Communications in Mathematical Physics

, Volume 219, Issue 2, pp 465–480

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

  • Dario Bambusi
  • Sandro Graffi

DOI: 10.1007/s002200100426

Cite this article as:
Bambusi, D. & Graffi, S. Commun. Math. Phys. (2001) 219: 465. doi:10.1007/s002200100426


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 H0Pt) 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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Dario Bambusi
    • 1
  • Sandro Graffi
    • 2
  1. 1.Dipartimento di Matematica “F. Enriques”, Università di Milano, Via Saldini 50, 20133 Milano, Italy.¶E-mail: bambusi@mat.unimi.itIT
  2. 2.Dipartimento di Matematica, Università di Bologna, Piazza di Porta S Donato 5, 40127 Bologna, Italy.¶E-mail: graffi@dm.unibo.itIT

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