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 $\omega$ and $\nu$, where $W(\phi ,\varphi )$ is a smooth map from ${\mathbb{T}}^d\times {\mathbb{T}}^n$ to $\mathbb{R}$ and odd in $\phi$.

48 pages, 6 figures

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• Time quasi-periodic gravity water waves in finite depth
• On time dependent Schr{ö}dinger equations: global well-posedness and growth of Sobolev norms
• Reducibility of 1-d Schrödinger equation with unbounded time quasiperiodic perturbations, III
• On reducibility of Quantum Harmonic Oscillator on ${\mathbb{R}}^d$ with quasiperiodic in time potential

Cited by in corpus (1)

• Reducibility of 1-D quantum harmonic oscillator with new unbounded oscillatory perturbations