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[We consider the heat semigroup \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$e^{-\frac{t}{\hbar}H}, t > 0,\,\, {\rm on}\,\, \mathbb{R}^d$$ \end{document} with generator H corresponding to a potential growing polynomially at infinity. Its trace for positive times is represented as an analytically continued infinite-dimensional oscillatory integral. The asymptotics in the small parameter _ is exhibited by using Laplace’s method in infinite dimensions in the case of a degenerate phase (this corresponds to the limit from quantum mechanics to classical mechanics, in a situation where the Euclidean action functional has a degenerate critical point).]
Published: Feb 4, 2011
Keywords: Heat kernels; polynomial potential; infinite-dimensional oscillatory integrals; Laplace method; degenerate phase; asymptotics; semiclassical limit
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