Skip to main content
SpringerLink
Log in

Asymptotic Completeness for N-Body Quantum Systems with Long-Range Interactions in a Time-Periodic Electric Field

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We show the asymptotic completeness for N-body quantum systems with long-range interactions in a time-periodic electric field whose mean in time is non-zero, where N ≥ 2. One of the main ingredients of this paper is to give some propagation estimates for physical propagators generated by time-periodic Hamiltonians which govern the systems under consideration.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Adachi T. (1994). Long-range scattering for three-body Stark Hamiltonians. J. Math. Phys. 35: 5547–5571

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. Adachi T. (1995). Propagation estimates for N-body Stark Hamiltonians. Ann. Inst. H. Poincaré Phys. Théor. 62: 409–428

    MATH  MathSciNet  Google Scholar 

  3. Adachi T. (2001). Scattering theory for N-body quantum systems in a time-periodic electric field. Funkcial. Ekvac. 44: 335–376

    MATH  MathSciNet  Google Scholar 

  4. Adachi T. and Tamura H. (1995). Asymptotic Completeness for Long-Range Many-Particle Systems with Stark Effect. J. Math. Sci., The Univ. of Tokyo 2: 77–116

    MATH  MathSciNet  Google Scholar 

  5. Adachi T. and Tamura H. (1996). Asymptotic Completeness for Long-Range Many-Particle Systems with Stark Effect, II. Commun. Math. Phys. 174: 537–559

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. Avron J.E. and Herbst I.W. (1977). Spectral and scattering theory of Schrödinger operators related to the Stark effect. Commun. Math. Phys. 52: 239–254

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Brezis H. (1983). Analyse fonctionnelle, Théorie et applications. Masson, Paris

    MATH  Google Scholar 

  8. Cycon H., Froese R.G., Kirsch W. and Simon B. (1987). Schrödinger Operators with Application to Quantum Mechanics and Global Geometry. Springer-Verlag, Berlin-Heidelberg-New York

    MATH  Google Scholar 

  9. Dereziński J. (1993). Asymptotic completeness of long-range N-body quantum systems. Ann. of Math. 138: 427–476

    Article  MATH  MathSciNet  Google Scholar 

  10. Dereziński J. and Gérard C. (1997). Scattering theory of classical and quantum mechanical N-particle systems. Springer-Verlag, Berlin-Heidelberg-New York

    Google Scholar 

  11. Dereziński J. and Gérard C. (1997). Long-range scattering in the position representation. J. Math. Phys. 38: 3925–3942

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. Enss V. and Veselić K. (1983). Bound states and propagating states for time-dependent Hamiltonians. Ann. Inst. H. Poincaré Sect. A (N.S.) 39: 159–191

    MATH  Google Scholar 

  13. Gérard C. (1992). Sharp propagation estimates for N-particle systems. Duke Math. J. 67: 483–515

    Article  MATH  MathSciNet  Google Scholar 

  14. Gérard, C., Łaba, I.: Multiparticle Quantum Scattering in Constant Magnetic Fields. Providence. RI: Amer. Math. Soc., 2002

  15. Graf G.M. (1990). Phase space analysis of the charge transfer model. Helv. Phys. Acta 64: 107–138

    MathSciNet  Google Scholar 

  16. Graf G.M. (1990). Asymptotic completeness for N-body short-range quantum systems: a new proof. Commun. Math. Phys. 132: 73–101

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. Graf G.M. (1991). A remark on long-range Stark scattering. Helv. Phys. Acta 64: 1167–1174

    MathSciNet  Google Scholar 

  18. Hasegawa, T.: On Applications of the Avron-Herbst Formula. Master Thesis, Kobe University, 2005 (in Japanese)

  19. Helffer, B., Sjöstrand, J.: Equation de Schrödinger avec champ magnétique et équation de Harper. In: Lecture Notes in Physics 345, Berlin-Heidelberg-New York: Springer-Verlag, 1989, 118–197

  20. Herbst I., Møller J.S. and Skibsted E. (1995). Spectral analysis of N-body Stark Hamiltonians. Commun. Math. Phys. 174: 261–294

    Article  MATH  ADS  Google Scholar 

  21. Herbst I., Møller J.S. and Skibsted E. (1996). Asymptotic completeness for N-body Stark Hamiltonians. Commun. Math. Phys. 174: 509–535

    Article  MATH  ADS  Google Scholar 

  22. Howland J.S. (1974). Scattering theory for time-dependent Hamiltonians. Math. Ann. 207: 315–335

    Article  MATH  MathSciNet  Google Scholar 

  23. Howland J.S. (1979). Scattering theory for Hamiltonians periodic in time. Indiana Univ. Math. J. 28: 471–494

    Article  MATH  MathSciNet  Google Scholar 

  24. Hunziker W. (1966). On the space-time behavior of Schrödinger wavefunctions. J. Math. Phys. 7: 300–304

    Article  MATH  ADS  MathSciNet  Google Scholar 

  25. Jensen A. and Ozawa T. (1993). Existence and non-existence results for wave operators for perturbations of the Laplacian. Rev. Math. Phys. 5: 601–629

    Article  MATH  MathSciNet  Google Scholar 

  26. Jensen A. and Yajima K. (1991). On the long-range scattering for Stark Hamiltonians. J. Reine Angew. Math. 420: 179–193

    MATH  MathSciNet  Google Scholar 

  27. Kimura, T.: Long-range quantum scattering theory in a time-periodic electric field. Master Thesis, Kobe University, 2006 (in Japanese)

  28. Kitada H. and Yajima K. (1982). Scattering theory for time-dependent long-range potentials. Duke Math. J. 49: 341–376

    Article  MATH  MathSciNet  Google Scholar 

  29. Møller J.S. (2000). Two-body short-range systems in a time-periodic electric field. Duke Math. J. 105: 135–166

    Article  MATH  MathSciNet  Google Scholar 

  30. Møller J.S. and Skibsted E. (2004). Spectral theory of time-periodic many-body systems. Adv. Math. 188: 137–221

    Article  MATH  MathSciNet  Google Scholar 

  31. Mourre E. (1981). Absence of singular continuous spectrum for certain self-adjoint operators. Commun. Math. Phys. 78: 391–408

    Article  MATH  ADS  MathSciNet  Google Scholar 

  32. Nakamura S. (1986). Asymptotic completeness for three-body Schrödinger equations with time-periodic potentials. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33: 379–402

    MATH  MathSciNet  Google Scholar 

  33. Perry P.A. (1983). Scattering Theory by the Enss Method. Math. Rep. Vol. 1. Harwood Academic, Chur-London-Paris- Utrecht-New York

    Google Scholar 

  34. Radin C. and Simon B. (1978). Invariant Domains for the Time-Dependent Schrödinger Equation. J. Differ. Eqs. 29: 289–296

    Article  MATH  MathSciNet  Google Scholar 

  35. Reed, M., Simon, B.: Methods of Modern Mathematical Physics I–IV, New York: Academic Press, 1972, 1975, 1977, 1978

  36. Shimizu, Y.: Long-Range Scattering Theory for Schrödinger Operators in a Time-Periodic Electric Field. Master Thesis, Kobe University, 2006

  37. Sigal I.M. and Soffer A. (1987). The N-particle scattering problem: asymptotic completeness for short-range systems. Ann. of Math. 125: 35–108

    Article  MathSciNet  Google Scholar 

  38. Skibsted E. (1991). Propagation estimates for N-body Schroedinger Operators. Commun. Math. Phys. 142: 67–98

    Article  MATH  ADS  MathSciNet  Google Scholar 

  39. White D. (1992). Modified wave operators and Stark Hamiltonians. Duke Math. J. 68: 83–100

    Article  MATH  MathSciNet  Google Scholar 

  40. Yafaev D.R. (1980). Wave operators for the Schrödinger equation. Theoret. and Math. Phys. 45: 992–998

    Article  ADS  MathSciNet  Google Scholar 

  41. Yajima K. (1977). Schrödinger equations with potentials periodic in time. J. Math. Soc. Japan 29: 729–743

    MATH  MathSciNet  Google Scholar 

  42. Yajima K. (1987). Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys. 110: 415–426

    Article  MATH  ADS  MathSciNet  Google Scholar 

  43. Yajima K. and Kitada H. (1983). Bound states and scattering states for time periodic Hamiltonians. Ann. Inst. H. Poincaré Sect. A (N.S.) 39: 145–157

    MATH  MathSciNet  Google Scholar 

  44. Yokoyama K. (1998). Mourre theory for time-periodic systems. Nagoya Math. J. 149: 193–210

    MATH  MathSciNet  Google Scholar 

  45. Yokoyama K. (1999). Asymptotic completeness for Hamiltonians with time-dependent electric fields. Osaka J. Math. 36: 63–85

    MATH  MathSciNet  Google Scholar 

  46. Zielinski L. (1994). A proof of asymptotic completeness for N-body Schrödinger operators. Commun. PDE 19: 455–522

    Article  MATH  MathSciNet  Google Scholar 

  47. Zorbas J. (1978). Scattering theory for Stark Hamiltonians involving long-range potentials. J. Math. Phys. 19: 577–580

    Article  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science, Kobe University, 1-1, Rokkodai-cho, Nada-ku, Kobe-shi, Hyogo, 657-8501, Japan

    Tadayoshi Adachi

Authors
  1. Tadayoshi Adachi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tadayoshi Adachi.

Additional information

Communicated by B. Simon

Research partially supported by the Grant-in-Aid for Young Scientists of MEXT #17740078.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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). https://doi.org/10.1007/s00220-007-0305-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-007-0305-4

Keywords

Search

Navigation