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


Author: Dario Bambusi
Journal: Trans. Amer. Math. Soc. 370 (2018), 1823-1865
MSC (2010): Primary 35J10, 35S05, 37K55
DOI: https://doi.org/10.1090/tran/7135
Published electronically: October 24, 2017
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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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Additional Information

Dario Bambusi
Affiliation: Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, I-20133 Milano, Italy
Email: dario.bambusi@unimi.it

DOI: https://doi.org/10.1090/tran/7135
Received by editor(s): June 14, 2016
Published electronically: October 24, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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