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Lecture notes in applied differential equations of mathematical physics /
Functional analysis is a well-established powerful method in mathematical physics, especially those mathematical methods used in modern non-perturbative quantum field theory and statistical turbulence. This book presents a unique, modern treatment of solutions to fractional random differential equat...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Singapore ; Hackensack, N.J. :
World Scientific Pub. Co.,
©2008.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Ch. 1. Elementary aspects of potential theory in mathematical physics. 1.1. Introduction. 1.2. The Laplace differential operator and the Poisson-Dirichlet potential problem. 1.3. The Dirichlet problem in connected planar regions: a conformal transformation method for green functions in string theory. 1.4. Hilbert spaces methods in the Poisson problem. 1.5. The abstract formulation of the Poisson problem. 1.6. Potential theory for the wave equation in R[symbol]
- Kirchhoff potentials (spherical means). 1.7. The Dirichlet problem for the diffusion equation
- seminar exercises. 1.8. The potential theory in distributional spaces
- The Gelfand-Chilov method.
- Ch. 2. Scattering theory in non-relativistic one-body short-range quantum mechanics: Möller wave operators and asymptotic completeness. 2.1. The wave operators in one-body quantum mechanics asymptotic properties of states in the continuous spectra of the Enss Hamiltonian. 2.2. The Enss proof of the non-relativistic one-body quantum mechanical scattering
- ch. 3. On the Hilbert space integration method for the wave equation and some applications to wave physics. 3.1. Introduction. 3.2. The abstract spectral method
- the nondissipative case. 3.3. The abstract spectral method
- the dissipative case. 3.4. The wave equation "path-integral" propagator. 3.5. On the existence of wave-scattering operators. 3.6. Exponential stability in two-dimensional magneto-elasticity: a proof on a dissipative medium. 3.7. An abstract semilinear Klein Gordon wave equation
- existence and uniqueness.
- Ch. 4. Nonlinear diffusion and wave-damped propagation: weak solutions and statistical turbulence behavior. 4.1. Introduction. 4.2. The theorem for parabolic nonlinear diffusion. 4.3. The hyperbolic nonlinear damping. 4.4. A path-integral solution for the parabolic nonlinear diffusion. 4.5. Random anomalous diffusion, a semigroup approach
- ch. 5. Domains of Bosonic functional integrals and some applications to the mathematical physics of path-integrals and string theory. 5.1. Introduction. 5.2. The Euclidean Schwinger generating functional as a functional Fourier transform. 5.3. The support of functional measures
- the Minlos theorem. 5.4. Some rigorous quantum field path-integral in the analytical regularization scheme. 5.5. Remarks on the theory of integration of functionals on distributional spaces and Hilbert-Banach spaces.
- Ch. 6. Basic integral representations in mathematical analysis of Euclidean functional integrals. 6.1. On the Riesz-Markov theorem. 6.2. The L. Schwartz representation theorem on [symbol] (distribution theory). 6.3. Equivalence of Gaussian measures in Hilbert spaces and functional Jacobians. 6.4. On the weak Poisson problem in infinite dimension. 6.5. The path-integral triviality argument. 6.6. The loop space argument for the thirring model triviality
- ch. 7. Nonlinear diffusion in R[symbol] and Hilbert spaces: a path-integral study. 7.1. Introduction. 7.2. The nonlinear diffusion. 7.3. The linear diffusion in the space [symbol].
- Ch. 8. On the Ergodic theorem. 8.1. Introduction. 8.2. On the detailed mathematical proof of the RAGE theorem. 8.3. On the Boltzmann Ergodic theorem in classical mechanics as a result of the RAGE theorem. 8.4. On the invariant Ergodic functional measure for some nonlinear wave equations. 8.5. An Ergodic theorem in Banach spaces and applications to stochastic-Langevin dynamical systems. 8.6. The existence and uniqueness results for some nonlinear wave motions in 2D
- ch. 9. Some comments on sampling of Ergodic process: an Ergodic theorem and turbulent pressure fluctuations. 9.1. Introduction. 9.2. A rigorous mathematical proof of the Ergodic theorem for wide-sense stationary stochastic process. 9.3. A sampling theorem for Ergodic process. 9.4. A model for the turbulent pressure fluctuations (random vibrations transmission).
- Ch. 10. Some studies on functional integrals pepresentations for fluid motion with random conditions. 10.1. Introduction. 10.2. The functional integral for initial fluid velocity random conditions. 10.3. An exactly soluble path-integral model for stochastic Beltrami fluxes and its string properties. 10.4. A complex trajectory path-integral representation or the Burger-Beltrami fluid flux
- ch. 11. The Atiyah-Singer index theorem: a heat kernel (PDE's) proof.