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Periodic Gibbs states

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Abstract

Periodic Gibbs states for quantum lattice systems are investigated. We formulate the definition of the periodic Gibbs states and the measures associated with them. Theorems of existence are proved for these states. We also prove the existence of the critical temperature for the system of anharmonic quantum oscillators with pairwise interaction.

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References

  1. S. Albeverio and R. Høegh-Krohn, “Homogeneous random fields and statistical mechanics,”J. Func. Anal.,19, No. 2, 242–272 (1975).

    Google Scholar 

  2. S. A. Globa and Yu. G. Kondrat'ev, “Construction of Gibbs states of quantum lattice systems,” in:Application of the Functional Analytical Methods to the Problems of Mathematical Physics [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1987), pp. 4–16.

    Google Scholar 

  3. M. Reed and B. Simon,Methods of Modern Mathematical Physics, Vol. 2.Fourier Analysis. Self-Adjointness, Academic Press, San Diego, CA (1975).

    Google Scholar 

  4. V. S. Barbulyak and Yu. G. Kondrat'ev, “Criteria of existence of periodic Gibbs states of quantum lattice systems,” in:Methods of Functional Analysis in the Problems of Mathematical Physics [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1990), pp. 30–41.

    Google Scholar 

  5. I. J. Fröhlich, R. Israel, E. Lieb, and B. Simon, “Phase transition and reflection-positivity,”Commun. Math. Phys.,62, No. 1, 1–37 (1978).

    Google Scholar 

  6. S. B. Shlosman, “The method of reflection positivity in the mathematical theory of phase transitions of the first kind,”Usp. Mat. Nauk,41, Issue 3, 69–111 (1986).

    Google Scholar 

  7. P. Billingsley,Convergence of Probability Measures, Wiley, New York (1968).

    Google Scholar 

  8. F. J. Dyson, E. H. Lieb, and B. Simon, “Phase transitions in quantum spin systems with isotropic and nonisotropic interactions,”J. Stat. Phys.,18, No. 4, 335–383 (1978).

    Google Scholar 

  9. L. A. Pastur and B. A. Khoruzhenko, “Phase transitions in quantum models of rotators in segnetoelectrics,”Tear. Mat. Fiz.,73, No. 1, 111–124(1987).

    Google Scholar 

  10. W. Driessler, L. Landau, and J. F. Perez, “Estimates of critical lengths and critical temperatures for classical and quantum lattice systems,”J. Stat. Phys.,20, No. 2, 123–162 (1979).

    Google Scholar 

  11. B. Simon,The P(ϕ) 2 Euclidean (Quantum) Field Theory, Princeton Univ. Press, Princeton, NJ (1974).

    Google Scholar 

  12. B. Simon, “Semiclassical analysis of low lying eigenvalues, II. Tunneling,”Ann. Math.,120, No. 1, 89–118 (1984).

    Google Scholar 

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Authors and Affiliations

  1. Lvov University, Lvov

    V. S. Barbulyak

  2. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev

    Yu. G. Kondrat'ev

Authors
  1. V. S. Barbulyak
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  2. Yu. G. Kondrat'ev
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 4, pp. 451–458, April, 1993.

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Barbulyak, V.S., Kondrat'ev, Y.G. Periodic Gibbs states. Ukr Math J 45, 481–489 (1993). https://doi.org/10.1007/BF01062943

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  • DOI: https://doi.org/10.1007/BF01062943

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