J Stat Phys (2007) 129: 121–149 DOI 10.1007/s10955-007-9363-4 Spectral Analysis of a Stochastic Ising Model in Continuum Yu. Kondratiev · E. Zhizhina Received: 16 September 2006 / Accepted: 9 June 2007 / Published online: 25 July 2007 © Springer Science+Business Media, LLC 2007 Abstract We consider an equilibrium stochastic dynamics of spatial spin systems in R d involving both a birth-and-death dynamics and a spin flip dynamics as well. Using a gen- eral approach to the spectral analysis of corresponding Markov generator, we estimate the spectral gap and construct one-particle invariant subspaces for the generator. Keywords Birth-and-death process · Continuous system · Gibbs measure · Glauber dynamics · Continuous Ising model 1 Introduction In this paper we study an equilibrium stochastic dynamics of continuous spin systems in- volving a birth-and-death process as well as a spin flip dynamics. The dynamics is a natural generalization of the stochastic Ising model and a Glauber-type dynamics of continuous gas which has been under consideration in [2, 4, 6]. The generator of this dynamics is a self- adjoint operator in L 2 -space w.r.t. an equilibrium measure. The main goal of the present paper is to study the structure of the low-lying spectrum of the infinite volume dynam- ics generator: to estimate the spectral gap, to construct leading invariant subspaces of the Dedicated to our admired teacher and friend Robert Minlos on occasion of his 75th birthday. The financial support of SFB-701, Bielefeld University, is gratefully acknowledged. The work is partially supported by RFBR grant 05-01-00449, Scientific School grant 934.2003.1, CRDF grant RUM1-2693-MO-05. Yu. Kondratiev ( ) Dept. of Mathematics and BiBoS, Bielefeld University, Bielefeld, Germany e-mail: kondrat@math.uni-bielefeld.de Yu. Kondratiev NaUKMA, Kiev, Ukraine E. Zhizhina IITP, Russian Acad. Sci., Moscow, Russia e-mail: ejj@iitp.ru