Borel summation of adiabatic invariants
  Borel summation techniques are developed to obtain exact invariants from formal adiabatic invariants (given as divergent series in a small parameter) for a class of differential equations, under assumptions of analyticity of the coefficients; the method relies on the study of associated partial differential equations in the complex plane. The type and location of the singularities of these associated functions, important in determining exponentially small corrections to formal invariants are also briefly discussed. 
O. Costin, L. Dupaigne, M. D. Kruskal