Abstract
This paper is devoted to an asymptotic formula for splitting of the lowest eigenvalues of a two-dimensional Schrodinger operator with a potential having two symmetric wells. We rigorously prove the corresponding formula, obtained earlier in a paper by J. Bruning, S. Yu. Dobrokhotov, and E. S. Semenov [“Unstable Closed Trajectories, Librations and Splitting of the Lowest Eigenvalues in Quantum Double Well Problem,” Regul. Chaotic Dyn. 11 (2), 167–180 (2006)] at the physical level of rigor. The crucial role in our considerations is played by an asymptotic formula for the Maupertuis action (as a function of energy) on a periodic trajectory of the classical system (a libration) lying near a doubly asymptotic trajectory.
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