  We study a quantum analogue of the iterative perturbation theory by
Kolmogorov used in the proof of the Kolmogorov-Arnold-Moser (KAM) theorem. The
method is based on sequent canonical transformations with a "running" coupling
constant $ \lm,\lm^{2},\lm^{4} $ etc. The proposed scheme, as its classical
predecessor, is "superconvergent" in the sense that after the n-th step, a
theory is solved to the accuracy of order $ \lm^{2^{n-1}} $. It is shown that
the Kolmogorov technique corresponds to an infinite resummation of the usual
perturbative series. The corresponding expansion is convergent for the quantum
anharmonic oscillator due to the fact that it turns out to be identical to the
Pade series. The method is easily generalizable to many-dimensional cases. The
Kolmogorov technique is further applied to a non-perturbative treatment of
Yang-Mills quantum mechanics. A controllable expansion for the wave function
near the origin is constructed. For large fields, we build an asymptotic
adiabatic expansion in inverse powers of the field. This asymptotic solution
contains arbitrary constants which are not fixed by the boundary conditions at
infinity. To find them, we approximately match the two expansions in an
intermediate region. We also discuss some analogies between this problem and
the method of QCD sum rules.
