338 3. Symmetric Unperturbed Operators Page 340 4. Pseudo-Friedrichs Extensions Page 341 § 4. Quadratic Forms and the Schrödinger Operators Page 343 1. Ordinary Differential Operators Page 343 2. The Dirichlet Form and the Laplace Operator Page 346 3. The Schrödinger Operators in R³ Page 348 4. Bounded Regions Page 352 § 5. The Spectral Theorem and Perturbation of Spectral Families Page 353 1. Spectral Families Page 353 2. The Selfadjoint Operator Associated with a Spectral Family Page 356 3. The Spectral Theorem Page 360 4. Stability Theorems for the Spectral Family Page 361 § 1. Analytic Families of Operators Page 365 1. Analyticity of Vector- and Operator-Valued Functions Page 365 2. Analyticity of a Family of Unbounded Operators Page 366 3. Separation of the Spectrum and Finite Systems of Eigenvalues Page 368 4. Remarks on Infinite Systems of Eigenvalues Page 371 5. Perturbation Series Page 372 6. A Holomorphic Family Related to a Degenerate Perturbation Page 373 § 2. Holomorphic Families of Type (A) Page 375 1. Definition Page 375 2. A Criterion for Type (A) Page 377 3. Remarks on Holomorphic Families of Type (A) Page 379 4. Convergence Radii and Error Estimates Page 381 5. Normal Unperturbed Operators Page 383 § 3. Selfadjoint Holomorphic Families Page 385 1. General Remarks Page 385 2. Continuation of the Eigenvalues Page 387 3. The Mathieu, Schrödinger, and Dirac Equations Page 389 4. Growth Rate of the Eigenvalues Page 390 5. Total Eigenvalues Considered Simultaneously Page 392 § 4. Holomorphic Families of Type (B) Page 393 1. Bounded-Holomorphic Families of Sesquilinear Forms Page 393 2. Holomorphic Families of Forms of Type (A) and Holomorphic Families of Operators of Type (B) Page 395 3. A Criterion for Type (B) Page 398 4. Holomorphic Families of Type (B₀) Page 401 5. The Relationship Between Holomorphic Families of Types (A) and (B) Page 403 6. Perturbation Series for Eigenvalues and Eigenprojections Page 404 7. Growth Rate of Eigenvalues and the Total System of Eigenvalues Page 407 8. Application to Differential Operators Page 408 9. The Two-Electron Problem Page 410 § 5. Further Problems of Analytic Perturbation Theory Page 413 1. Holomorphic Families of Type (C) Page 413 2. Analytic Perturbation of the Spectral Family Page 414 § 6. Eigenvalue Problems in the Generalized Form Page 416 1. General Considerations Page 416 2. Perturbation Theory Page 419 3. Holomorphic Families of Type (A) Page 421 4. Holomorphic Families of Type (B) Page 422 5. Boundary Perturbation Page 423 § 1. Strong Convergence in the Generalized Sense Page 427 1. Strong Convergence of the Resolvent Page 427 2. Generalized Strong Convergence and Spectra Page 431 3. Perturbation of Eigenvalues and Eigenvectors Page 433 4. Stable Eigenvalues Page 437 § 2. Asymptotic Expansions Page 441 1. Asymptotic Expansion of the Resolvent Page 441 2. Remarks on Asymptotic Expansions Page 444 3. Asymptotic Expansions of Isolated Eigenvalues and Eigenvectors Page 445 4. Further Asymptotic Expansions Page 448 § 3. Generalized Strong Convergence of Sectorial Operators Page 453 1. Convergence of a Sequence of Bounded Forms Page 453 2. Convergence of Sectorial Forms "From Above" Page 455 3. Nonincreasing Sequences of Symmetric Forms Page 459 4. Convergence from Below Page 461 5. Spectra of Converging Operators Page 462 § 4. Asymptotic Expansions for Sectorial Operators Page 463 1. The Problem. The Zeroth Approximation for the Resolvent Page 465 3. The First and Higher Order Approximations for the Resolvent Page 466 4. Asymptotic Expansions for Eigenvalues and Eigenvectors Page 470 § 5. Spectral Concentration Page 473 1. Unstable Eigenvalues Page 473 2. Spectral Concentration Page 474 3. Pseudo-Eigenvectors and Spectral Concentration Page 475 4. Asymptotic Expansions Page 476 § 1. One-Parameter Semigroups and Groups of Operators Page 479 1. The Problem Page 479 2. Definition of the Exponential Function Page 480 3. Properties of the Exponential Function Page 482 4. Bounded and Quasi-Bounded Semigroups Page 486 5. Solution of the Inhomogeneous Differential Equation Page 488 6. Holomorphic Semigroups Page 489 7. The Inhomogeneous Differential Equation for a Holomorphic Semigroup Page 493 § 2. Perturbation of Semigroups Page 497 1. Analytic Perturbation of Quasi-Bounded Semigroups Page 499 2. Analytic Perturbation of Holomorphic Semigroups Page 501 3. Perturbation of Contraction Semigroups Page 502 4. Convergence of Quasi-Bounded Semigroups in a Restricted Sense Page 503 5. Strong Convergence of Quasi-Bounded Semigroups Page 506 6. Asymptotic Perturbation of Semigroups Page 509 § 3. Approximation by Discrete Semigroups Page 509 1. Discrete Semigroups Page 511 2. Approximation of a Continuous Semigroup by Discrete Semigroups Page 513 3. Approximation Theorems Page 514 4. Variation of the Space Page 514 § 1. The Continuous Spectrum of a Selfadjoint Operator Page 516 1. The Point and Continuous Spectra Page 516 2. The Absolutely Continuous and Singular Spectra Page 518 3. The Trace Class Page 521 4. The Trace and Determinant Page 523 § 2. Perturbation of Continuous Spectra Page 525 1. A Theorem of Weyl-Von Neumann Page 525 2. A Generalization Page 527 § 3. Wave Operators and the Stability of Absolutely Continuous Spectra Page 529 1. Introduction Page 529 2. Generalized Wave Operators Page 531 3. A Sufficient Condition for the Existence of the Wave Operator Page 535 4. An Application to Potential Scattering Page 536 § 4. Existence and Completeness of Wave Operators Page 537 1. Perturbations of Rank One (Special Case) Page 537 2. Perturbations of Rank One (General Case) Page 540 3. Perturbations of the Trace Class Page 542 4. Wave Operators for Functions of Operators Page 545 5. Strengthening of the Existence Theorems Page 549 6. Dependence of W+ (H₂, H₁) on H₁ and H₂ Page 553 § 5. A Stationary Method Page 553 1. Introduction Page 553 2. The I Operations Page 555 3. Equivalence with the Time-Dependent Theory Page 557 4. The I Operations on Degenerate Operators Page 558 5. Solution of the Integral Equation for Rank A = 1 Page 560 6. Solution of the Integral Equation for a Degenerate A Page 563 7. Application to Differential Operators Page 565 Chapter I. Page 568 Chapter II. Page 568 Chapter III. Page 569 Chapter IV. Page 570 Chapter V. Page 570 Chapter VI. Page 573 Chapter VII. Page 574 Chapter VIII. Page 574 Chapter IX. Page 575 Chapter X. Page 576 Bibliography Page 583 Articles Page 583 Books and Monographs Page 593 Supplementary Bibliography Page 596 Articles Page 596 Notation Index Page 606 Author Index Page 608 Subject Index Page 612