Communications in Mathematical Physics

, Volume 242, Issue 1–2, pp 221–250 | Cite as

Quasi-Periodic Solutions for Two-Level Systems

  • Guido Gentile


We consider the Schrödinger equation for a class of two-level atoms in a quasi-periodic external field in the case in which the spacing 2ɛ between the two unperturbed energy levels is small, and we study the problem of finding quasi-periodic solutions of a related generalized Riccati equation. We prove the existence of quasi-periodic solutions of the latter equation for a Cantor set ℰ of values of ɛ around the origin which is of positive Lebesgue measure: such solutions can be obtained from the formal power series by a suitable resummation procedure. The set ℰ can be characterized by requesting infinitely many Diophantine conditions of Mel’nikov type.


Energy Level Power Series Lebesgue Measure External Field Riccati Equation 
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© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Guido Gentile
    • 1
  1. 1.Dipartimento di MatematicaUniversità di Roma TreRomaItaly

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