Lie-series approach to the evolution of resonant and nonresonant anharmonic oscillators in quantum mechanics

Abstract

A version of a normal-form method of Kummer and Gompa, analogous to the Lie-series methods of classical mechanics, is used to rigorously construct uniform long-time approximations to the quantum evolution of a system of resonant or nonresonant anharmonic oscillators. The Hamiltonian Hε governing the system is a self-adjoint operator in the Hilbert space H=L2(Rν) (ν⩾2), given formally by H0+εV. Here, H0 denotes the Hamiltonian of ν one-dimensional simple harmonic oscillators and εV their anharmonic coupling, ε⩾0 being a small parameter and V an operator of multiplication by a smooth function of polynomial growth at infinity. We treat the case in which an arbitrary number ρ⩾1 of the frequencies of these oscillators are rationally independent, imposing a standard diophantine condition on the independent frequencies if ρ⩾2. Under these assumptions, stated precisely in the paper, an Nth-order approximant ψN(t,ε) to the exact solution ψ(t,ε) of the Schrödinger equation idψ(t,ε)/dt=Hεψ(t,ε), satisfying the initial condition ψ(0,ε)=ψ0, is constructed recursively, where the initial state ψ0 belongs to a family of smooth functions dense in ℋ. The main result is that ψN(t,ε) differs in ℋ-norm from ψ(t,ε) by ⩽const εN+1|t| for t∈R and ε in an arbitrary compact interval [0,ε0]. This result is obtained by an approach simpler and quite different from that of Kummer and Gompa, and extends their work to oscillators with nonpolynomial couplings, under very general resonance conditions.