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.
Adjoint47 Adjoint operator251 Almost periodic functions224 Angle between two vectors189 Baire space448 Banach, reflexive, examples of381 Bilinear form455 Bilinear form, continuous455 Bilinear form, separately continuous455 Boundary value154 Bounded linear function17 Bounded pointwise241449 Bounded uniformly or norm241 Breit — Wigner formula37 Calculus of variations373 Caratheodory functions420 Carrier163 Cauchy estimates120 Cauchy sequence in 35 Cauchy — Riemann equations126 Cauchy's integral formula I118 Cauchy's integral formula II119 Cauchy's principal value33 Chain rule55 Change of variables54 Class 390 Coercive383 Commutation relations of Heisenberg280 Compact, locally237 Compact, relatively381 Compact, sequentially380 complete190214 Complete, sequentially16 Completeness relation214 Completion of a normed space443 Completion of an inner product space443 Constraint403 Continuous at 17 Continuous on X17 Convergence of sequences of distributions35 Convergence of series of distributions40 Convex396 Convex minimization382 Convolution - product of functions83 Convolution in 85 Convolution of distributions92 Convolution, equation99 Core of a linear operator257 Core of a quadratic form266 Delta function, Dirac's3 Delta sequences36 Density matrix or statistical operator308 Derivative388 Derivative of a distribution48 Derivative, weak or distributional64 Dido374 Dido's problem409 Differentiable, twice390 Direct method of the calculus of variations376 Direct orthogonal sum201 Direct sum of Hilbert spaces228 Dirichlet boundary condition376 Dirichlet boundary conditions418 Dirichlet form267 Dirichlet integral375 Dirichlet problem375417 Dirichlet, Laplacian419 Distribution with support in 57 Distribution, Dirac's delta33 Distribution, local order30 Distribution, order30 Distribution, periodic55 Distribution, regular5 Distribution, tempered42 Distributions29 Distributions of compact support2943 Distributions, regular3132 Distributions, tempered29 Divergence type423 Domain of a linear operator248 Dual, algebraic9 Dual, topological1827 Eigenvalue problem, nonlinear423 Elementary solution, advanced resp. retarded112 Equi-continuous449 Euclidean spaces193 Euler373 Euler — Lagrange equation376401 Euler — Lagrange, necessary condition of394 Existence of a minimum380 Extension of a linear operator249 Extension of a quadratic form266 Fock space, Boson234 Fock space, fermion234 Form domain266 Form norm266 Form sum273 Fourier expansion214 Fourier hyperfunction164 Fourier transform129 Fourier transformation129 Fourier transformation on 142 Fourier transformation on 137 Fourier transformation on 136 Fourier transformation, inverse133 Frechet differentiable388 Fredholm alternative330 Friedrich's extension272 Fubini's theorem for distributions79 Function, Heaviside49 Function, strongly decreasing20 Functional375 Functional calculus of self-adjoint operators356 Functional, analytic163 Fundamental theorem of algebra121 Fundamental theorem of calculus392 Gateaux derivative398 Gateaux differential397 Generalized functions5 Generalized functions, Gelfand type 162 Gram determinant204205 Gram — Schmidt orthonormalization213 Green's function106 Hadamard's principal value34 Hamilton operator, free261 Heat equation110147 Heisenberg's uncertainty relation311 Helmholtz differential operator146 Hermite functions221 Hermite polynomials221 Hilbert cube304 Hilbert space191 Hilbert space basis212 Hilbert space, separable211212 Hilbert sum or direct sum of Hilbert spaces228 Hilbert transform96 Hilbert — Schmidt operator308 Hlctvs, metrizable445 Holomorphic117 Hyperfunctions164 Hypo-elliptic107 Hypo-ellipticity of 116 Inequality, Bessel's188 Inequality, Cauchy — Schwarz — Bunjakowski189 Inequality, Schwarz'188 Inequality, triangle189 Infinitesimal generator299 Inner product space187 Integral equation179 Integral of functions391
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