ANNALS OF PHYSICS 159, 199-219 (1985) Asymptotic #i i/2-Expansions in Classic Particle Limit of Non-relativistic Quantum Mechanics R. ZUCCHINI Department of Physics, New York University, New York, New York 10003 Received February 17, 1984 The unitary group describing the evolution of the quantum fluctuation around any classic’ phase orbit has a perturbative strongly asymptotic expansion in the small parameter h”‘. Likewise, the mean values of the observables position and momentum in suitable time- dependent states have an asymptotic expansion in h”‘, whose zeroth-order term satisfies the classic canonical equations. Similar expansions are found for the squared quantum dispersions. ‘E’ 1985 Academic Press, Inc. 1. INTRODUCTION When dealing with the Classical Limit two natural questions arise: first, understanding in what sense Classical Mechanics turns out to be the limit of Quantum Mechanics when ZI + 0 and putting the limit procedure on a firm mathematical basis, and second, expanding the quantum fluctuation around the solutions of the classical equations in a power series of the small parameter h, in order to obtain approximate expressions. Many authors have analyzed the first problem in the context of relativistic and non-relativistic particle and field theories [l-7]. A solution for the second problem is given by the famous WKB method [8]. However, there are alternative methods. J. Ginibre and G. Velo found an asymptotic expansion in h “* for the unitary group describing the quantum fluctuation around the solution of the classical field equations for non-relativistic bosons which in some cases is Bore1 summable [9]. The aim of this paper consists in investigating whether analogous results can be obtained in the context of a non-relativistic particle theory. The problem is partly simplified because the structure of the Hilbert space of the quantum states is simpler in a particle theory, and because the classical equations are local. However, the interaction is generally more complicated, so that a more complicated combinatorics is expected. Consider a quantum mechanical system with v degrees of freedom and with an analogous classical system. In Quantum Mechanics the states of the system are represented by rays of some Hilbert space P and the observables by selfadjoint operators in 5!. Conversely, in Classical Mechanics the states are points of a real 2v- dimensional vector space 8, called phase space, and the observables real measurable functions defined on 8. Suppose that the A-dependent operator A, and the phase 199 OOO3-4916/85 $7.50 Copyright @ 1985 by Academic Press, Inc. All rights of reproduction in any form reserved.