Reducibility of 1-d Schrödinger equation with unbounded oscillation perturbations

Abstract

We build a new estimate for the normalized eigenfunctions of the operator $-{\partial }_{xx}+\mathcal{V}(x)$ based on the oscillatory integrals and Langer's turning point method, where $\mathcal{V}(x)\sim |x{|}^{2\ell }$ at infinity with $\ell >1$. From it and an improved reducibility theorem we show that the equation \[\textstyle {\rm i}\partial_t ψ=-\partial_x^2 ψ+\mathcal V(x) ψ+ε\langle x\rangle^μ W(νx,ωt)ψ,\quad ψ=ψ(t,x),~x\in\mathbb R, ~μ<\min\left\{\ell-\frac23,\frac{\sqrt{4\ell^2-2\ell+1}-1}2\right\},\] can be reduced in $L^2(\mathbb{R})$ to an autonomous system for most values of the frequency vector $\ddot{I}‰$ and $\hat{I}½$, where $W(\ddot{I}†,\ddot{I}•)$ is a smooth map from ${\mathbb{T}}^d\times {\mathbb{T}}^n$ to $\mathbb{R}$ and odd in $\ddot{I}†$.

48 pages, 6 figures