We prove the existence of quasi-periodic solutions for Schrodinger equations with a multiplicative potential on the d-dimensional torus T^d, d ≥ 1, finitely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are infinitely many times differentiable so are the solutions. The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized operators (“Green functions”) along scales of Sobolev spaces. The key off-diagonal decay estimates of the Green functions are proved via a new multiscale inductive analysis. The main novelty concerns the measure and “complexity” estimates.

Quasi-periodic solutions with Sobolev regularity of NLS on T^d with a multiplicative potential / Berti, Massimiliano; Bolle, P.. - In: JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY. - ISSN 1435-9855. - 15:1(2013), pp. 229-286. [10.4171/JEMS/361]

Quasi-periodic solutions with Sobolev regularity of NLS on T^d with a multiplicative potential

Berti, Massimiliano;
2013-01-01

Abstract

We prove the existence of quasi-periodic solutions for Schrodinger equations with a multiplicative potential on the d-dimensional torus T^d, d ≥ 1, finitely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are infinitely many times differentiable so are the solutions. The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized operators (“Green functions”) along scales of Sobolev spaces. The key off-diagonal decay estimates of the Green functions are proved via a new multiscale inductive analysis. The main novelty concerns the measure and “complexity” estimates.
2013
15
1
229
286
https://hal.archives-ouvertes.fr/hal-00621262/document
https://arxiv.org/abs/1012.1427
Berti, Massimiliano; Bolle, P.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/11920
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