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| Debnath L., Mikusinski P. — Introduction to Hilbert Spaces with Applications |
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| Предметный указатель |
 -norm 51
 -norm 75
Abel’s formula 260
Abel’s integral equation 247
Abel’s problem 432
Absolutely convergent series 21
Abstract minimization problem 443
action 367
Adjoint boundary conditions 251
adjoint of a differential operator 250
Adjoint operator 149 204
Admissible set 425
Admittance 269
Angular momentum operator 396 400
Annihilation operator 392
Anomalous Zeeman effect 400
Anti-derivative of a distribution 293
Anti-Hermitian operator 155
Anti-linear functional 124
Appolonius’ identity 128
Approximate eigenvalue 186
Approximation theory 453
Associated Legendre functions 254 376
Associated Legendre operator 254
Asymptotically stable solution 461
Atomic units 372
Autonomous dynamical system 461
Banach fixed point theorem 30
Banach space 19
Basic representation 38
Basis 10
Beppo Levi 59
Bessel functions 256
Bessel operator 256
Bessel’s equality and inequality 104
Best approximation 453
Bifurcation 465
Bifurcation, diagram 466 468 470
Bifurcation, points 466
Biharmonic equation 300 323
Bilinear concomitant 251
Bilinear functional 143
Born 367
Bosons 402
Bounded bilinear functional 143
Bounded linear mapping 24
Bounded operator 138
Bounded quadratic form 144
Cartesian product 8
Cartesian product of vector spaces 8
Cauchy sequence 18
Cayley transform 213
Chain rule 419
Change of variable theorem 66
Characteristic function 38
Chebyshev operator 254
Chebyshev polynomials 131 254 460
Classical quantities 353
Classical solution 295
Closed balls 14
Closed graph theorem 208
Closed operator 208
Closed set 15
Closed unbounded operator 208
Closest point property 118
Closure 16
Coercive functionals 147
Commutation relations 354
Commutator 350
Commuting operators 141
Compact operator 171
Compact sets 17
Compact-normal resolvent 212
Compatibility theorem 407
Compatible observables 351
Complementary observables 351
Complementary projection operator 168
Complete normed space 19
Complete sequence 107
Completely continuous operator 171 176
Completion of a normed space 28
Complex vector space 4
Conjugate variables 339
Conjugate-linear functional 124
Conservation law of energy 384
Conservative 334
Constant of motion 335 387
Continuity equation 366
Continuous mapping 24
Continuous spectrum 177
Contraction 30
Contraction mapping 30
Contraction mapping theorem 30 225
Control function 447
Convergence in norm 53
Convergence in normed space 12
Convergence of test function 286
Convergence, absolute 21
Convergence, almost everywhere 55
Convergence, pointwise 12
Convergence, strong 25 97
Convergence, uniform 12 25
Convergence, weak 97
Convergence, weak distributional 289
Convex functions 423
Convex sets 118
Convolution 79
Convolution theorem 196
Correspondence principle 353
Cost functional 447
Creation operator 392
de Broglie wave 370
Degenerate eigenvalue 178
Degree of degeneracy 178
Dense subsets 17
Densely defined operator 204
Derivative, Frechet 416
Derivative, Gateaux 412
Derivative, second Frechet 420
Derivatives of distributions 288
Differential operator 139
Diffusion equation 299 316
DIMENSION 10 348
Dirac delta distribution 288
Dirichlet integral 303
Dirichlet problem 310
dispersion relation 368
Distributional solution 296
Distributions 287 294
Domain 23
Domain of a differential operator 250
Dual space 27
d’Alembert solution 318
Ehrenfest’s theorem 375 383
Eigenfunction 176 253
Eigenvalue 176 253
Eigenvalue space 178
Eigenvector 176
Elliptic functional 147
Elliptic operator 310
Equality almost everywhere 54
Equation of continuity 366
Equation, Abel integral 247
Equation, Bessel 256
Equation, biharmonic 300 323
Equation, continuity 366
Equation, diffusion 299
Equation, Euler — Lagrange 428 430 435
Equation, Fredholm integral 224 232 233
Equation, Hamilton 340
| Equation, Hamilton — Jacobi 367
Equation, Heisenberg 386
Equation, Helmholtz 299 300 310
Equation, Hermite 257
Equation, homogeneous integral 224
Equation, integral 223
Equation, Klein — Gordon 300
Equation, Lagrange 336 338 431
Equation, Laguerre 255
Equation, Laplace 298 319 320
Equation, Legendre 254
Equation, matrix Riccati differential 451
Equation, Navier — Stokes 322
Equation, Newton 334
Equation, non-homogeneous integral 224
Equation, non-homogeneous wave 299
Equation, nonlinear Fredholm integral 233
Equation, ordinary differential 248 268
Equation, partial differential 295
Equation, Poisson 299 305
Equation, Schrodinger 300 362 375
Equation, state 452
Equation, Sturm — Liouville 257
Equation, telegrapher 299
Equation, Volterra integral 223 224 236 237 239 246
Equation, vorticity transport 322
Equation, wave 299 300
Equations of motion 345
Equilibrium solution 461
Equivalence of norms 13
Euclidean norm 11
Euclidean space 92
Euler — Lagrange equations 428 431 435 436 439
Existence and uniqueness of solution 228 229 232 233
expectation value 355
Extension of operators 204
Extremum 425
Fatou’s Lemma 61
Fejer’s kernel 114
Fermat’s principle 431
fermions 402
Finite dimensional operator 173
Finite dimensional vector space 10
Fixed point 29 224
Fixed point theorem 30
Formally self-adjoint differential operator 251
Fourier coefficients 117
Fourier series 117
Fourier transform 193 198
Frechet derivative 416
Fredholm alternative 229
Fredholm alternative for self-adjoint compact operators 229
Fredholm equations 224 232 233 274
Fredholm operator 151
Friedrich’s first inequality 311
Fubini’s Theorem 78
Function of an operator 191
Function spaces 5
Function, associated Legendre 254
Function, Bessel 256
Function, characteristic 38
Function, control 447
Function, convex 423
Function, Dirac delta 269 288
Function, eigen- 176 253
Function, Green’s 263 298
Function, Hamilton’s 339 431
Function, Hamilton’s principal 367
Function, Heaviside 288
Function, input 269
Function, Lagrangian 335
Function, Lebesgue integrable 43 75
Function, locally integrable complex valued 74
Function, measurable 70
Function, momentum density 372
Function, momentum wave 371
Function, null 52
Function, output 269
Function, Rademacher 110
Function, series of integrable 50
Function, smooth 285
Function, square integrable 74 76
Function, state 349 447
Function, step 38 75
Function, tent 83
Function, test 285
Function, transfer 271
Function, Walsh 111
Function, wave 349
Function, weight 253
Functional 27
Functional, anti-linear 124
Functional, bilinear 143
Functional, bounded bilinear 143
Functional, coercive 147
Functional, conjugate linear 124
Functional, cost 447
Functional, elliptic 147
Functional, linear 27
Functional, positive bilinear 143
Functional, quadratic 441
Functional, sesquilinear 143
Functional, strictly positive 143
Functional, symmetric bilinear 143
Fundamental matrix 448
Fundamental solution 297
Gateaux derivative 412
Gegenbauer polynomials 132
Generalized coordinates 336
Generalized force 339
Generalized Fourier coefficients 106
Generalized Fourier series 106
Generalized momentum 339
Generator 345 380
Geodesic 437
Gradient 418
Gradient of a functional 413
Gram — Schmidt orthonormalization process 56
Graph 23 208
Graph of an operator 23 208
Green’s first identity 301
Green’s function 263 298
Green’s second identity 301
Ground state energy 392 393
Group velocity 369
Hamilton — Jacobi’s equation 367
Hamiltonian 339 353 431
Hamiltonian operator 362
Hamilton’s canonical equation 340 386
Hamilton’s function 339 431
Hamilton’s principal function 367
Hamilton’s variational principle 342
Hammerstein operator 418
Heaviside function 288
Heisenberg commutation relation 351 354
Heisenberg operator 385
Heisenberg picture 378 384
Heisenberg’s equation of motion 385
Heisenberg’s uncertainty principle 359
Helmholtz equation 299 300 306 308
Hermite operator 256
Hermite polynomials 102 256
Hermitian operator 150
hilbert 87
Hilbert space 93
Hilbert space isomorphism 126
Hilbert transform 275
Hilbert — Schmidt theorem 187
Holder’s Inequality 7
Holonomic system 336
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