On the relationship between large order graphs and instantons for the double well oscillator

The double well oscillator is used as a QCD-like model for studying the rel ationship between large order graphs and the instanton-anti-instanton solut ion. We derive an equation fur the perturbative coefficients of the ground state energy when the number of 3 and/or 4-vertices is fixed and large. The se coefficients are determined in terms of an exact "bounce" solution. When the number of 4-vertices is analytically continued to be near the negative of half the number of 3-vertices the bounce solution approaches the instan ton-anti-instanton solution and determines leading Borel singularity. (C) 2 001 Elsevier Science B.V. All rights reserved.