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Dirichlet Forms and Dirichlet Operators for Gibbs Measures of Quantum Unbounded Spin Systems: Essential Self-Adjointness and Log-Sobolev Inequality

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Abstract

For each γ ∈ [0, 1] we consider the Dirichlet form ℰ\(_\mu ^\gamma\) and the associated Dirichlet operator \(H_\mu ^\gamma\) for the Gibbs measure μ of quantum unbounded spin systems interacting via superstable and regular potential. The Gibbs measure μ is related to the Gibbs state of the system via a (functional) Euclidean integral procedure. The configuration space for the spin systems is given by \(\Omega : = E^{\mathbb{Z}^v } ,\) \(E: = \left\{ {\omega \in C\left( {\left[ {0,1} \right];\mathbb{R}^d } \right):\omega \left( 0 \right) = \omega \left( 1 \right)} \right\}.\) We formulate Dirichlet forms in the framework of rigged Hilbert spaces which are related to the space Ω. Under appropriate conditions on the potential, we show that the Dirichlet operator \(H_\mu ^{\left( \gamma \right)}\) is essentially self-adjoint on the domain of smooth cylinder functions. We give sufficient conditions on the potential so that the corresponding Gibbs measure is uniformly log-concave (ULC). This property gives the spectral gap of the Dirichlet operator \(H_\mu ^{\left( \gamma \right)}\) at the lower end of the spectrum. Furthermore, we prove that under the conditions of (ULC), the unique Gibbs measure μ satisfies the log-Sobolev inequality (LS). We use an approximate argument used in the study of the same subjects for loop spaces, which in turn is a modification of the method originally developed by S. Albeverio, Yu. G. Kondratiev, and M. Röckner.

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Authors and Affiliations

  1. Department of Mathematics, Sogang University, Seoul, 121-742, Korea;

    Hye Young Lim

  2. Department of Mathematics and Institute for Mathematical Sciences, Yonsei University, Seoul, 120-749, Korea;

    Yong Moon Park

  3. Institute for Mathematical Sciences, Yonsei University, Seoul, 120-749, Korea;

    Hyun Jae Yoo

  4. Institut für Mathematik, Ruhr-Universität-Bochum, D-44780, Bochum, Germany;

    Hyun Jae Yoo

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  1. Hye Young Lim
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Lim, H.Y., Park, Y.M. & Yoo, H.J. Dirichlet Forms and Dirichlet Operators for Gibbs Measures of Quantum Unbounded Spin Systems: Essential Self-Adjointness and Log-Sobolev Inequality. Journal of Statistical Physics 90, 949–1002 (1998). https://doi.org/10.1023/A:1023245508048

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