Chapter One Operator theory in finite-dimensional vector spaces § 1 Vector spaces and normed vector spaces . . . . . . . . . . . . . 1 1. Basic notions . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Linear manifolds . . . . . . . . . . . . . . . . . . . . . . 3 4. Convergence and norms . . . . . . . . . . . . . . . . . . . 4 5. Topological notions in a normed space . . . . . . . . . . . . . 6 6. Infinite series of vectors . . . . . . . . . . . . . . . . . . 7 7. Vector-valued functions . . . . . . . . . . . . . . . . . . . 8 § 2. Linear forms and the adjoint space . . . . . . . . . . . . . . . . 10 1. Linear forms . . . . . . . . . . . . . . . . . . . . . . . . 10 2. The adjoint space . . . . . . . . . . . . . . . . . . . . . . 11 3. The adjoint basis . . . . . . . . . . . . . . . . . . . . . . 12 4. The adjoint space of a normed space . . . . . . . . . . . . . . 13 5. The convexity of balls . . . . . . . . . . . . . . . . . . . . 14 6. The second adjoint space . . . . . . . . . . . . . . . . . . . 15 § 3. Linear operators . . . . . . . . . . . . . . . . . . . . . . . . 16 1. Definitions. Matrix representations . . . . . . . . . . . . . . 16 2. Linear operations on operators . . . . . . . . . . . . . . . . 18 3. The algebra of linear operators . . . . . . . . . . . . . . 19 4. Projections. Nilpotents . . . . . . . . . . . . . . . . . . . . 20 5. Invariance. Decomposition . . . . . . . . . . . . . . . . . . 22 6. The adjoint operator . . . . . . . . . . . . . . . . . . . . . 23 § 4. Analysis with operators . . . . . . . . . . . . . . . . . . . . . 25 1. Convergence and norms for operators . . . . . . . . . . . . . 25 2. The norm of T" . . . . . . . . . . . . . . . . . . . . . . . 27 3. Examples of norms . . . . . . . . . . . . . . . . . . . . . 28 4. Infinite series of operators . . . . . . . . . . . . . . . . . . 29 5. Operator-valued functions . . . . . . . . . . . . . . . . . . 31 6. Pairs of projections . . . . . . . . . . . . . . . . . . . . . 32 § 5. The eigenvalue problem . . . . . . . . . . . . . . . . . . . . . 34 1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 34 2. The resolvent . . . . . . . . . . . . . . . . . . . . . . . . 36 3. Singularities of the resolvent . . . . . . . . . . . . . . . . . 38 4. The canonical form of an operator . . . . . . . . . . . . . . . 40 5. The adjoint problem . . . . . . . . . . . . . . . . . . . . . 43 6. Functions of an operator . . . . . . . . . . . . . . . . . . . 44 7. Similarity transformations . . . . . . . . . . . . . . . . . . 46 § 6. Operators in unitary spaces . . . . . . . . . . . . . . . . . . . 47 1. Unitary spaces . . . . . . . . . . . . . . . . . . . . . . . 47 2. The adjoint space . . . . . . . . . . . . . . . . . . . . . . 48 3. Orthonormal families . . . . . . . . . . . . . . . . . . . . 49 4. Linear operators . . . . . . . . . . . . . . . . . . . . . . 51 5. Symmetric forms and symmetric operators . . . . . . . . . . . 52 6. Unitary, isometric and normal operators . . . . . . . . . . . . 54 7. Projections . . . . . . . . . . . . . . . . . . . . . . . . . 55 8. Pairs of projections . . . . . . . . . . . . . . . . . . . . . 56 9. The eigenvalue problem . . . . . . . . . . . . . . . . . . . 58 10. The minimax principle . . . . . . . . . . . . . . . . . . . . 60 Chapter Two Perturbation theory in a finite-dimensional space 62 § 1. Analytic perturbation of eigenvalues . . . . . . . . . . . . . . . 63 1. The problem . . . . . . . . . . . . . . . . . . . . . . . . 63 2. Singularities of the eigenvalues . . . . . . . . . . . . . . . . 65 3. Perturbation of the resolvent . . . . . . . . . . . . . . . . . 66 4. Perturbation of the eigenprojections . . . . . . . . . . . . . . 67 5. Singularities of the eigenprojections . . . . . . . . . . . . . . 69 6. Remarks and examples . . . . . . . . . . . . . . . . . . . . 70 7. The case of T (x) linear in x . . . . . . . . . . . . . . . . . 72 8. Summary . . . . . . . . . . . . . . . . . . . . . . . . . 73 § 2. Perturbation series . . . . . . . . . . . . . . . . . . . . . . . 74 1. The total projection for the A-group . . . . . . . . . . . . . . 74 2. The weighted mean of eigenvalues . . . . . . . . . . . . . . . 77 3. The reduction process . . . . . . . . . . . . . . . . . . . . 81 4. Formulas for higher approximations . . . . . . . . . . . . . . 83 5. A theorem of Motzkin-Taussky . . . . . . . . . . . . . . . 85 6. The ranks of the coefficients of the perturbation series . . . . . . 86 § 3. Convergence radii and error estimates . . . . . . . . . . . . . . 88 1. Simple estimates . . . . . . . . . . . . . . . . . . . . . . 88 2. The method of majorizing series . . . . . . . . . . . . . . . . 89 3. Estimates on eigenvectors . . . . . . . . . . . . . . . . . . 91 4. Further error estimates . . . . . . . . . . . . . . . . . . . 93 5. The special case of a normal unperturbed operator . . . . . . . . 94 6. The enumerative method . . . . . . . . . . . . . . . . . . . 97 § 4. Similarity transformations of the eigenspaces and eigenvectors . . . . 98 1. Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . 98 2. Transformation functions . . . . . . . . . . . . . . . . . . . 99 3. Solution of the differential equation . . . . . . . . . . . . . . 102 4. The transformation function and the reduction process . . . . . . 104 5. Simultaneous transformation for several projections . . . . . . . 104 6. Diagonalization of a holomorphic matrix function . . . . . . . . 106 § 5. Non-analytic perturbations . . . . . . . . . . . . . . . . . . . 106 1. Continuity of the eigenvalues and the total projection . . . . . . 106 2. The numbering of the eigenvalues . . . . . . . . . . . . . . . 108 3. Continuity of the eigenspaces and eigenvectors . . . . . . . . . 110 4. Differentiability at a point . . . . . . . . . . . . . . . . . . 111 5. Differentiability in an interval . . . . . . . . . . . . . . . . 113 6. Asymptotic expansion of the eigenvalues and eigenvectors . . . . 115 7. Operators depending on several parameters . . . . . . . . . . . 116 8. The eigenvalues as functions of the operator . . . . . . . . . . 117 § 6. Perturbation of symmetric operators . . . . . . . . . . . . . . . 120 1. Analytic perturbation of symmetric operators . . . . . . . . . . 120 2. Orthonormal families of eigenvectors . . . . . . . . . . . . . . 121 3. Continuity and differentiability . . . . . . . . . . . . . . . . 122 4. The eigenvalues as functions of the symmetric operator . . . . . 124 5. Applications. A theorem of Lidskii . . . . . . . . . . . . . . 124 Chapter Three Introduction to the theory of operators in Banach spaces § 1. Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . 127 1. Normed spaces . . . . . . . . . . . . . . . . . . . . . . . 127 2. Banach spaces . . . . . . . . . . . . . . . . . . . . . . . 129 3. Linear forms . . . . . . . . . . . . . . . . . . . . . . . . 132 4. The adjoint space . . . . . . . . . . . . . . . . . . . . . . 134 5. The principle of uniform boundedness . . . . . . . . . . . . . 136 6. Weak convergence . . . . . . . . . . . . . . . . . . . . . . 137 7. Weak* convergence . . . . . . . . . . . . . . . . . . . . . 140 8. The quotient space . . . . . . . . . . . . . . . . . . . . . 140 § 2. Linear operators in Banach spaces . . . . . . . . . . . . . . . . 142 1. Linear operators. The domain and range . . . . . . . . . . . . 142 2. Continuity and boundedness . . . . . . . . . . . . . . . . . 145 3. Ordinary differential operators of second order. . . . . . . . . . 146 § 3. Bounded operators . . . . . . . . . . . . . . . . . . . . . . . 149 1. The space of bounded operators . . . . . . . . . . . . . . . 149 2. The operator algebra .1 (X) . . . . . . . . . . . . . . . . . . 153 3. The adjoint operator . . . . . . . . . . . . . . . . . . . . . 154 4. Projections . . . . . . . . . . . . . . . . . . . . . . . . . 155 § 4. Compact operators . . . . . . . . . . . . . . . . . . . . . . . 157 1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . 157 2. The space of compact operators . . . . . . . . . . . . . . . . 158 3. Degenerate operators. The trace and determinant . . . . . . . . 160 § 5. Closed operators . . . . . . . . . . . . . . . . . . . . . . . . 163 1. Remarks on unbounded operators . . . . . . . . . . . . . . . 163 2. Closed operators . . . . . . . . . . . . . . . . . . . . . . . 164 3. Closable operators . . . . . . . . . . . . . . . . . . . . . . 165 4. The closed graph theorem .. . . . . . . . . . . . . . . . . . 166 5. The adjoint operator . . . . . . . . . . . . . . . . . . . . . 167 6. Commutativity and decomposition . . . . . . . . . . . . . . . 171 § 6. Resolvents and spectra . . . . . . . . . . . . . . . . . . . . . 172 1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 172 2. The spectra of bounded operators . . . . . . . . . . . . . . . 176 3. The point at infinity . . . . . . . . . . . . . . . . . . . . . 176 4. Separation of the spectrum . . . . . . . . . . . . . . . . . . 178 5. Isolated eigenvalues . . . . . . . . . . . . . . . . . . . . . 180 6. The resolvent of the adjoint . . . . . . . . . . . . . . . . . 183 7. The spectra of compact operators . . . . . . . . . . . . . . . 185 8. Operators with compact resolvent . . . . . . . . . . . . . . . 187 Chapter Four Stability theorems § 1. Stability of closedness and bounded invertibility . . . . . . . . . . 189 1. Stability of closedness under relatively bounded perturbation . . . 189 2. Examples of relative boundedness . . . . . . . . . . . . . . . 191 3. Relative compactness and a stability theorem . . . . . . . . . . 194 4. Stability of bounded invertibility . . . . . . . . . . . . . . . 196 § 2. Generalized convergence of closed operators . . . . . . . . . . . . 197 1. The gap between subspaces . . . . . . . . . . . . . . . . . . 197 2. The gap and the dimension . . . . . . . . . . . . . . . . . . 199 3. Duality . . . . . . . . . . . . . . . . . . . . . . . . . . 200 4. The gap between closed operators . . . . . . . . . . . . . . . 201 5. Further results on the stability of bounded invertibility . . . . . 205 6. Generalized convergence . . . . . . . . . . . . . . . . . . . 206 § 3. Perturbation of the spectrum . . . . . . . . . . . . . . . . . . 208 1. Upper semicontinuity of the spectrum . . . . . . . . . . . . . 208 2. Lower semi-discontinuity of the spectrum . . . . . . . . . . . 209 3. Continuity and analyticity of the resolvent . . . . . . . . . . . 210 4. Semicontinuity of separated parts of the spectrum . . . . . . . . 212 5. Continuity of a finite system of eigenvalues . . . . . . . . . . . 213 6. Change of the spectrum under relatively bounded perturbation . 214 7. Simultaneous consideration of an infinite number of eigenvalues 215 8. An application to Banach algebras. Wiener's theorem . . . . . . 216 § 4. Pairs of closed linear manifolds . . . . . . . . . . . . . . . . . 218 1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 218 2. Duality . . . . . . . . . . . . . . . . . . . . . . . . . . 221 3. Regular pairs of closed linear manifolds . . . . . . . . . . . . 223 4. The approximate nullity and deficiency . . . . . . . . . . . . 225 5. Stability theorems . . . . . . . . . . . . . . . . . . . . . . 227 § 5. Stability theorems for semi-Fredholm operators . . . . . . . . . . 229 1. The nullity, deficiency and index of an operator . . . . . . . . . 229 2. The general stability theorem . . . . . . . . . . . . . . . . . 232 3. Other stability theorems . . . . . . . . . . . . . . . . . . . 236 4. Isolated eigenvalues . . . . . . . . . . . . . . . . . . . . . 239 5. Another form of the stability theorem . . . . . . . . . . . . . 241 6. Structure of the spectrum of a closed operator . . . . . . . . . 242 § 6. Degenerate perturbations . . . . . . . . . . . . . . . . . . . . 244 1. The Weinstein-Aronszajn determinants . . . . . . . . . . . . . 244 2. The W-A formulas . . . . . . . . . . . . . . . . . . . . . . 246 3. Proof of the W-A formulas . . . . . . . . . . . . . . . . . . 248 4. Conditions excluding the singular case . . . . . . . . . . . . . 249 Chapter Five Operators in Hilbert spaces § 1. Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . 251 1. Basic notions . . . . . . . . . . . . . . . . . . . . . . . . 251 2. Complete orthonormal families . . . . . . . . . . . . . . . . 254 § 2. Bounded operators in Hilbert spaces . . . . . . . . . . . . . . . 256 1. Bounded operators and their adjoints . . . . . . . . . . . . . 256 2. Unitary and isometric operators . . . . . . . . . . . . . . . . 257 3. Compact operators . . . . . . . . . . . . . . . . . . . . . 260 4. The Schmidt class . . . . . . . . . . . . . . . . . . . . . . 262 5. Perturbation of orthonormal families . . . . . . . . . . . . . . 264 § 3. Unbounded operators in Hilbert spaces . . . . . . . . . . . . . . 267 1. General remarks . . . . . . . . . . . . . . . . . . . . . . 267 2. The numerical range . . . . . . . . . . . . . . . . . . . . 267 3. Symmetric operators . . . . . . . . . . . . . . . . . . . . 269 4. The spectra of symmetric operators . . . . . . . . . . . . . . 270 5. The resolvents and spectra of selfadjoint operators . . . . . . . 272 6. Second-order ordinary differential operators . . . . . . . . . . 274 7. The operators T*T . . . . . . . . . . . . . . . . . . . . . 275 8. Normal operators . . . . . . . . . . . . . . . . . . . . . . 276 9. Reduction of symmetric operators . . . . . . . . . . . . . . 277 10. Semibounded and accretive operators . . . . . . . . . . . . . 278 11. The square root of an m-accretive operator . . . . . . . . . . 281 § 4. Perturbation of selfadjoint operators . . . . . . . . . . . . . . . 287 1. Stability of selfadjointness . . . . . . . . . . . . . . . . . . 287 2. The case of relative bound 1 . . . . . . . . . . . . . . . . . 289 3. Perturbation of the spectrum . . . . ... . . . . . . . . . . . 290 4. Semibounded operators . . . . . . . . . . . . . . . . . . . 291 5. Completeness of the eigenprojections of slightly non-selfadjoint operators . 293 § 5. The Schrödinger and Dirac operators . . . . . . . . . . . . . . . 297 1. Partial differential operators . . . . . . . . . . . . . . . . . 297 2. The Laplacian in the whole space . . . . . . . . . . . . . . . 299 3. The Schrödinger operator with a static potential . . . . . . . . 302 4. The Dirac operator . . . . . . . . . . . . . . . . . . . . . 305 Chapter Six Sesquilinear forms in Hilbert spaces and associated operators § 1. Sesquilinear and quadratic forms . . . . . . . . . . . . . . . . . 308 1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 308 2. Semiboundedness . . . . . . . . . . . . . . . . . . . . . . 310 3. Closed forms . . . . . . . . . . . . . . . . . . . . . . . . 313 4. Closable forms . . . . . . . . . . . . . . . . . . . . . . . 315 5. Forms constructed from sectorial operators . . . . . . . . . . . 318 6. Sums of forms . . . . . . . . . . . . . . . . . . . . . . . 319 7. Relative boundedness for forms and operators . . . . . . . . . . 321 § 2. The representation theorems . . . . . . . . . . . . . . . . . . 322 1. The first representation theorem . . . . . . . . . . . . . . . . 322 2. Proof of the first representation theorem . . . . . . . . . . . . 323 3. The Friedrichs extension . . . . . . . . . . . . . . . . . . . 325 4. Other examples for the representation theorem . . . . . . . . . 326 5. Supplementary remarks . . . . . . . . . . . . . . . . . . . 328 6. The second representation theorem . . . . . . . . . . . . . . 331 7. The polar decomposition of a closed operator . . . . . . . . . . 334 § 3. Perturbation of sesquilinear forms and the associated operators . . . 336 1. The real part of an m-sectorial operator . . . . . . . . . . . . . 336 2. Perturbation of an m-sectorial operator and its resolvent . . . . . 338 3. Symmetric unperturbed operators . . . . . . . . . . . . . . . 340 4. Pseudo-Friedrichs extensions . . . . . . . . . . . . . . . . . 341 § 4. Quadratic forms and the Schrödinger operators . . . . . . . . . . 343 1. Ordinary differential operators . . . . . . . . . . . . . . . . 343 2. The Dirichlet form and the Laplace operator . . . . . . . . . . 346 3. The Schrödinger operators in R3 . . . . . . . . . . . . . . . . 348 4. Bounded regions . . . . . . . . . . . . . . . . . . . . . . 352 § 5. The spectral theorem and perturbation of spectral families . . . . . 353 1. Spectral families . . . . . . . . . . . . . . . . . . . . . . 353 2. The selfadjoint operator associated with a spectral family . . . . 356 3. The spectral theorem . . . . . . . . . . . . . . . . . . . . 360 4. Stability theorems for the spectral family . . . . . . . . . . . . 361 Chapter Seven Analytic perturbation theory § 1. Analytic families of operators . . . . . . . . . . . . . . . . . . 365 1. Analyticity of vector- and operator-valued functions . . . . . . . 365 2. Analyticity of a family of unbounded operators . . . . . . . . . 366 3. Separation of the spectrum and finite systems of eigenvalues . . . 368 4. Remarks on infinite systems of eigenvalues . . . . . . . . . . . 371 5. Perturbation series . . . . . . . . . . . . . . . . . . . . 372 6. A holomorphic family related to a degenerate perturbation . . . . 373 § 2. Holomorphic families of type (A) . . . . . . . . . . . . . . . . 375 1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . 375 2. A criterion for type (A) . . . . . . . . . . . . . . . . . . . 377 3. Remarks on holomorphic families of type (A) . . . . . . . . . . 379 4. Convergence radii and error estimates . . . . . . . . . . . . . 381 5. Normal unperturbed operators . . . . . . . . . . . . . . . . 383 § 3. Selfadjoint holomorphic families . . . . . . . . . . . . . . . . . 385 1. General remarks . . . . . . . . . . . . . . . . . . . . . . . 385 2. Continuation of the eigenvalues . . . . . . . . . . . . . . . . 387 3. The Mathieu, Schrodinger, and Dirac equations . . . . . . . . . 389 4. Growth rate of the eigenvalues . . . . . . . . . . . . . . . . 390 5. Total eigenvalues considered simultaneously . . . . . . . . . . 392 § 4. Holomorphic families of type (B) . . . . . . . . . . . . . . . . 393 1. Bounded-holomorphic families of sesquilinear forms . . . . . . . 393 2. Holomorphic families of forms of type (a) and holomorphic families of operators of type (B) . 395 3. A criterion for type (B) . . . . . . . . . . . . . . . . . . . 398 4. Holomorphic families of type (B) . . . . . . . . . . . . . . . 401 5. The relationship between holomorphic families of types (A) and (B) 403 6. Perturbation series for eigenvalues and eigenprojections . . . . . 404 7. Growth rate of eigenvalues and the total system of eigenvalues . . . 407 8. Application to differential operators . . . . . . . . . . . . . . 408 9. The two-electron problem . . . . . . . . . . . . . . . . . . 410 § 5. Further problems of analytic perturbation theory . . . . . . . . . 413
§ 6. Eigenvalue problems in the generalized form . . . . . . . . . . . 416 1. General considerations . . . . . . . . . . . . . . . . . . . . 416 2. Perturbation theory . . . . . . . . . . . . . . . . . . . . . 419 3. Holomorphic families of type (A) . . . . . . . . . . . . . . . 421 4. Holomorphic families of type (B) . . . . . . . . . . . . . . . 422 5. Boundary perturbation . . . . . . . . . . . . . . . . . . . . 423 Chapter Eight Asymptotic perturbation theory § 1. Strong convergence in the generalized sense . . . . . . . . . . . . 427 1. Strong convergence of the resolvent . . . . . . . . . . . . . . 427 2. Generalized strong convergence and spectra . . . . . . . . . . . 431 3. Perturbation of eigenvalues and eigenvectors . . . . . . . . . . 433 4. Stable eigenvalues . . . . . . . . . . . . . . . . . . . . . . 437 § 2. Asymptotic expansions . . . . . . . . . . . . . . . . . . . . . 441 1. Asymptotic expansion of the resolvent . . . . . . . . . . . . . 441 2. Remarks on asymptotic expansions . . . . . . . . . . . . . . 444 3. Asymptotic expansions of isolated eigenvalues and eigenvectors . . 445 4. Further asymptotic expansions . . . . . . . . . . . . . . . . 448 § 3. Generalized strong convergence of sectorial operators . . . . . . . . 453 1. Convergence of a sequence of bounded forms . . . . . . . . . . 453 2. Convergence of sectorial forms "from above" . . . . . . . . . . 455 3. Nonincreasing sequences of symmetric forms . . . . . . . . . . 459 4. Convergence from below . . . . . . . . . . . . . . . . . . . 461 5. Spectra of converging operators . . . . . . . . . . . . . . . . 462 § 4. Asymptotic expansions for sectorial operators . . . . . . . . . . . 463 1. The problem. The zeroth approximation for the resolvent . . . . . 463 2. The 1/2-order approximation for the resolvent . . . . . . . . . 465 3. The first and higher order approximations for the resolvent . . . . 466 4. Asymptotic expansions for eigenvalues and eigenvectors . . . . . 470 § 5. Spectral concentration . . . . . . . . . . . . . . . . . . . . . 473 1. Unstable eigenvalues . . . . . . . . . . . . . . . . . . . . . 473 2. Spectral concentration . . . . . . . . . . . . . . . . . . . . 474 3. Pseudo-eigenvectors and spectral concentration . . . . . . . . . 475 4. Asymptotic expansions . . . . . . . . . . . . . . . . . . . . 476 Chapter Nine Perturbation theory for semigroups of operators § 1. One-parameter semigroups and groups of operators . . . . . . . . . 479 1. The problem . . . . . . . . . . . . . . . . . . . . . . . . 479 2. Definition of the exponential function . . . . . . . . . . . . . 480 3. Properties of the exponential function . . . . . . . . . . . . . 482 4. Bounded and quasi-bounded semigroups . . . . . . . . . . . . 486 5. Solution of the inhomogeneous differential equation . . . . . . . 488 6. Holomorphic semigroups . . . . . . . . . . . . . . . . . . . 489 7. The inhomogeneous differential equation for a holomorphic semigroup 493 8. Applications to the heat and Schrodinger equations . . . . . . . 495 § 2. Perturbation of semigroups . . . . . . . . . . . . . . . . . . . 497 1. Analytic perturbation of quasi-bounded semigroups . . . . . . . 497 2. Analytic perturbation of holomorphic semigroups . . . . . . . . 499 3. Perturbation of contraction semigroups . . . . . . . . . . . . 501 4. Convergence of quasi-bounded semigroups in a restricted sense . . . 502 5. Strong convergence of quasi-bounded semigroups . . . . . . . . 503 6. Asymptotic perturbation of semigroups . . . . . . . . . . . . 506 § 3. Approximation by discrete semigroups . . . . . . . . . . . . . . 509 1. Discrete semigroups . . . . . . . . . . . . . . . . . . . . 509 2. Approximation of a continuous semigroup by discrete semigroups . 511 3. Approximation theorems . . . . . . . . . . . . . . . . . . . 513 4. Variation of the space . . . . . . . . . . . . . . . . . . . . 514 Chapter Ten Perturbation of continuous spectra and unitary equivalence § 1. The continuous spectrum of a selfadjoint operator . . . . . . . . . 516 1. The point and continuous spectra . . . . . . . . . . . . . . . 516 2. The absolutely continuous and singular spectra . . . . . . . . . 518 3. The trace class . . . . . . . . . . . . . . . . . . . . . . . 521 4. The trace and determinant . . . . . . . . . . . . . . . . . . 523 § 2. Perturbation of continuous spectra . . . . . . . . . . . . . . . . 525 1. A theorem of Weyl-von Neumann . . . . . . . . . . . . . . 525 2. A generalization . . . . . . . . . . . . . . . . . . . . . . . 527 § 3. Wave operators and the stability of absolutely continuous spectra . . . 529 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 529 2. Generalized wave operators . . . . . . . . . . . . . . . . . . 531 3. A sufficient condition for the existence of the wave operator . . . 535 4. An application to potential scattering . . . . . . . . . . . . . 536 § 4. Existence and completeness of wave operators . . . . . . . . . . . 537 1. Perturbations of rank one (special case) . . . . . . . . . . . . 537 2. Perturbations of rank one (general case) . . . . . . . . . . . . 540 3. Perturbations of the trace class . . . . . . . . . . . . . . . . 542 4. Wave operators for functions of operators . . . . . . . . . . . 545 5. Strengthening of the existence theorems . . . . . . . . . . . . 549 6. Dependence of W± (H2, H1) on H1 and H2 . . . . . . . . . . . . 553 § 5. A stationary method . . . . . . . . . . . . . . . . . . . . . . 553 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 553 2. The r operations . . . . . . . . . . . . . . . . . . . . . . 555 3. Equivalence with the time-dependent theory . . . . . . . . . . 557 4. The r operations on degenerate operators . . . . . . . . . . . 558 5. Solution of the integral equation for rank A = I . . . . . . . . 560 6. Solution of the integral equation for a degenerate A . . . . . . . 563 7. Application to differential operators . . . . . . . . . . . . . . 565 Supplementary Notes Chapter I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 Chapter II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 Chapter III . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 Chapter IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 Chapter V . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 Chapter VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 Chapter VII . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 Chapter VIII . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 Chapter IX . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 Chapter X . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 Books and monographs . . . . . . . . . . . . . . . . . . . . . 593 Supplementary Bibliography . . . . . . . . . . . . . . . . . . . . 596 Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . 596 Notation index . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 Author index . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 Subject index . . . . . . . . . . . . . . . . . . . . . . . . . . . 612
| TItle | Perturbation Theory for Linear Operators |
| Author | Tosio Kato |
| Publisher | Springer Verlag |
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Welcome to Marinkyo's School! > Books about mathmatics > Tosio Kato : Perturbation Theory for Linear Operators
