Blanchard P., Bruening E. — Mathematical Methods in Physics: Distributions, Hilbert Space Operators, and Variational Method :: Электронная библиотека попечительского совета мехмата МГУ

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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.

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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

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Предметный указатель

Sobolev embeddings      417 424
Sobolev inequality      417
Sobolev space      417
Sobolev spaces      196
Sokhotski — Plemelji formula      37
Solution, classical or strong      50 105
Solution, distributional or weak      50 105
Solution, fundamental      102 106
Spectral family      341
Spectrum      318
Spectrum, continuous      321
Spectrum, discrete      321
Spectrum, essential      362
Spectrum, point      321
Square root lemma      288
State, bound      365
State, scattering      365
Stone's formula      370
Sturm — Liouville problem      426
Subset of first category      447
Subset of second category      447
Subset, meager      447
Subset, nonmeager      447
Subset, nowhere dense      447
Subspace, absolutely continuous      359
Subspace, invariant      351
Subspace, reducing      351
Subspace, singular      359
Subspace, singularly continuous      359
Support condition      91
Support of a distribution      41
Support of a spectral family      341
Support of an analytic functional      163
Support of Fourier hyperfunctions      166
Support, singular      42
Tangent space, existence of      405
Taylor expansion with remainder      392
Tensor product for distributions      79
Tensor product of functions      72
Tensor product of Hilbert spaces      231
Tensor product of Hilbert spaces, totally anti-symmetric      233
Tensor product of Hilbert spaces, totally symmetric      233
Tensor product of vector spaces      230
Tensor product, projective, of E, F      74
Tensor product, projective, of p, q      74
Test function space, $\mathcal{D}(\Omega)$       19
Test function space, $\mathcal{E}(\Omega)$       21
Test function space, $\mathcal{S}(\Omega)$       21
Theorem of F. Riesz      236
Theorem of residues      125
Theorem, Baire, version 1      447
Theorem, Baire, version 2      448
Theorem, Baire, version 3      448
Theorem, Banach — Saks      243
Theorem, Banach — Steinhaus      241 449

Theorem, Cauchy      118
Theorem, closed graph      453
Theorem, convolution      141
Theorem, de Figueiredo — Karlovitz      193
Theorem, extension of linear functionals      207
Theorem, Frechet — von Neumann — Jordan      192
Theorem, Hellinger — Toeplitz      277
Theorem, Hilbert — Schmidt      329
Theorem, Hoermander      143
Theorem, identity of holomorphic functions      122
Theorem, Inverse mapping      453
Theorem, Kakutani      193
Theorem, Kato — Rellich      314
Theorem, Liouville      121
Theorem, open mapping      452
Theorem, Plancherel      142
Theorem, Riesz — Fischer      196
Theorem, Riesz — Frechet      206
Theorem, Riesz — Schauder      328
Theorem, Spectral      347
Theorem, Stone      298
Theorem, Weyl      363
Topological complement      406
Topological space      8
Topological space, hausdorff      12 13
topology      7
Topology of uniform convergence      16
Topology on $\mathfrak{B}(\mathcal{H})$ , norm or uniform      284
Topology on $\mathfrak{B}(\mathcal{H})$ , strong      284
Topology on $\mathfrak{B}(\mathcal{H})$ , weak      284
Topology, defined by semi-norms      11
Total subset      203
Trace norm      306
Trace of trace class operators      306
Ultradifferentiable functions      168
Ultradifferential operator      169
Ultradistributions      168
Uniform boundedness principle      241 450
Unitary operator      297
Unitary operators, n-parameter group      300
Unitary operators, one-parameter group      298
Upper semi-continuity      379
Variation, nth      400
Vector space, locally convex topological      7
Vector space, topological      7
Wave operator      111
Weak Cauchy sequence      238
Weak convergence      238
Weak limit      238
Weak topology      237
Weak topology $\mathcal{D'}(\Omega)$       35
Weierstrass theorem, Generalized I      382
Weierstrass theorem, Generalized II      383
Weierstrass theorems, generalized      378
Weyl's criterium      321
Wiener — Hopf operators      280

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