On the large order asymptotics of the wave function perturbation theory

The problem of finding the large order asymptotics for the eigenfunction perturbation theory in quantum mechanics is studied. The relation between the wave function arguemnt x and the number of perturbation theory order k that allows us to construct the asymptotics by the saddle-point technique is found: xk12 = const, k → ∞. Classical euclidean solutions starting from the classical vacuum play an important role in constructing such asymptotics. The correspondence between the trajectory end and the parameter xk12 is found. The obtained results can be applied to the calculation of the main values of the observables depending on k in the k-th order of perturbation theory at large k and, probably, to the mulitparticle production problem.