We prove an abstract theorem giving a (t)ϵ bound (for all ϵ > 0) on the growth of the Sobolev norms in linear Schrodinger equations of the form i Ψ = H0ψ + V(t)ψ as t → ∞. The abstract theorem is applied to several cases, including the cases where (i) H0 is the Laplace operator on a Zoll manifold and V (t) a pseudodifferential operator of order smaller than 2; (ii) H0 is the (resonant or nonresonant) harmonic oscillator in Rd and V (t) a pseudodifferential operator of order smaller than that of H0 depending in a quasiperiodic way on time. The proof is obtained by first conjugating the system to some normal form in which the perturbation is a smoothing operator and then applying the results of [MR17].

Growth of Sobolev norms for abstract linear Schrodinger equations / Bambusi, D.; Grebert, B.; Maspero, A.; Robert, D.. - In: JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY. - ISSN 1435-9855. - 23:2(2021), pp. 557-583. [10.4171/JEMS/1017]

Growth of Sobolev norms for abstract linear Schrodinger equations

Maspero, A.;
2021

Abstract

We prove an abstract theorem giving a (t)ϵ bound (for all ϵ > 0) on the growth of the Sobolev norms in linear Schrodinger equations of the form i Ψ = H0ψ + V(t)ψ as t → ∞. The abstract theorem is applied to several cases, including the cases where (i) H0 is the Laplace operator on a Zoll manifold and V (t) a pseudodifferential operator of order smaller than 2; (ii) H0 is the (resonant or nonresonant) harmonic oscillator in Rd and V (t) a pseudodifferential operator of order smaller than that of H0 depending in a quasiperiodic way on time. The proof is obtained by first conjugating the system to some normal form in which the perturbation is a smoothing operator and then applying the results of [MR17].
23
2
557
583
10.4171/JEMS/1017
https://arxiv.org/abs/1706.09708
Bambusi, D.; Grebert, B.; Maspero, A.; Robert, D.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/127370
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