Reducibility of 1-D Quantum Harmonic Oscillator with Decaying Conditions on the Derivative of Perturbation Potentials

Abstract

We prove the reducibility of 1-D quantum harmonic oscillators in R perturbed by a quasi-periodic in time potential V (x, ωt) under the following conditions, namely there is a C > 0 such thatfor any θ ∈ T n σ and i, j ≥ 1. A new reducibility theorem is set up under this kind of decay in the perturbation matrix element P j i (θ) as well as the discrete difference matrix element P j+1 i+1 (θ) − P j i (θ). For the proof the novelty is that we use the decay in the discrete difference matrix element to control the measure estimates for the thrown parameter sets.