Doublewell Tunneling via the Feynman-Kac Formula

Author: Hardarson, Askell

Year: 1988

Degree: Dissertation (Ph.D.)

Advisor: Simon, Barry M.

Committee Members: Simon, Barry M.; Luxemburg, W. A. J.; Katok, Anatole; Koonin, Steven E.

Option: Mathematics

DOI: 10.7907/34ks-xy63

Abstract

We discuss asymptotics of the heat kernel [equation; see abstract in scanned thesis for details] and its x-derivatives when T, λ → ∞ and (T/λ) → 0 where H(λ) = - ((Δ/2) + λ²V) and where V is a double well. When the groundstate is localized in both wells for λ large we derive, by the Feynman-Kac formula, W.K.B. expansions of the groundstate, the first excited state and their gradients.

As a consequence we get a general asymptotic formula for the splitting of the two lowest eigenvalues, E₀(λ) and E₁(λ).

This formula allows us, in principle, always to go beyond the leading order given by [equation; see abstract in scanned thesis for details] where C is the action of a classical instanton.

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