Reducibility of 1-d Schroedinger equation with time quasiperiodic unbounded perturbations, I

TL;DR

This paper proves that certain 1D Schrödinger equations with polynomial potentials and small quasiperiodic perturbations can be simplified to a reducible form, ensuring bounded Sobolev norms, using pseudodifferential calculus and KAM theory.

Contribution

It establishes reducibility for Schrödinger equations with polynomial potentials under specific growth conditions on perturbations, extending previous results to borderline cases.

Findings

System is reducible if perturbation grows slower than a critical rate.

Boundedness of Sobolev norms is achieved for the system.

Extensions to cases with maximal growth rate are provided.

Abstract

We study the Schr\"odinger equation on $\R$ with a polynomial potential behaving as $x^{2l}$ at infinity, $1\leq l\in\N$ and with a small time quasiperiodic perturbation. We prove that if the symbol of the perturbation grows at most like $(\xi^2+x^{2l})^{\beta/(2l)}$, with $\beta<l+1$, then the system is reducible. Some extensions including cases with $\beta=2l$ are also proved. The result implies boundedness of Sobolev norms. The proof is based on pseudodifferential calculus and KAM theory.