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

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. Rockner.