Di Persio, Luca (2007) Asymptotic Expansions of Integrals: Statistical Mechanics and Quantum Theory. UNSPECIFIED thesis, UNSPECIFIED.

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Abstract

This work deals with the subject of asymptotic expansions for both finite and infinite dimensional integrals. We first discuss a long standing problem related to the formation of crystals at zero temperature. The majority of the techniques used in this part come from the classical theory of Laplace Integrals in many dimensions and from the theory of Cluster Expansions in Probability Theory. We then move to the Quantum scenario in order to study the Caldeira-Legget model by the rigorous definition of the Influence Functional introduced by Feynman and Vernon. We make use of the theory of Feynman Path Integrals, providing the possibility to exploit the infinite dimensional generalization of the Stationary Phase method to study the asymptotics of the integrals characterizing the Caldeira-Legget model. An analogous study is made for a problem related to the semiclassical limit for the stochastic Schrödinger equation introduced by Belavkin (white noise given by a Brownian motion). Moreover we give an overview of the results related to the asymptotic expansions of integrals spanning from the unidimensional, real case, to the infinite dimensional environment and including Stokes phenomena, detailed multidimensional expansions, uniform asymptotics and asymptotics for coalescing saddle points. Angefertigt mit der Genehmigung der Mathematisch-Naturwissenschaftlichen Fakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn.

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