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Blanchard P., Bruening E. — Mathematical Methods in Physics: Distributions, Hilbert Space Operators, and Variational Method
Blanchard P., Bruening E. — Mathematical Methods in Physics: Distributions, Hilbert Space Operators, and Variational Method

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Название: Mathematical Methods in Physics: Distributions, Hilbert Space Operators, and Variational Method

Авторы: Blanchard P., Bruening E.

Аннотация:

Physics has long been regarded as a wellspring of mathematical problems. "Mathematical Methods in Physics" is a self-contained presentation, driven by historic motivations, excellent examples, detailed proofs, and a focus on those parts of mathematics that are needed in more ambitious courses on quantum mechanics and classical and quantum field theory. A comprehensive bibliography and index round out the work. Key Topics:

- Part I: A brief introduction to (Schwartz) distribution theory; Elements from the theories of ultra distributions and hyperfunctions are given in addition to some deeper results for Schwartz distributions, thus providing a rather comprehensive introduction to the theory of generalized functions. Basic properties of and basic properties for distributions are developed with applications to constant coefficient ODEs and PDEs; the relation between distributions and holomorphic functions is developed as well.

- Part II: Fundamental facts about Hilbert spaces and their geometry. The theory of linear (bounded and unbounded) operators is developed, focusing on results needed for the theory of Schroedinger operators. The spectral theory for self-adjoint operators is given in some detail.

- Part III: Treats the direct methods of the calculus of variations and their applications to boundary- and eigenvalue-problems for linear and nonlinear partial differential operators, concludes with a discussion of the Hohenberg — Kohn variational principle.

- Appendices: Proofs of more general and deeper results, including completions, metrizable Hausdorff locally convex topological vector spaces, Baire's theorem and its main consequences, bilinear functionals. Aimed primarily at a broad community of graduate students in mathematics, mathematical physics, physics and engineering, as well as researchers in these disciplines. Requisite knowledge for the reader includes differential and integral calculus, linear algebra, and some topology. Some basic knowledge of ordinary and partial differential equations will enhance the appreciation of the presented material.


Язык: en

Рубрика: Физика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2003

Количество страниц: 471

Добавлена в каталог: 09.04.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Adjoint      47
Adjoint operator      251
Almost periodic functions      224
Angle between two vectors      189
Baire space      448
Banach, reflexive, examples of      381
Bilinear form      455
Bilinear form, continuous      455
Bilinear form, separately continuous      455
Boundary value      154
Bounded linear function      17
Bounded pointwise      241 449
Bounded uniformly or norm      241
Breit — Wigner formula      37
Calculus of variations      373
Caratheodory functions      420
Carrier      163
Cauchy estimates      120
Cauchy sequence in $\mathcal{D'}(\Omega)$      35
Cauchy — Riemann equations      126
Cauchy's integral formula I      118
Cauchy's integral formula II      119
Cauchy's principal value      33
Chain rule      55
Change of variables      54
Class $\mathcal{C}^n$      390
Coercive      383
Commutation relations of Heisenberg      280
Compact, locally      237
Compact, relatively      381
Compact, sequentially      380
complete      190 214
Complete, sequentially      16
Completeness relation      214
Completion of a normed space      443
Completion of an inner product space      443
Constraint      403
Continuous at $x_0$      17
Continuous on X      17
Convergence of sequences of distributions      35
Convergence of series of distributions      40
Convex      396
Convex minimization      382
Convolution - product of functions      83
Convolution in $\mathcal{S}(\mathbb{R}^n)$      85
Convolution of distributions      92
Convolution, equation      99
Core of a linear operator      257
Core of a quadratic form      266
Delta function, Dirac's      3
Delta sequences      36
Density matrix or statistical operator      308
Derivative      388
Derivative of a distribution      48
Derivative, weak or distributional      64
Dido      374
Dido's problem      409
Differentiable, twice      390
Direct method of the calculus of variations      376
Direct orthogonal sum      201
Direct sum of Hilbert spaces      228
Dirichlet boundary condition      376
Dirichlet boundary conditions      418
Dirichlet form      267
Dirichlet integral      375
Dirichlet problem      375 417
Dirichlet, Laplacian      419
Distribution with support in $x_0$      57
Distribution, Dirac's delta      33
Distribution, local order      30
Distribution, order      30
Distribution, periodic      55
Distribution, regular      5
Distribution, tempered      42
Distributions      29
Distributions of compact support      29 43
Distributions, regular      31 32
Distributions, tempered      29
Divergence type      423
Domain of a linear operator      248
Dual, algebraic      9
Dual, topological      18 27
Eigenvalue problem, nonlinear      423
Elementary solution, advanced resp. retarded      112
Equi-continuous      449
Euclidean spaces      193
Euler      373
Euler — Lagrange equation      376 401
Euler — Lagrange, necessary condition of      394
Existence of a minimum      380
Extension of a linear operator      249
Extension of a quadratic form      266
Fock space, Boson      234
Fock space, fermion      234
Form domain      266
Form norm      266
Form sum      273
Fourier expansion      214
Fourier hyperfunction      164
Fourier transform      129
Fourier transformation      129
Fourier transformation on $L^2(\mathbb{R}^n)$      142
Fourier transformation on $\mathcal{S'}(\mathbb{R}^n)$      137
Fourier transformation on $\mathcal{S}(\mathbb{R}^n)$      136
Fourier transformation, inverse      133
Frechet differentiable      388
Fredholm alternative      330
Friedrich's extension      272
Fubini's theorem for distributions      79
Function, Heaviside      49
Function, strongly decreasing      20
Functional      375
Functional calculus of self-adjoint operators      356
Functional, analytic      163
Fundamental theorem of algebra      121
Fundamental theorem of calculus      392
Gateaux derivative      398
Gateaux differential      397
Generalized functions      5
Generalized functions, Gelfand type $\mathcal{S}_a^b$      162
Gram determinant      204 205
Gram — Schmidt orthonormalization      213
Green's function      106
Hadamard's principal value      34
Hamilton operator, free      261
Heat equation      110 147
Heisenberg's uncertainty relation      311
Helmholtz differential operator      146
Hermite functions      221
Hermite polynomials      221
Hilbert cube      304
Hilbert space      191
Hilbert space basis      212
Hilbert space, separable      211 212
Hilbert sum or direct sum of Hilbert spaces      228
Hilbert transform      96
Hilbert — Schmidt operator      308
Hlctvs, metrizable      445
Holomorphic      117
Hyperfunctions      164
Hypo-elliptic      107
Hypo-ellipticity of $\bar{\partial}$      116
Inequality, Bessel's      188
Inequality, Cauchy — Schwarz — Bunjakowski      189
Inequality, Schwarz'      188
Inequality, triangle      189
Infinitesimal generator      299
Inner product space      187
Integral equation      179
Integral of functions      391
Integral operators of Hilbert — Schmidt      278
Integral with respect to a spectral family      344
Isometry      297
Isoperimetric problem      374
Kato perturbation      314
Kato perturbation of free Hamiltonian      316
Knotensatz      220
Lagrange multiplier      403
Lagrange multipliers, existence of      407
Laguerre functions      222
Laguerre polynomials      221
Laplace operator      145
Laplace operator, fundamental solution      108
Laplace transform of distributions      158
Laplace transform of functions      158
Laurent expansion      123
Lebesgue space      195
Legendre polynomials      222
Leibniz      373
Leibniz formula      53
Lemma of Riemann — Lebesgue      129
Lemma of Riesz      329
Lemma of Weyl      105
Lemma of Wielandt      281
Length      187
Level surface      404
Linear hull or span      200
Linear operator from $\mathcal{H}$ into $\mathcal{K}$      248
Linear operator of multiplication      259
Linear operator, bounded      250 275
Linear operator, closable      255
Linear operator, closed      253
Linear operator, closure of      255
Linear operator, core of      257
Linear operator, essentially self-adjoint      256 258
Linear operator, positive      288
Linear operator, product or composition      250
Linear operator, self-adjoint      256 258
Linear operator, symmetric      256
Linear operator, unbounded      250 276
Local extrema, necessary and sufficient conditions for      394
Locally integrable      15
Lower semi-continuous      379
Matrix spaces      193
Maximal      214
Maximal self-adjoint part of A      351
Maxwell's equations in vacuum      112
Metric      18
Metric space      441
Metric space, completion      442
Metrizable, HLCTVS      18
Minimax principle      415
Minimizer, existence      380
Minimizer, uniqueness      380
Minimizing point      396
Minimizing sequence      376
Minimum (maximum), global      394
Minimum (maximum), local      393
Modulus of an operator      289
Momentum operator      260
Monotone      396
Muliplicator space      53
Multiplication operator, bounded      277
Multiplication operator, unbounded      278
Neumann series      287
Newton      373
Norm      8
Norm of a bounded linear operator      275
Norm topology      190
Norm, Hilbertian      192
Norm, induced by an inner product      189
o      388
ONB, characterization of      213
Open ball      441
Operator of finite rank      302
Operator, compact      301
Operator, completely continuous      301
Operator, inverse      248
Operator, trace      305
Operator, trace class      304
Orthogonal      187
Orthogonal complement $M^{\bot}$      199
Orthonormal      187
Orthonormal basis      212
Orthonormal polynomials      219
Orthonormal system      187
p-ball, open      10
Parallelogram law      192
Parseval relation      214
Partial differential operator, linear constant coefficients      50
Partial differential operator, linear elliptic      420
Pauli matrices      234
Poisson equation      109
Polar decomposition      289
Polarization identity      192
Pole of finite order      124
Pre-Hilbert space      187
Product of distributions and functions      52
Product, inner or scalar      186
Product, rule      52
Projection theorem for closed subspaces      201
Projection theorem for convex sets      413
Projector or projection operator      293
Pseudo function      35
Pythagoras, Theorem of      188
Quadratic form      265
Quadratic form, closable      266
Quadratic form, closed      266
Quadratic form, continuous      266
Quadratic form, densely defined      265
Quadratic form, first representation theorem      269
Quadratic form, lower bound      266
Quadratic form, minimization      383
Quadratic form, positive      266
Quadratic form, second representation theorem      271
Quadratic form, semi-bounded, from below      265
Quadratic form, symmetric      265
Quadratic functional      383
Radon measure      67
Regular (critical) point      394
Regular points of an operator A      318
Regularization of distributions      87
Regularizing sequence      88
Renormalization      58
Residue      124
Resolution of the identity      295 341
Resolvent      318
Resolvent, identity      319
Resolvent, set      318
Restriction of a linear operator      249
Riemann integral      311
Rodrigues' formula      221
Schroedinger operator free      148
Schwartz distributions      159
Self-adjoint, geometric characterization      339
Semi-metric      18
Semi-norm      8
Semi-norm, comparable      11
Semi-norm, smaller than      10
Semi-norms, system of      11
Semi-norms, system of, filtering      11
Semi-norms, systems of, equivalent      12
Sequence space      194
Sequence, Cauchy      15 190
Sequence, converges      15 190
Set, open      8
Singularity, essential      124
Singularity, isolated      124
Singularity, removable      124
Sobolev constant      417
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