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 ${\rho }_{\lambda }$, ${\displaystyle {\rho }_{\lambda }={\int }_{\mathrm{\partial }\mathrm{\Omega }}(\mathrm{\nabla }\overline{{\phi }_{\lambda ,0}^{\mathrm{\Omega }}}{\phi }_{\lambda ,0}^{-\mathrm{\Omega }}-\overline{{\phi }_{\lambda ,0}^{\mathrm{\Omega }}}\mathrm{\nabla }{\phi }_{\lambda ,0}^{-\mathrm{\Omega }})\cdot \nu \mathrm{.}}$ On the exponentially long time scale $\beta =N\mathrm{/}\mid {\rho }_{\lambda }\mid$, the number of passages converges, for every fixed $N>0$, to a Poisson random variable of mean $N$. We identify $-\frac{1}{\lambda }\log \mid {\rho }_{\lambda }\mid \to S(d,-d)$ and obtain ${\displaystyle E_1(\lambda )-E_0(\lambda )=2\mid {\rho }_{\lambda }\mid (1+o(1))\mathrm{.}}$ Thus the familiar instanton expansion of the double-well eigenvalue splitting emerges directly from a factorization of the heat-kernel trace.