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 relationship between large order graphs and the instanton-antiinstanton solution. We derive an equation for the perturbative coefficients of the ground state energy when the number of 3 and/or 4-vertices is fixed and large. These 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 instanton-antiinstanton solution and detremines leading Borel singularity.