Advertisement

Mathematical Theory of the Wigner-Weisskopf Atom

  • V. Jakšić
  • E. Kritchevski
  • C.-A. Pillet
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 695)

Abstract

In these lectures we shall study an “atom”, S, described by finitely many energy levels, coupled to a “radiation field”, R, described by another set (typically continuum) of energy levels. More precisely, assume that S and R are described, respectively, by the Hilbert spaces hS, hR and the Hamiltonians h S , h R . Let h = hS ⊕ hR and h 0 = h S h R . If ν is a self-adjoint operator on h describing the coupling between S and R, then the Hamiltonian we shall study is h λh 0 + λν, where λ ∈ ℝ is a coupling constant.

Keywords

Point Spectrum Borel Probability Measure Open Quantum System Weak Coupling Limit Pure Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aguilar, J., Combes, J.M.: A class of analytic perturbations for one-body Schrödinger Hamiltonians. Commun. Math. Phys. 22, 269 (1971). Mathematical Theory of the Wigner-Weisskopf Atom 213zbMATHMathSciNetCrossRefADSGoogle Scholar
  2. 2.
    Aschbacher, W., Jakšić, V., Pautrat, Y., Pillet, C.-A.: Topics in quantum statistical mechanics. In Open Quantum Systems III. Lecture Notes of the Summer School on Open Quantum Systems held in Grenoble, June 16–July 4, 2003. To be published in Lecture Notes in Mathematics, Springer, New York.Google Scholar
  3. 3.
    Aschbacher, W., Jakšić, V., Pautrat, Y., Pillet, C.-A.: Transport properties of ideal Fermi gases. In preparation.Google Scholar
  4. 4.
    Aizenstadt, V.V., Malyshev, V.A.: Spin interaction with an ideal Fermi gas. J. Stat. Phys. 48, 51 (1987).MathSciNetCrossRefADSGoogle Scholar
  5. 5.
    Arai, A.: On a model of a harmonic oscillator coupled to a quantized, massless, scalar field. J. Math. Phys. 21, 2539 (1981).MathSciNetCrossRefADSGoogle Scholar
  6. 6.
    Balslev, E., Combes, J.M.: Spectral properties of many-body Schrödinger operators with dilation analytic interactions. Commun. Math. Phys. 22, 280 (1971).zbMATHMathSciNetCrossRefADSGoogle Scholar
  7. 7.
    Braun, E.: Irreversible behavior of a quantum harmonic oscillator coupled to a heat bath. Physica A 129, 262 (1985).MathSciNetCrossRefADSGoogle Scholar
  8. 8.
    Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics 1. Springer, Berlin (1987).zbMATHGoogle Scholar
  9. 9.
    Bratteli, O., Robinson, D. W.: Operator Algebras and Quantum Statistical Mechanics 2. Springer, Berlin (1996).zbMATHGoogle Scholar
  10. 10.
    Baez, J.C., Segal, I.E., Zhou, Z.: Introduction to algebraic and constructive quantum field theory. Princeton University Press, Princeton NJ (1991).zbMATHGoogle Scholar
  11. 11.
    Cycon, H., Froese, R., Kirsch,W., Simon, B.: Schrödinger Operators. Springer, Berlin (1987).zbMATHGoogle Scholar
  12. 12.
    Carmona, R., Lacroix, J.: Spectral Theory of Random Schrödinger Operators Birkhauser, Boston (1990).zbMATHGoogle Scholar
  13. 13.
    Cohen-Tannoudji, C., Dupont-Roc, J., Grynberg, G.: Atom-Photon Interactions. John Wiley, New York (1992).Google Scholar
  14. 14.
    Davies, E.B.: The harmonic oscillator in a heat bath. Commun. Math. Phys. 33, 171 (1973).CrossRefADSGoogle Scholar
  15. 15.
    Davies, E.B.: Markovian master equations. Commun. Math. Phys. 39, 91 (1974).zbMATHCrossRefADSGoogle Scholar
  16. 16.
    Davies, E.B.: Markovian master equations II. Math. Ann. 219, 147 (1976).zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Davies, E.B.: Dynamics of a multilevel Wigner-Weisskopf atom. J. Math. Phys. 15, 2036 (1974).CrossRefADSGoogle Scholar
  18. 18.
    De Bivre, S.: Local states of free Bose fields. Lect. Notes Phys. 695, 17-63 (2006).Google Scholar
  19. 19.
    Dereziński, J.: Introduction to representations of canonical commutation and anticommutation relation. Lect. Notes Phys. 695. 65–145 (2006).CrossRefGoogle Scholar
  20. 20.
    Dereziński, J.: Fermi golden rule, Feshbach method and embedded point spectrum. Séminaire Equations aux Dérivées Partielles, 1998–1999, Exp. No. XXIII, 13 pp., Sémin. Equation Dériv. Partielles, Ecole Polytech., Palaiseau, (1999).Google Scholar
  21. 21.
    Dereziński, J.: Fermi golden rule and open quantum systems. In Open Quantum Systems III. Lecture Notes of the Summer School on Open Quantum Systems held in Grenoble, June 16–July 4, 2003. To be published in Lecture Notes in Mathematics, Springer, New York.Google Scholar
  22. 22.
    del Rio, R., Fuentes S., Poltoratskii, A.G.: Coexistence of spectra in rank-one perturbation problems. Bol. Soc. Mat. Mexicana 8, 49 (2002).zbMATHMathSciNetGoogle Scholar
  23. 23.
    Dirac, P.A.M.: The quantum theory of the emission and absorption of radiation. Proc. Roy. Soc. London, Ser. A 114, 243 (1927).zbMATHADSCrossRefGoogle Scholar
  24. 24.
    Davidson, R., Kozak, J.J.: On the relaxation to quantum-statistical equilibrium of the Wigner-Weisskopf Atom in a one-dimensional radiation field: a study of spontaneous emission. J. Math. Phys. 11, 189 (1970).CrossRefADSGoogle Scholar
  25. 25.
    del Rio, R., Makarov, N., Simon, B.: Operators with singular continuous spectrum: II. Rank one operators. Commun. Math. Phys. 165, 59 (1994).zbMATHCrossRefADSGoogle Scholar
  26. 26.
    del Rio, R., Simon, B.: Point spectrum and mixed spectral types for rank one perturbations. Proc. Amer. Math. Soc. 125, 3593 (1997).zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Fermi, E.: Nuclear Physics. Notes compiled by Orear J., Rosenfeld A.H. and Schluter R.A. The University of Chicago Press, Chicago, (1950).Google Scholar
  28. 28.
    Friedrichs, K.O.: Perturbation of Spectra in Hilbert Space. AMS, Providence (1965).zbMATHGoogle Scholar
  29. 29.
    Frigerio, A., Gorini, V., Pulé, J.V.: Open quasi free systems. J. Stat. Phys. 22, 409 (1980).CrossRefADSGoogle Scholar
  30. 30.
    Ford, G.W., Kac, M., Mazur, P.: Statistical mechanics of assemblies of coupled oscillators. J. Math. Phys. 6, 504 (1965).zbMATHMathSciNetCrossRefADSGoogle Scholar
  31. 31.
    Gordon, A.: Pure point spectrum under 1-parameter perturbations and instability of Anderson localization. Commun. Math. Phys. 164, 489 (1994).zbMATHCrossRefADSGoogle Scholar
  32. 32.
    Haag, R.: Local Quantum Physics. Springer-Verlag, New York (1993).zbMATHGoogle Scholar
  33. 33.
    Haake, F.: Statistical treatment of open systems by generalized master equation. Springer Tracts in Modern Physics 66, Springer, Berlin (1973).Google Scholar
  34. 34.
    Heitler, W.: The Quantum Theory of Radiation. Oxford, Oxford University Press (1954).zbMATHGoogle Scholar
  35. 35.
    Herbst, I.: Exponential decay in the Stark effect. Commun. Math. Phys. 75, 197 (1980).zbMATHMathSciNetCrossRefADSGoogle Scholar
  36. 36.
    Howland, J.S.: Perturbation theory of dense point spectra. J. Func. Anal. 74, 52 (1987).zbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Jakšić, V.: Topics in spectral theory. In Open Quantum Systems I. Lecture Notes of the Summer School on Open Quantum Systems held in Grenoble, June 16-July 4, 2003. To be published in Lecture Notes in Mathematics, Springer, New York.Google Scholar
  38. 38.
    Jakšić, V., Last, Y.: A new proof of Poltoratskii's theorem. J. Func. Anal. 215, 103 (2004).CrossRefzbMATHGoogle Scholar
  39. 39.
    Jakšić, V., Pillet, C.-A.: On a model for quantum friction II: Fermi's golden rule and dynamics at positive temperature. Commun. Math. Phys. 176, 619 (1996).CrossRefADSzbMATHGoogle Scholar
  40. 40.
    Jakšić, V., Pillet, C.-A.: On a model for quantum friction III: Ergodic properties of the spin-boson system, Commun. Math. Phys. 178, 627 (1996).CrossRefADSzbMATHGoogle Scholar
  41. 41.
    Jakšić, V., Pillet, C.-A.: Statistical mechanics of the FC oscillator. In preparation.Google Scholar
  42. 42.
    Kato, T.: Perturbation Theory for Linear Operators. Second edition. Springer, Berlin (1976).zbMATHGoogle Scholar
  43. 43.
    Katznelson, A.: An Introduction to Harmonic Analysis. Dover, New York, (1976).Google Scholar
  44. 44.
    Koosis, P.: Introduction to Hp Spaces. Second edition. Cambridge University Press, New York (1998).zbMATHGoogle Scholar
  45. 45.
    Kritchevski, E.: Ph.D. Thesis, McGill University. In preparation.Google Scholar
  46. 46.
    Maassen, H.: Quantum stochastic calculus with integral kernels. II. The Wigner-Weisskopf atom. Proceedings of the 1stWorld Congress of the Bernoulli Society, Vol. 1(Tashkent, 1986), 491–494, VNU Sci. Press, Utrecht (1987). Mathematical Theory of the Wigner-Weisskopf Atom 215Google Scholar
  47. 47.
    Kindenberg, K., West, B.J.: Statistical properties of quantum systems: the linear oscillator. Phys. Rev. A 30, 568 (1984).MathSciNetCrossRefADSGoogle Scholar
  48. 48.
    Mehta, M.L.: Random Matrices. Second edition. Academic Press, New York (1991).zbMATHGoogle Scholar
  49. 49.
    Messiah, A.: Quantum Mechanics. Volume II. John Wiley & Sons, New York.Google Scholar
  50. 50.
    Okamoto, T., Yajima, K.: Complex scaling technique in non-relativistic qed, Ann. Inst. H. Poincare 42, 311 (1985).zbMATHMathSciNetGoogle Scholar
  51. 51.
    Pillet, C.-A.: Quantum dynamical systems. In Open Quantum Systems I. Lecture Notes of the Summer School on Open Quantum Systems held in Grenoble, June 16-July 4, 2003. To be published in Lecture Notes in Mathematics, Springer, New York.Google Scholar
  52. 52.
    Poltoratskii, A.G.: The boundary behavior of pseudocontinuable functions. St. Petersburg Math. J. 5, 389 (1994).MathSciNetGoogle Scholar
  53. 53.
    Paloviita, A., Suominen, K-A., Stenholm, S.: Weisskopf-Wigner model for wave packet excitation. J. Phys. B 30, 2623 (1997).CrossRefADSGoogle Scholar
  54. 54.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, I. Functional Analysis. Second edition. Academic Press, London (1980).zbMATHGoogle Scholar
  55. 55.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, II. Fourier Analysis, Self-Adjointness. Academic Press, London (1975).zbMATHGoogle Scholar
  56. 56.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, III. Scattering Theory. Academic Press, London (1978).zbMATHGoogle Scholar
  57. 57.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, IV. Analysis of Operators. Academic Press, London, (1978).zbMATHGoogle Scholar
  58. 58.
    Rudin, W.: Real and Complex Analysis. 3rd edition. McGraw-Hill (1987).Google Scholar
  59. 59.
    Simon, B.: Spectral analysis of rank one perturbations and applications. CRM Lecture Notes Vol. 8, pp. 109–149, AMS, Providence, RI (1995).Google Scholar
  60. 60.
    Simon, B.: Resonances in N-body quantum systems with dilation analytic potential and foundations of time-dependent perturbation theory. Ann. Math. 97, 247 (1973).zbMATHCrossRefGoogle Scholar
  61. 61.
    Simon, B., Wolff, T.: Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians. Comm. Pure Appl. Math. 39, 75 (1986).zbMATHMathSciNetCrossRefGoogle Scholar
  62. 62.
    Van Hove, L.: Master equation and approach to equilibrium for quantum systems. In Fundamental problems in statistical mechanics, compiled by E.G.D. Cohen, North-Holland, Amsterdam (1962).Google Scholar
  63. 63.
    Weder, R.A.: On the Lee model with dilatation analytic cutoff. function. J. Math. Phys. 15, 20 (1974).MathSciNetCrossRefADSGoogle Scholar
  64. 64.
    Weisskopf, V., Wigner, E.: Berechnung der natürlichen Linienbreite auf Grund der Diracschen Lichttheorie. Zeitschrift für Physik 63, 54 (1930).zbMATHCrossRefADSGoogle Scholar
  65. 65.
    De Groot, S.R., Mazur, P.: Non-Equilibrium Thermodynamics. North-Holland, Amsterdam (1969).Google Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • V. Jakšić
    • 1
  • E. Kritchevski
    • 1
  • C.-A. Pillet
    • 2
  1. 1.Department of Mathematics and StatisticsMcGill UniversityCanada
  2. 2.CPT-CNRS, UMR 6207Université de ToulonLa Garde CedexFrance

Personalised recommendations