Instantons in a Double-Well are Poisson Distributed

Abstract

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

34 pages, no figures