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Journal of Statistical Physics

, Volume 153, Issue 1, pp 70–92 | Cite as

Monotonicity of the Polaron Energy II: General Theory of Operator Monotonicity

  • Tadahiro Miyao
Article

Abstract

We construct a general theory of operator monotonicity and apply it to the Fröhlich polaron hamiltonian. This general theory provides a consistent viewpoint of the Fröhlich model.

Keywords

Operator inequality Positivity preserving semigroup Polaron Self-dual cone 

Notes

Acknowledgements

I would like to thank H. Spohn for useful discussions. Financial support by KAKENHI (20554421) is gratefully acknowledged.

References

  1. 1.
    Anapolitanos, I., Landon, B.: The ground state energy of the multi-polaron in the strong coupling limit. arXiv:1212.3571
  2. 2.
    Bach, V., Fröhlich, J., Sigal, I.M.: Quantum electrodynamics of confined nonrelativistic particles. Adv. Math. 137(2), 299–395 (1998) MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Benguria, R.D., Bley, G.A.: Exact asymptotic behavior of the Pekar-Tomasevich functional. J. Math. Phys. 52, 052110 (2011) MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Bös, W.: Direct integrals of selfdual cones and standard forms of von Neumann algebras. Invent. Math. 37, 241–251 (1976) MathSciNetADSCrossRefMATHGoogle Scholar
  5. 5.
    Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics. 2. Equilibrium States. Models in Quantum Statistical Mechanics, 2nd edn. Texts and Monographs in Physics. Springer, Berlin (1997) MATHGoogle Scholar
  6. 6.
    Devreese, J., Alexandrov, S.: Fröhlich polaron and bipolaron: recent developments. Rep. Prog. Phys. 72, 066501 (2009) ADSCrossRefGoogle Scholar
  7. 7.
    Donsker, M., Varadhan, S.R.S.: Asymptotics for the polaron. Commun. Pure Appl. Math. 36, 505–528 (1983) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dybalski, W., Moller, J.S.: The translation invariant massive Nelson model: III. Asymptotic completeness below the two-boson threshold. arXiv:1210.6645
  9. 9.
    Eckmann, J.P.: A model with persistent vacuum. Commun. Math. Phys. 18, 247–264 (1970) MathSciNetADSCrossRefMATHGoogle Scholar
  10. 10.
    Faris, W.G.: Invariant cones and uniqueness of the ground state for fermion systems. J. Math. Phys. 13, 1285–1290 (1972) MathSciNetADSCrossRefMATHGoogle Scholar
  11. 11.
    Feynman, R.P.: Slow electrons in a polar crystal. Phys. Rev. 97, 660–665 (1955) ADSCrossRefMATHGoogle Scholar
  12. 12.
    Feynman, R.P.: Statistical Mechanics: A Set of Lectures. Advanced Book Classics. Westview Press (1998) MATHGoogle Scholar
  13. 13.
    Frank, R.L., Lieb, E.H., Seiringer, R., Thomas, L.E.: Stability and absence of binding for multi-polaron systems. Publ. Math. IHES 113, 39–67 (2011) MathSciNetMATHGoogle Scholar
  14. 14.
    Frank, R.L., Lieb, E.H., Seiringer, R.: Binding of Polarons and Atoms at Threshold. Commun. Math. Phys. 313, 405–424 (2012) MathSciNetADSCrossRefMATHGoogle Scholar
  15. 15.
    Fröhlich, H.: Electrons in lattice fields. Adv. Phys. 3, 325 (1954) ADSCrossRefGoogle Scholar
  16. 16.
    Fröhlich, J.: On the infrared problem in a model of scalar electrons and massless, scalar bosons. Ann. Inst. H. Poincaré Sect. A (NS) 19, 1–103 (1973) Google Scholar
  17. 17.
    Fröhlich, J.: Existence of dressed one electron states in a class of persistent models. Fortschr. Phys. 22, 150–198 (1974) CrossRefGoogle Scholar
  18. 18.
    Gerlach, B., Löwen, H.: Analytical properties of polaron systems or: do polaronic phase transitions exist or not? Rev. Mod. Phys. 63, 63–90 (1991) ADSCrossRefGoogle Scholar
  19. 19.
    Glimm, J., Jaffe, A.: Singular perturbations of selfadjoint operators. Commun. Pure Appl. Math. 22, 401–414 (1969) MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Griesemer, M., Hantsch, F., Wellig, D.: On the Magnetic Pekar Functional and the Existence of Bipolarons. arXiv:1111.1624
  21. 21.
    Griesemer, M., Møller, J.S.: Bounds on the minimal energy of translation invariant N-polaron systems. Commun. Math. Phys. 297, 283–297 (2010) ADSCrossRefMATHGoogle Scholar
  22. 22.
    Gross, L.: A relativistic polaron without cutoffs. Commun. Math. Phys. 31, 25–73 (1973) ADSCrossRefMATHGoogle Scholar
  23. 23.
    Haagerup, U.: The standard form of von Neumann algebras. Math. Scand. 37, 271–283 (1975) MathSciNetGoogle Scholar
  24. 24.
    Kishimoto, A., Robinson, D.W.: Subordinate semigroups and order properties. J. Aust. Math. Soc. Ser. A 31, 59–76 (1981) MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Kishimoto, A., Robinson, D.W.: Positivity and monotonicity properties of C 0-semigroups. I. Commun. Math. Phys. 75, 67–84 (1980). Comm. Math. Phys., 75, 1980, 85–101 MathSciNetADSCrossRefMATHGoogle Scholar
  26. 26.
    Lee, T.D., Low, F., Pines, D.: The motion of slow electrons in a polar crystal. Phys. Rev. 90, 297–302 (1953) MathSciNetADSCrossRefMATHGoogle Scholar
  27. 27.
    Lewin, M., Rougerie, N.: On the binding of small polarons in a mean-field quantum crystal. arXiv:1202.5103
  28. 28.
    Lieb, E.H., Thomas, L.E.: Exact ground state energy of the strong-coupling polaron. Commun. Math. Phys. 183, 511–519 (1997). Commun. Math. Phys., 188, 1997, 499–500 MathSciNetADSCrossRefMATHGoogle Scholar
  29. 29.
    Lieb, E.H., Yamazaki, K.: Ground-State Energy and Effective Mass of the Polaron. Phys. Rev. 111, 728–733 (1958) ADSCrossRefMATHGoogle Scholar
  30. 30.
    Miura, Y.: On order of operators preserving selfdual cones in standard forms. Far East J. Math. Sci.: FJMS 8, 1–9 (2003) MathSciNetMATHGoogle Scholar
  31. 31.
    Miyao, T., Spohn, H.: The bipolaron in the strong coupling limit. Ann. Henri Poincaré 8, 1333–1370 (2007) MathSciNetADSCrossRefMATHGoogle Scholar
  32. 32.
    Miyao, T.: Nondegeneracy of ground states in nonrelativistic quantum field theory. J. Oper. Theory 64, 207–241 (2010) MathSciNetADSMATHGoogle Scholar
  33. 33.
    Miyao, T.: Self-dual cone analysis in condensed matter physics. Rev. Math. Phys. 23, 749–822 (2011) MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Miyao, T.: Monotonicity of the polaron energy. arXiv:1211.0344
  35. 35.
    Miyao, T.: Note on the one-dimensional Holstein-Hubbard model. J. Stat. Phys. 147, 436–447 (2012) MathSciNetADSCrossRefMATHGoogle Scholar
  36. 36.
    Miyao, T.: Ground state properties of the SSH model. J. Stat. Phys. 149, 519–550 (2012) MathSciNetADSCrossRefMATHGoogle Scholar
  37. 37.
    Møller, J.S.: The polaron revisited. Rev. Math. Phys. 18, 485–517 (2006) MathSciNetCrossRefGoogle Scholar
  38. 38.
    Nelson, E.: Interaction of nonrelativistic particles with a quantized scalar field. J. Math. Phys. 5, 1190–1197 (1964) ADSCrossRefGoogle Scholar
  39. 39.
    Pizzo, A.: One-particle (improper) states in Nelson’s massless model. Ann. Henri Poincaré 4, 439–486 (2003) MathSciNetADSCrossRefMATHGoogle Scholar
  40. 40.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. II. Academic Press, New York (1975) Google Scholar
  41. 41.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. IV. Academic Press, New York (1978) Google Scholar
  42. 42.
    Sloan, A.D.: The polaron without cutoffs in two space dimensions. J. Math. Phys. 15, 190 (1974) MathSciNetADSCrossRefGoogle Scholar
  43. 43.
    Spohn, H.: Effective mass of the polaron: a functional integral approach. Ann. Phys. 175, 278–318 (1987) MathSciNetADSCrossRefGoogle Scholar
  44. 44.
    Spohn, H.: The polaron at large total momentum. J. Phys. A 21(5), 1199–1211 (1988) MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsHokkaido UniversitySapporoJapan

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