Instanton-anti-instanton interaction and asymptotics of the perturbation theory expansion for the double well oscillator

Abstract

An instanton-anti-instanton pair is considered as a source of singularity in the Borel transform for the ground state energy of the anharmonic oscillator. The problem of defining the short range instanton-anti-instanton interaction reduces to a calculation of the smooth part of the Borel function, which on the other hand is determined by the first few terms of ordinary perturbation theory. The preasymptotics (∼ 1 n ) of the large order perturbation theory contribution to the ground state energy of the anharmonic oscillator is found analytically. To this end the subleading correction to the long range classical instanton-anti-instanton interaction, the one-loop quantum contribution to instanton-anti-instanton interaction, and the second quantum correction to a single instanton density have been taken into account.