Reducibility Of D-dimensional Quantum Harmonic Oscillators Under Smooth Perturbations

This paper mainly discusses the reducibility of the high-dimensional Schrodinger equation under the smooth perturbation with time t.Specifically,consider the non-autonomous equation defined by d-dimensional quantum harmonic oscillation:(?),where Hε(ωt)=-Δ+(?)lvj2xj2+εW(ωt,x,-i▽),vj>0,ω∈D=(0,2π)n,and W(θ,x,ξ)=(?)ω(θ)xm1ξm2,m1,m2∈N,m1+m2=2,(θ,x,ξ)∈Tn×Rd×Rd,ω(θ)∈Cs(Tn,R).Suppose ε*>0 sufficiently small and s*=max{12(n+1),48n},for any |ε|<ε*,s>S*;if ω(θ)∈ Cs(Tn,R),then there exists a Cantor set Dε(?)(0,2π)n with its measure satisfying meas((0,27r)n\De)≤ Cε1/32,such that if ω∈Dε the Schrodinger equation is reducible.More precisely,there exists a quasi-periodic mapping Uω(ωt)with t.Denote Uω(ωt)φ=ψ,then φ satisfies the autonomous equation:iφ=H∞φ,where H∞=(?)vj∞(xj2-(?)xj2)is a positive definitely diagonal operator with |vj∞-vj|≤Cε,j=1,2,...,d.Meanwhile,if m>0 and ψ0 ∈ Hm we have the estimates on Hm norm for the solutions:Cm ‖ψ0‖m≤‖Uω*(ωt)e-itH∞Uω(0)ψ0‖m≤Cm‖ψ0‖m,(?)t ∈R.Furthermore,we get that all the spectrum of the Floquet operator K:=-iω·(?)-Δ+(?)vj2xj2+εW(θ,x,-i▽)are pure point spectrum.Since the perturbation considered in this paper is finite-order smooth,we will apply the KAM theory in the finite-order smooth case.The method presented in this paper is as follows:By Weyl quantization method,we correspond the high-dimensional quantum harmonic equation to the classical Hamiltonian function with finite degree of freedom.Subsequently,we apply Moser-Jackson-Zehnder Lemma[5,18]to construct a series of complex analytic functions to approximate the smooth perturbation function W.Therefore we can formulate KAM iteration to obtain the reducible result for the non-autonomous Hamiltonian function with finite degree of freedom.This corresponds to the reducibility of the high-dimensional quantum harmonic equation.It should be emphasized that the reducibility of the problem on high-dimensional Schrodinger equation for general perturbations(non-quadratic polynomials)is stil-1 open.The key idea of this paper is to use the Weyl quantization method to correspond the high-dimensional quantum harmonic equations to the classical finite dimensional Hamiltonian function.The idea was inspired by[3].