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

Bambusi, Dario

doi:10.1090/tran/7135

Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations. I
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Trans. Amer. Math. Soc. 370 (2018), 1823-1865 Request permission

Abstract:

We study the Schrödinger equation on $\mathbb {R}$ with a polynomial potential behaving as $x^{2l}$ at infinity, $1\leq l\in \mathbb {N}$, and with a small time quasiperiodic perturbation. We prove that if the symbol of the perturbation grows at most like $(\xi ^2+x^{2l})^{\beta /(2l)}$, 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

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