Time Quasi-Periodic Unbounded Perturbations¶of Schrödinger Operators and KAM Methods

By Dario Bambusi, S. Graffi. 2000

153 citations recorded by the source corpus.

Abstract: We eliminate by KAM methods the time dependence in a class of linear differential equations in ℓ2 subject to an unbounded, quasi-periodic forcing. This entails the pure-point nature of the Floquet spectrum of the operator H0+εP(ωt) for ε small. Here H0 is the one-dimensional Schrödinger operator p2+V, V(x)∼|x|α, α <2 for |x|→∞, the time quasi-periodic perturbation P may grow as |x|β, β <(α−2)/2, and the frequency vector ω is non resonant. The proof extends to infinite dimensional spaces the result valid for quasiperiodically forced linear differential equations and is based on Kuksin's estimate of solutions of homological equations with non-constant coefficients.

View Paper: https://www.semanticscholar.org/paper/4fe34f715b071ba81f776f430bc56af39e04d629

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Citations in the corpus

• Reducibility of first order linear operators on tori via Moser's theorem (2018, background): “ coefficients. This is a problem which is interesting on itself and has been studied for PDEs both on compact and non-compact domains, mostly in a perturbative regime. We mention among others [15], [8],[9],[24],[10],[33] for the case of linear equations. Regarding reducible KAM theory for non linear P”
• A KAM Theorem for Hamiltonian Partial Differential Equations with Unbounded Perturbations (2011, background): “Another KAM theorem with unbounded linear Hamiltonian perturbation is due to Bambusi-Graffi [2] where the spectrum property is investigated for the time dependent linear Schrödinger equation iẏ(t) = (A+ εF(t))y(t) with d− d̃ > 1 also.”
• Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type (2012, background): “See also Bambusi & Graffi [4] for a related linear result that corresponds to a gap γ > 1.”