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The trace formula for the heat semigroup with polynomial potential. (English) Zbl 1248.35042

Dalang, Robert C. (ed.) et al., Seminar on stochastic analysis, random fields and applications VI. Centro Stefano Franscini, Ascona, Italy, May 19–23, 2008. Basel: Birkhäuser (ISBN 978-3-0348-0020-4/pbk; 978-3-0348-0021-1/ebook). Progress in Probability 63, 3-21 (2011).
Summary: We consider the heat semigroup \(e^{-\frac{t}{\hbar}^{H}}, t > 0\), on \(\mathbb R^d\) 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).
For the entire collection see [Zbl 1213.60017].

MSC:

35C20 Asymptotic expansions of solutions to PDEs
35K05 Heat equation
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
35C15 Integral representations of solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
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