We prove an abstract theorem giving a < t >(epsilon) bound (for all epsilon > 0) on the growth of the Sobolev norms in linear Schrodinger equations of the form i(psi)over dot = H-0 psi V(t)psi as t -> infinity. The abstract theorem is applied to several cases, including the cases where (i) H-0 is the Laplace operator on a Zoll manifold and V(t) a pseudodifferential operator of order smaller than 2; (ii) H-0 is the (resonant or nonresonant) harmonic oscillator in R-d and V(t) a pseudodifferential operator of order smaller than that of H-0 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 / D.P. Bambusi, B. Gr??bert, A. Maspero, D. Robert. - In: JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY. - ISSN 1435-9855. - 23:2(2020), pp. 557-583. [10.4171/jems/1017]
Growth of Sobolev norms for abstract linear Schrodinger equations
D.P. Bambusi
Primo
;
2020
Abstract
We prove an abstract theorem giving a < t >(epsilon) bound (for all epsilon > 0) on the growth of the Sobolev norms in linear Schrodinger equations of the form i(psi)over dot = H-0 psi V(t)psi as t -> infinity. The abstract theorem is applied to several cases, including the cases where (i) H-0 is the Laplace operator on a Zoll manifold and V(t) a pseudodifferential operator of order smaller than 2; (ii) H-0 is the (resonant or nonresonant) harmonic oscillator in R-d and V(t) a pseudodifferential operator of order smaller than that of H-0 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].File | Dimensione | Formato | |
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