Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations. I
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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 theoryReferences
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Additional Information
- Dario Bambusi
- Affiliation: Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, I-20133 Milano, Italy
- MR Author ID: 239364
- Email: dario.bambusi@unimi.it
- Received by editor(s): June 14, 2016
- Published electronically: October 24, 2017
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 1823-1865
- MSC (2010): Primary 35J10, 35S05, 37K55
- DOI: https://doi.org/10.1090/tran/7135
- MathSciNet review: 3739193