Double wells: Nevanlinna analyticity, distributional borel sum and asymptotics

We consider the energy levels of a Stark family, in the parameter j, of quartic double wells with perturbation parameter g: H(g, j) = p(2) + x(2)(1 - gx)(2) - j (gx - 1/2). For non-even j (and for the symmetric case j = 0) we prove analyticity in the full Nevanlinna disk Rg(-2) > R(-1) of the g(2)-plane, as predicted by Crutchfield. By means of an approximation we give a heuristic estimate of the asymptotic small g behaviour, showing the relation between the avoided crossings and the failure of the usual perturbation series