• Journal of the Korean Mathematical Society
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• v.37 no.4
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• pp.579-592
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• 2000

We study the variational principle for quantum unbounded spin systems interacting via superstable and regular interactions. We show that the (weak) KMS state constructed via the thermodynamic limit of finite volume Green's functions satisfies the Gibbs variational equality.

• Journal of the Korean Mathematical Society
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• v.33 no.4
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• pp.823-855
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• 1996

We study Dirichlet forms and the associated diffusion processes for the Gibbs measures related to the quantum unbounded spin systems (lattice boson systems) interacting via superstable and regular potentials. This work is a continuation of the author's previous study on the classical systems [LPY] to the quantum cases. In [LPY], we constructed Dirichlet forms and the associated diffusion processes for the Gibbs measures of classical unbounded spin systems. Furthermore, we also showed the essential self-adjointness of the Dirichlet operator and the log-Sobolev inequality for any Gibbs measure under appropriate conditions on the potentials. In this atudy we try to extend the results of the classical systems to the quantum cases. Because of some technical difficulties, we are only able to construct a Dirichlet form and the associated diffusion process for any Gibbs measure of the quantum systems. We utilize the general scheme of the previous work on the theory in infinite dimensional spaces [AH-K1-2, AKR, AR1-2, Kus, MR, Ro, Sch] and the ideas we employed in our study of the calssical systems ]LPY].

• Journal of the Korean Mathematical Society
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• v.22 no.1
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• pp.43-74
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• 1985

• Communications of the Korean Mathematical Society
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• v.9 no.3
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• pp.711-730
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• 1994

We consider Gibbs measures on $(R \times W_{0,0})^{Z^\nu}, W_{0,0} = {\omega \in C[0,1] : \omega(0) = \omega(1)}$, which are associated to an interaction between particles in lattice boson systems (quantum unbounded spin systems). In [4], the Gibbs measures were introduced in the study of equilibrium states of interacting lattice boson systems and were characterized by means of the equilibrium conditions. In this paper we utilize the techniques of the stochastic calculus of variations and the infinite dimensional Ito integral to derive stochastic equations which we call the equilibrium equations. We show that under appropriate conditions the equilibrium conditions and the equilibrium equations are equivalent. The lattice boson systems with superstable and regular interactions, which we studied in [4], are typical examples.