Transseries for the ground state density and generalized Bloch equation: Double-well potential case

Based on the generalized Bloch equation, the transseries expansion for the phase (exponent) of the ground state density for double-well potential is constructed. It is shown that the leading and next-to-leading terms in semiclassical expansion are still defined by the flucton trajectory (its classical action) and quadratic fluctuations (the determinant), respectively, while the next-to-next-to-leading term (at large distances) is of nonperturbative nature. It comes from the fact that all flucton classical trajectories modified by multi-instanton, instanton-anti-instanton additions lead to the same classical action behavior at large distances. This correction is proportional to sum of all leading instanton contributions to energy gap.