Reducibility of 1-d Schrödinger equation with unbounded oscillation perturbations

We build a new estimate for the normalized eigenfunctions of operator −∂xx + V(x) based on oscillatory integrals and Langer’s turning point method, where ∼ ∣x∣2ℓ at infinity with ℓ > 1. From this an improved reducibility theorem we show that equation $$\matrix{\hfill {{\rm{i}}{\partial _t}\psi = - \partial _x^2\psi V\left(x \right)\psi {{\left\langle x \right\rangle}^\mu}W\left({\nu x,\omega t} \right)\psi,\,\,\,\,\psi \psi \left({t,x} \right),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \cr \hfill {x \in \mathbb{R},\,\,\,\,\,\,\,\mu < \min \left\{{\ell {2 \over 3},{{\sqrt {4{\ell ^2} 2\ell 1} 2}} \right\},} \cr}$$ can be reduced in L2 (ℝ) to autonomous system most values frequency vector ω ν, W(φ, ϕ) is smooth map from $${\mathbb{T}^d} \times {\mathbb{T}^n}$$ ℝ odd φ.