Reducibility of 1D quantum harmonic oscillator with decaying conditions on the derivative of perturbation potentials

Abstract

Abstract We prove the reducibility of 1D 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 that | V ( x , θ ) | ⩽ C , | x ∂ x V ( x , θ ) | ⩽ C , ∀ ( x , θ ) ∈ R × T σ n . The corresponding perturbation matrix ( P i j ( θ ) ) is proved to satisfy ( 1 + | i − j | ) | P i j ( θ ) | ⩽ C and i j | P i + 1 j + 1 ( θ ) − P i j ( θ ) | ⩽ C for any θ ∈ T σ n and i, j ⩾ 1. A new reducibility theorem is set up under this kind of decay in the perturbation matrix element P i j ( θ ) as well as the discrete difference matrix element P i + 1 j + 1 ( θ ) − P i j ( 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.