The problem of large order behaviour of perturbation theory for quantum mechanical systems is considered. A new approach to it is developed. An explicit mechanism showing the connection between large order recursive relations and classical euclidean equations of motion is found. Large order asymptotics of the solution to the recursive relations is constructed. The developed method is applicable to the excited states, as well as to the ground state. Singular points of the obtained asymptotics of the perturbation series for eigenfunctions and density matrices are investigated and formulas being valid near such points are obtained.