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.
