Librations and ground-state splitting in a multidimensional double-well problem

摘要

We derive an asymptotic formula for the splitting of the lowest eigenvalues of the multidimensional Schr?dinger operator with a symmetric double-well potential. Unlike the well-known formula of Maslov, Dobrokhotov, and Kolosoltsov, the obtained formula has the form A(h)e~(-S/h)(1 + o(1)), where S is the action on a periodic trajectory (libration) of the classical system with the inverted potential and not the action on the doubly asymptotic trajectory. In this expression, the principal term of the pre-exponential factor takes a more elegant form. In the derivation, we merely transform the asymptotic formulas in the mentioned work without going beyond the framework of classical mechanics.