On the Mourre estimates for Floquet Hamiltonians

  • Tadayoshi AdachiEmail author
  • Amane Kiyose


In the spectral and scattering theory for a Schrödinger operator with a time-periodic potential \(H(t)=p^2/2+V(t,x)\), the Floquet Hamiltonian \(K=-i\partial _t+H(t)\) associated with H(t) plays an important role frequently, by virtue of the Howland–Yajima method. In this paper, we introduce a new conjugate operator for K in the standard Mourre theory, that is different from the one due to Yokoyama, in order to relax a certain smoothness condition on V.


Mourre estimates Floquet Hamiltonians Schrödinger operator with time-periodic potentials AC Stark Hamiltonians 

Mathematics Subject Classification

81U05 81Q10 



The first author is partially supported by the Grant-in-Aid for Scientific Research (C) #17K05319 from JSPS. The authors are grateful to the referees for many valuable comments and suggestions.


  1. 1.
    Adachi, T.: Scattering theory for \(N\)-body quantum systems in a time-periodic electric field. Funkcial. Ekvac. 44, 335–376 (2001)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Adachi, T.: Asymptotic completeness for \(N\)-body quantum systems with long-range interactions in a time-periodic electric field. Commun. Math. Phys. 275, 443–477 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Adachi, T.: On the Mourre estimates for three-body Floquet Hamiltonians, preprint. arXiv:1904.10190
  4. 4.
    Adachi, T., Kimura, T., Shimizu, Y.: Scattering theory for two-body quantum systems with singular potentials in a time-periodic electric field. J. Math. Phys. 51, 032103 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Amrein, W.O., Boutet de Monvel, A., Georgescu, V.: \(C_0\)-Groups, Commutator Methods and Spectral Theory of \(N\)-body Hamiltonians, Progress in Mathematics, vol. 135. Birkhäuser Verlag, Basel (1996)CrossRefzbMATHGoogle Scholar
  6. 6.
    Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics. Springer, Berlin (1987)zbMATHGoogle Scholar
  7. 7.
    Howland, J.S.: Stationary scattering theory for time-dependent Hamiltonians. Math. Ann. 207, 315–335 (1974) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Howland, J.S.: Scattering theory for Hamiltonians periodic in time. Indiana Univ. Math. J. 28, 471–494 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kitada, H., Yajima, K.: A scattering theory for time-dependent long-range potentials. Duke Math. J. 49, 341–376 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Korotyaev, E.L.: Scattering theory for a three-particle system with two-body interactions periodic in time. Theor. Math. Phys. 62, 163–170 (1985)CrossRefGoogle Scholar
  11. 11.
    Kuwabara, Y., Yajima, K.: The limiting absorption principle for Schrödinger operators with long-range time-periodic potentials. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34, 833–851 (1987)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Mourre, E.: Absence of singular continuous spectrum for certain selfadjoint operators. Commun. Math. Phys. 78, 391–408 (1981)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Møller, J.S.: Two-body short-range systems in a time-periodic electric field. Duke Math. J. 105, 135–166 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Møller, J.S., Skibsted, E.: Spectral theory of time-periodic many-body systems. Adv. Math. 188, 137–221 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Nakamura, S.: Asymptotic completeness for three-body Schrödinger equations with time-periodic potentials. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33, 379–402 (1986)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Perry, P., Sigal, I.M., Simon, B.: Spectral analysis of \(N\)-body Schrödinger operators. Ann. Math. (2) 114, 519–567 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics I. Academic Press, New York (1972) zbMATHGoogle Scholar
  18. 18.
    Sigal, I.M., Soffer, A.: Long-range many-body scattering. Asymptotic clustering for Coulomb-type potentials. Invent. Math 99, 115–143 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Simon, B.: Phase space analysis of simple scattering systems: extensions of some work of Enss. Duke Math. J. 46, 119–168 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Skibsted, E.: Spectral analysis of \(N\)-body systems coupled to a bosonic field. Rev. Math. Phys. 10, 989–1026 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tamura, H.: Principle of limiting absorption for \(N\)-body Schrödinger operators—a remark on the commutator method. Lett. Math. Phys. 17, 31–36 (1989)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Yajima, K.: Scattering theory for Schrödinger equations with potentials periodic in time. J. Math. Soc. Jpn. 29, 729–743 (1977)CrossRefzbMATHGoogle Scholar
  23. 23.
    Yajima, K.: Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys. 110, 415–426 (1987)ADSCrossRefzbMATHGoogle Scholar
  24. 24.
    Yokoyama, K.: Mourre theory for time-periodic systems. Nagoya Math. J. 149, 193–210 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Course of Mathematical Science, Department of Human Coexistence, Graduate School of Human and Environmental StudiesKyoto UniversityKyoto-shiJapan
  2. 2.Department of Mathematics, Graduate School of ScienceKobe UniversityKobe-shiJapan

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