Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barata, J.C.A.: On formal quasi-periodic solutions of the Schrödinger equation for a two-level system with a Hamiltonian depending quasi-periodically on time. Rev. Math. Phys. 12(1), 25–64 (2000)
Barata, J.C.A.: Convergent perturbative solutions of the Schrödinger equation for two-level systems with Hamiltonians depending periodically on time. Ann. Henri Poincaré 2(5), 963–1005 (2001)
Blekher, P.M., Jauslin, H.R., Lebowitz, J.L.: Floquet spectrum for two-level systems in quasiperiodic time-dependent fields. J. Statist. Phys. 68(1–2), 271–310 (1992)
Duclos, P., Šťovíček, P.: Floquet Hamiltonians with Pure Point Spectrum. Commu. Math. Phys. 177(2), 327–347 (1996)
Duclos, P., Lev, O., Šťovíček, P., Vittot, M.: Weakly regular Floquet Hamiltonians with pure point spectrum. Rev. Math. Phys. 14(6), 531–568 (2002)
Eliasson, L.H.: Floquet solutions for the 1-dimensional quasiperiodic Schrödinger equation. Common. Math. Phys. 146(3), 447–482 (1992)
Eliasson, L.H.: Ergodic skew systems on T d× SO(3,R). Preprint, ETH Zürich, 1991
Eliasson, L.H.: Absolutely convergent series expansions for quasi-periodic motions. Math. Phys. Electron. J. 2 paper 4, 1–33 (1996). Preprint, 1988
Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman lectures on physics. Vol. 3: Quantum mechanics. Reading, MA: Addison-Wesley, 1965
Gallavotti, G.: Twistless KAM tori. Commun. Math. Phys. 164, 145–156 (1994)
Gallavotti, G., Gentile, G.: Hyperbolic low-dimensional invariant tori and summations of divergent series. Common. Math. Phys. 227(3), 421–460 (2002)
Gentile, G.: Diagrammatic techniques in perturbation theory, and applications. In Degasperis, A., Gaeta, G.: Proceedings of ‘‘Symmetry and Perturbation Theory II’‘ (Rome, 16–22 December 1998), (Eds). Singapore World Scientific, 1999, pp. 59–78
Gentile, G.: Pure point spectrum for two-level systems in a strong quasi-periodic filed. Preprint, 2003
Gentile, G., Mastropietro, V.: Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics. A review with some applications. Rev. Math. Phys. 8 393–444 (1996)
Krikorian, R.: Réductibilité presque partout des systèmes quasipériodiques dans le cas SO(3). C. R. Acad. Sci. Paris Sér. I Math. 321(8), 1039–1044 (1995)
Krikorian, R.: Réducibilité des systèmes produits-croisés à valeurs dans les groupes compacts. Astérisque 259, 1–216 (1999)
Krikorian, R.: Global density of reducible quasi-periodic cocycles on T 1× SU(2). Ann. of Math. 154(2), 269–326 (2001)
Nussenzveig, H. M.: Introduction to Quantum Optics. New York: Gordon and Breach, 1973
Whitney, H.: Analytic extensions of differential functions defined in closed sets. Trans. Am. Math. Soc. 36(1), 63–89 (1934)
Wreszinski, W.F., Casmeridis, S.: Models of two-level atoms in quasiperiodic external fields. J. Statist. Phys. 90(3–4), 1061–1068 (1998)
Author information
Authors and Affiliations
Additional information
Communicated by G. Gallavotti
Rights and permissions
About this article
Cite this article
Gentile, G. Quasi-Periodic Solutions for Two-Level Systems. Commun. Math. Phys. 242, 221–250 (2003). https://doi.org/10.1007/s00220-003-0943-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-003-0943-0