Instantons in a Double-Well are Poisson Distributed

Abstract

We give a rigorous realization of the dilute instanton picture for a semiclassical Schr\"odinger 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 $\rho_\lambda$, $\displaystyle \rho_\lambda = \int_{\partial\Omega} \left( \nabla\overline{\varphi_{\lambda,0}^{\Omega}}\, \varphi_{\lambda,0}^{-\Omega} - \overline{\varphi_{\lambda,0}^{\Omega}}\, \nabla\varphi_{\lambda,0}^{-\Omega} \right)\cdot\nu. $ On the exponentially long time scale $\beta=N/\lvert\rho_\lambda\rvert$, the number of passages converges, for every fixed $N>0$, to a Poisson random variable of mean $N$. We identify $-\frac{1}{\lambda}\log\lvert\rho_\lambda\rvert\to S(d,-d)$ and obtain $\displaystyle E_1(\lambda)-E_0(\lambda) = 2\lvert\rho_\lambda\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.