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978-3-8439-1289-1, Reihe Mathematik

Herry Pribawanto Suryawan A White Noise Approach to Self-intersection Local Times and Feynman Integrals for Quantum Particles in Random Media

141 Seiten, Dissertation Technische Universität Kaiserslautern (2013), Softcover, A5

This dissertation is motivated by a problem coming from quantum systems in disordered media. Using the Feynman path integral representation of quantum mechanics it is possible to derive a model of a quantum particle in a random system containing dense and weakly-coupled scatterers. The scattering potential employed in the model is assumed to be a Dirac delta distribution, i.e. the weak limit of Gaussian functions. The main goal of the dissertation is to give a mathematically rigorous realization of the corresponding Feynman integrand obtained in the model in dimension one. Our investigation is based on the theory of white noise analysis. The first part of the dissertation deals with the investigation of the concept of local times and self-intersection local times of Brownian bridge using classical stochastic analysis as well as white noise analysis. As a main result, we establish the existence and convergence of the exponential of the self-intersection local times of the one-dimensional Brownian bridges which plays essential roles regarding the potential energy part in the Feynman integrand. In the second part we apply a refinement of a Wick formula for the product of a square-integrable function of white noise with a Donsker's delta distribution combined with the complex scaling method of Cameron-Doss. We prove that the Feynman-Kac-Cameron-Doss integrand for the quantum particles in random media exists as a regular distribution of white noise. In particular, we obtain a neat formula for the propagator with identical start and end point of the quantum particle-scatterers interaction system. Thus, we have a well-defined mathematical object which is used to calculate the density of states of the system.