Instantons in a Double-Well are Poisson Distributed

We give a rigorous realization of the dilute instanton picture for a semiclassical Schrödinger operator with a symmetric double-well potential on $\mathbb{R}^n$. Using a localized Feynman--Kac representation, we decompose the heat-kernel trace according to the number of passages made by a Brownian bridge between shrinking neighborhoods of the two wells. We identify the weight of one passage with a hopping coefficient $ρ_λ$, $\displaystyle ρ_λ= \int_{\partialΩ} \left( \nabla\overline{φ_{λ,0}^Ω}\, φ_{λ,0}^{-Ω} - \overline{φ_{λ,0}^Ω}\, \nablaφ_{λ,0}^{-Ω} \right)\cdotν. $ On the exponentially long time scale $β=N/\lvertρ_λ\rvert$, the number of passages converges, for every fixed $N>0$, to a Poisson random variable of mean $N$. We identify $-\frac{1}λ\log\lvertρ_λ\rvert\to S(d,-d)$ and obtain $\displaystyle E_1(λ)-E_0(λ) = 2\lvertρ_λ\rvert\left(1+o(1)\right). $ Thus the familiar instanton expansion of the double-well eigenvalue splitting emerges directly from a factorization of the heat-kernel trace.