For a Gibbs system of one-dimensional quantum oscillators on a d-dimensional hypercubic lattice interacting via superstable pair and many-particle potentials of finite range, we prove the existence of a solution of the (lattice) Kirkwood–Salsburg equation for correlation functions depending on the Wiener paths. Some many-particle potentials may be nonpositive.
Similar content being viewed by others
References
W. I. Skrypnyk, “On the polymeric decompositions for equilibrium systems of oscillators with ternary interaction,” Ukr. Mat. Zh., 53, No. 11, 1532–1544 (2001).
S. Albeverio, Yu. G. Kondratiev, R. A. Minlos, and O. L. Rebenko, Small Mass Behaviour of Quantum Gibbs States for Lattice Models with Unbounded Spins, Preprint UMa-CCM 22/97, Uni. da Madeira (1997).
R. A. Minlos, A. Verbeure, and V. A. Zagrebnov, A Quantum Crystal Model in the Light-Mass Limit: Gibbs States, Preprint KULTP-97/16 (1997).
Y. M. Park and H. J. Yoo, “Uniqueness and clustering properties of Gibbs states for classical and quantum unbounded spin systems,” J. Stat. Phys., 80, No. 1/2, 223–271 (1995).
D. Ruelle, Statistical Mechanics. Rigorous Results, W. A. Benjamin, New York (1969).
G. E. Shilov and B. A. Gurevich, Integral, Measure, and Derivative [in Russian], Nauka, Moscow (1967).
M. Reed and B. Simon, Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-Adjointness, Academic Press, New York (1975).
H-H. Kuo, Gaussian Measures in Banach Spaces, Springer, Berlin (1975).
W. I. Skrypnik, “Long-range order in Gibbs lattice classical linear oscillator systems,” Ukr. Mat. Zh., 58, No. 3, 388–405 (2006).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 5, pp. 689–700, May, 2009.
Rights and permissions
About this article
Cite this article
Skrypnyk, W.I. Kirkwood–Salsburg equation for a quantum lattice system of oscillators with many-particle interaction potentials. Ukr Math J 61, 821–833 (2009). https://doi.org/10.1007/s11253-009-0239-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-009-0239-4