Reducibility of the quantum harmonic oscillator in d-dimensions with finitely differentiable perturbations

Abstract

In this paper, the d-dimensional quantum harmonic oscillator with a pseudo-differential time quasi-periodic perturbation iψ̇=(−Δ+V(x)+ϵW(ωt,x,−i∇))ψ,x∈Rd is considered, where ω ∈ (0,2π)n, V(x)≔∑j=1dvj2xj2,vj≥v0>0, and W(θ, x, ξ) is a real polynomial in (x, ξ) of degree at most two, with coefficients belonging to Cℓ in θ∈Tn for the order ℓ satisfying ℓ ≥ 2n + β, 0 < β < 1. Using the techniques developed by Bambusi et al. [Anal. PDE 11(3), 775–799 (2018)] and Rüssmann [“On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus,” in Dynamical Systems, Theory and Applications, Lecture Notes in Physics Vol. 38 (Rencontres, Battelle Research Institute, Seattle, WA, 1975), pp. 598–624], this paper shows that for any ϵ small enough, there is a set Dϵ⊂(0,2π)n with a big Lebesgue measure such that for any ω∈Dϵ, the system is reducible.