We prove a reducibility result for a quantum harmonic oscillator in arbitrary dimension with arbitrary frequencies perturbed by a linear operator which is a polynomial of degree two in xj, −i∂j with coefficients which depend quasiperiodically on time.
We prove the first bifurcation result of time quasi-periodic traveling wave solutions for space periodic water waves with vorticity. In particular, we prove the existence of small amplitude time quasi-periodic solutions of the gravity-capillary water waves equations with constant vorticity, for a bidimensional fluid over a flat bottom delimited by a space-periodic free interface. These quasi-periodic solutions exist for all the values of depth, gravity and vorticity, and restrict the surface tension to a Borel set of asymptotically full Lebesgue measure.
In this paper we consider time dependent Schrödinger linear PDEs of the form i∂tψ = L(t)ψ, where L(t) is a continuous family of self-adjoint operators. We give conditions for well-posedness and polynomial growth for the evolution in abstract Sobolev spaces.is a perturbation smooth in time and H is a self-adjoint positive operator whose spectrum can be enclosed in spectral clusters whose distance is increasing, we prove that the Sobolev norms of the solution grow at most as t ǫ when t → ∞, for any ǫ > 0. If V (t) is analytic in time we improve the bound to (log t) γ , for some γ > 0. The proof follows the strategy, due to Howland, Joye and Nenciu, of the adiabatic approximation of the flow. We recover most of known results and obtain new estimates for several models including 1-degree of freedom Schrödinger operators on R and Schrödinger operators on Zoll manifolds.
We prove an abstract theorem giving a t bound (for all > 0) on the growth of the Sobolev norms in linear Schrödinger equations of the form i ψ = H 0 ψ + V (t)ψ as t → ∞. 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].
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