Reducibility of 1-d Schrödinger equation with unbounded oscillation perturbations

Abstract

We build a new estimate for the normalized eigenfunctions of the operator −∂_xx + V(x) based on the oscillatory integrals and Langer’s turning point method, where V(x) ∼ ∣x∣^2ℓ at infinity with ℓ > 1. From this estimate and an improved reducibility theorem we show that the equation

$$\matrix{\hfill {{\rm{i}}{\partial _t}\psi = - \partial _x^2\psi + V\left(x \right)\psi + {{\left\langle x \right\rangle}^\mu}W\left({\nu x,\omega t} \right)\psi,\,\,\,\,\psi = \psi \left({t,x} \right),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \cr \hfill {x \in \mathbb{R},\,\,\,\,\,\,\,\mu < \min \left\{{\ell - {2 \over 3},{{\sqrt {4{\ell ^2} - 2\ell + 1} - 1} \over 2}} \right\},} \cr}$$

can be reduced in L² (ℝ) to an autonomous system for most values of the frequency vector ω and ν, where W(φ, ϕ) is a smooth map from ${\mathbb{T}^d} \times {\mathbb{T}^n}$ to ℝ and odd in φ.