Exactly Solvable Models

  • D. Ya. Petrina
Part of the Mathematical Physics Studies book series (MPST, volume 17)


In this section we describe four models of quantum statistical mechanics, namely, the BCS (Bardeen -Cooper-Schrieffer) model of superconductivity, the Bogolyubov model of superfluidity, the model of Huang, Yang, and Luttinger, and the Peierls-Frohlich model. We employ the following scheme: First, we define model Hamiltonians for systems of particles contained in a bounded region (cube) ∧ with periodic boundary conditions and then pass to the thermodynamic limit and study the corresponding limiting model Hamiltonian. The procedure of limit transition is not justified; it enables one to determine the model Hamiltonian of an infinite system which is then studied rigorously.


Thermodynamic Limit Difference Variable Selfadjoint Operator Bose Condensation Canonical Commutation Relation 
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  1. [1]
    Angelescu, N. A., Verbeure, A., and Zagrebnov, V. A. On Bogoliubov’s model of superfluidity, J. Phys. A, Math. Gen. (1992), 25, 3473–3491.MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. [1]
    Araki, H. and Woods, E. J. Representations of the canonical commutation relation. Describing a nonrelativistic infinite free Bose gas, J. Math. Phys. (1963), 4, 637–662.MathSciNetADSCrossRefGoogle Scholar
  3. [1]
    Araki, H. and Wyss, W. Representations of canonical anticommutation relations, Helv. Phys. Acta. (1964), 37, 136–159.MathSciNetzbMATHGoogle Scholar
  4. [1]
    Bardeen, J., Cooper, L. N., and Schrieffer, J. R. Microscopic theory of superconductivity, Phys. Rev. (1957), 106, 162– 164MathSciNetADSCrossRefGoogle Scholar
  5. [1a]
    Bardeen, J., Cooper, L. N., and Schrieffer, J. R.Theory of superconductivity, Phys. Rev. (1957), 108, 1175 –1204.MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. [1]
    Belokolos, E. D. and Petrina, D. Ya. On a relationship between the methods of approximating Hamiltonian and finite zone integration, Teor. Mat. Fiz. (1984), 58, No. 1,61–71.MathSciNetCrossRefGoogle Scholar
  7. [1]
    Billard, P. and Fano, G. An existence proof for the gap equation in the superconductivity theory, Comm. Math. Phys. (1968), 10, 274–219.zbMATHGoogle Scholar
  8. [1]
    Bogolyubov, N. N. To the theory of superfluidity, Izv. Akad. Nauk SSSR., Ser. Fiz. (1947), 11, No. 1, 77–90.MathSciNetGoogle Scholar
  9. [1a]
    Bogolyubov, N. N. Selected Papers [in Russian], Vol. 2, Naukova Dumka, Kiev (1970), pp. 210–224.Google Scholar
  10. [2]
    Bogolyubov, N. N. On the Model Hamiltonian in the Theory of Superconductivity [in Russian], Preprint JINR, No. R-511, Dubna, 1960.Google Scholar
  11. [2a]
    Bogolyubov, N. N. Selected Papers [in Russian], Vol. 3, Naukova Dumka, Kiev (1970), pp. 110–173.Google Scholar
  12. [3]
    Bogolyubov, N. N. Selected Papers [in Russian], Vol. 3, Naukova Dumka, Kiev (1970), pp. 174–243.Google Scholar
  13. [3a]
    Bogolyubov, N. N. Quasiaverages in the Problems of Statistical Mechanics [in Russian], Preprint JINR, No. R-1451, Dubna, 1963.Google Scholar
  14. [1]
    Bogolyubov, N. N., Zubarev, D. N., and Tserkovnikov, Yu. A. An asymptotically exact solution for the model Hamiltonian in the theory of superconductivity, Zh. Eksper. Teoret. Fiz. (1960), 39, issue 1, 120–129.Google Scholar
  15. [1]
    Bogolyubov, N. N. (jr.) A Method for Investigating Model Hamiltonians [in Russian], Nauka, Moscow, 1974.Google Scholar
  16. [1]
    Bogolyubov, N. N. (jr.), Brankov, J. G., Zagrebnov, V. A., Kurbatov, A. M., and Tonchev, N. S. Method of Approximating Hamiltonian in Statistical Physics [in Russian], Izd. Bolgar. Akad. Nauk, Sofia, 1981.Google Scholar
  17. [1]
    Bratelli, O. and Robinson, D. Operator Algebras and Quantum Statistical Mechanics, Springer, New York-Heidelberg-Berlin, 1979.Google Scholar
  18. [1]
    Dell’Antonio, G. F. Structure of the algebras of some free systems, Comm. Math. Phys. (1968), 9, 81–116.MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. [1]
    Emch, G. Algebraic Methods in Statistical Mechanics and Quantum Field Theory [Russian translation], Mir, Moscow, 1976.Google Scholar
  20. [1]
    Fannes, M., Spohn H., and Verbeure A. Equilibrium states for mean field models, J. Math. Phys. (1980), 21, 355 –358.MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. [1]
    Haag, R. The mathematical Structure of the Bardeen-Cooper-Schrieffer Model, Nuovo Cimento(1962),25,287.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [1]
    Hepp, K. and Lieb, H. Equilibrium statistical mechanics of matter interacting with the quantized radiation field, Phys. Rev. A. (1973), 8, 2517.MathSciNetADSCrossRefGoogle Scholar
  23. [2]
    Hepp, K. and Lieb, H. On the superradiant phase transition for molecules in a quantized radiation field: The Dicke maser model, Ann. Phys. (1973), 76, 360.MathSciNetADSCrossRefGoogle Scholar
  24. [1]
    Huang, K., Yang, C. N., and Luttinger, J. M. Imperfect Bose gas with hard-sphere interaction, Phys. Rev. (1957), 105, 776.MathSciNetADSzbMATHCrossRefGoogle Scholar
  25. [1]
    Hunziker, W. On the spectra of Schrödinger multiparticle Hamiltonians, Helv. Phys. Acta. (1966), 39, 451–462.MathSciNetzbMATHGoogle Scholar
  26. [1]
    Ginibre, J. On the asymptotic exactness of the Bogoliubov approximation for many boson systems, Comm. Math. Phys. (1968), 8, 26 –51.MathSciNetADSzbMATHCrossRefGoogle Scholar
  27. [1]
    GoderisD., VerbeureA., and Vets P. Relaxation of the Dicke maser model, J. Math. Phys. (1987), 28, 2250–2253.MathSciNetADSCrossRefGoogle Scholar
  28. [1]
    Gredzhuk, V. I. Justification of the method of approximating Hamiltonian in Bogolyubov’s theory of superfluidity, Dokl. Ukr. Akad. Nauk (1992), No. 1, 58–62.Google Scholar
  29. [1]
    Petrina, D. Ya. On Hamiltonians in quantum statistics and a model Hamiltonian in the theory of superconductivity, Teor. Mat. Fiz. (1970), 4, 394.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [2]
    Petrina, D. Ya. Exactly Solvable Models of Quantum Statistical Mechanics, Preprint 18/1992,Politecnico di Torino, Torino, 1992.Google Scholar
  31. [1]
    Petrina, D. Ya. and Yatsyshin, V. P. On a model Hamiltonian in the theory of superconductivity, Teor. Mat. Fiz. (1972), 10, 283.CrossRefGoogle Scholar
  32. [1]
    Reed, M. and Simon, B. Methods of Modern Mathematical Physics, Vols. 1–4, Academic Press, New York-London, 1972, 1975, 1978, 1979.zbMATHGoogle Scholar
  33. [1]
    Van den Berg, M., Lewis, J. T., and Pule, J. V. The Large Deviation Principle and Some Models of an Interacting Boson Gas, Comm. Math. Phys. (1988), 118, 61–85.MathSciNetADSzbMATHCrossRefGoogle Scholar
  34. [1]
    Van Winter, C. Theory of finite systems of particles. I, Mat.-Fys. Skr. Danske Vid. Selsk. (1964), 1,1–60.Google Scholar
  35. [1]
    Zagrebnov, V. A., Brankov, J. G., and Tonchev N. S. A rigorous result for the systems interacting with boson field, Dokl. Akad. Nauk SSSR (1975), 225, 71–73.MathSciNetGoogle Scholar
  36. [1]
    Zhislin, S. M. Investigation of the spectrum of the Schrödinger operator for a many-particle system, Trudy Mosk. Mat. Obshch. (1960), 9, 61 –128.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • D. Ya. Petrina
    • 1
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKievUkraine

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