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 H?(?t)=-?+(?)lvj2xj2+?W(?t,x,-i?),vj>0,??D=(0,2?)n,and W(?,x,?)=(?)?(?)xm1?m2,m1,m2?N,m1+m2=2,(?,x,?)?Tn×Rd×Rd,?(?)?Cs(Tn,R).Suppose ?*>0 sufficiently small and s*=max{12(n+1),48n},for any |?|<?*,s>S*;if ?(?)? Cs(Tn,R),then there exists a Cantor set D?(?)(0,2?)n with its measure satisfying meas((0,27r)n\De)? C?1/32,such that if ??D? the Schrodinger equation is reducible.More precisely,there exists a quasi-periodic mapping U?(?t)with t.Denote U?(?t)?=?,then ? satisfies the autonomous equation:i?=H??,where H?=(?)vj?(xj2-(?)xj2)is a positive definitely diagonal operator with |vj?-vj|?C?,j=1,2,...,d.Meanwhile,if m>0 and ?0 ? Hm we have the estimates on Hm norm for the solutions:Cm ??0?m??U?*(?t)e-itH?U?(0)?0?m?Cm??0?m,(?)t ?R.Furthermore,we get that all the spectrum of the Floquet operator K:=-i?·(?)-?+(?)vj2xj2+?W(?,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].