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Transactions of the Moscow Mathematical Society

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Existence and a priori estimates for Euclidean Gibbs states


Authors: S. Albeverio, Yu. Kondratiev, T. Pasurek and M. Röckner
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 67 (2006).
Journal: Trans. Moscow Math. Soc. 2006, 1-85
MSC (2000): Primary 82B10; Secondary 46G12, 60H30
DOI: https://doi.org/10.1090/S0077-1554-07-00158-6
Published electronically: January 8, 2007
MathSciNet review: 2301591
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Abstract: We prove a priori estimates and, as a sequel, the existence of Euclidean Gibbs states for quantum lattice systems. For this purpose we develop a new analytical approach, the main tools of which are: first, a characterization of the Gibbs states in terms of their Radon-Nikodým derivatives under shift transformations as well as in terms of their logarithmic derivatives through integration by parts formulae, and second, the choice of appropriate Lyapunov functionals describing stabilization effects in the system. The latter technique becomes applicable since on the basis of the integration by parts formulae the Gibbs states are characterized as solutions of an infinite system of partial differential equations. Our existence results generalize essentially all previous ones. In particular, superquadratic growth of the interaction potentials is allowed and $ N$-particle interactions for $ N\in\mathbb{N} \cup\{\infty\}$ are included. We also develop abstract frames both for the necessary single spin space analysis and for the lattice analysis apart from their applications to our concrete models. Both types of general results obtained in these two frames should also be of their own interest in infinite-dimensional analysis.


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Additional Information

S. Albeverio
Affiliation: Institut für Angewandte Mathematik, Universität Bonn, D-53155 Bonn, Germany; BiBoS Research Centre, Bielefeld, Germany; and CERFIM, Locarno, Switzerland
Email: albeverio@uni-bonn.de

Yu. Kondratiev
Affiliation: Fakultät für Mathematik and BiBoS Research Centre, Bielefeld Universität, D-33615 Bielefeld, Germany; and Institute of Mathematics, NASU, Kiev, Ukraine
Email: kondrat@mathematik.uni-bielefeld.de

T. Pasurek
Affiliation: BiBoS Research Centre, Bielefeld Universität, D-33615 Bielefeld, Germany
Email: pasurek@physik.uni-bielefeld.de

M. Röckner
Affiliation: Fakultät für Mathematik and BiBoS Research Centre, Bielefeld Universität, D-33615 Bielefeld, Germany
Email: roeckner@mathematik.uni-bielefeld.de

DOI: https://doi.org/10.1090/S0077-1554-07-00158-6
Keywords: Quantum lattice systems; Euclidean Gibbs states; smooth measures on infinite-dimensional spaces and their logarithmic derivatives; integration by parts formulae; Lyapunov functionals
Published electronically: January 8, 2007
Article copyright: © Copyright 2007 American Mathematical Society

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