Quantum KAM Technique and Yang-Mills Quantum Mechanics

Abstract

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.