Friday, July 5, 2013

1307.1383 (Martin Grothaus et al.)

A White Noise Approach to the Feynman Integrand for Electrons in Random

Martin Grothaus, Felix Riemann, Herry P. Suryawan
Using the Feynman path integral representation of quantum mechanics it is possible to derive a model of an electron in a random system containing dense and weakly-coupled scatterers, see [Proc. Phys. Soc. 83, 495-496 (1964)]. The main goal of this paper is to give a mathematically rigorous realization of the corresponding Feynman integrand in dimension one based on the theory of white noise analysis. We refine and apply a Wick formula for the product of a square-integrable function with Donsker's delta functions and use a method of complex scaling. As an essential part of the proof we also establish the existence of the exponential of the self-intersection local times of a one-dimensional Brownian bridge. As result we obtain a neat formula for the propagator with identical start and end point. Thus, we obtain a well-defined mathematical object which is used to calculate the density of states, see e.g. [Proc. Phys. Soc. 83, 495-496 (1964)].
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