Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations. I

Abstract

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