465 Dynamic compensation theory of an inversion symmetric spin-boson system is developed rigorously, describing the existence of two branches of localization and delocalization, the transition of ground state between them, excited states, thermodynamics and dynamics. This is achieved by introducing two kinds of displaced bosons. The oscillation of quantum coherence is described. We develop a rigorous theory of an inversion symmetric (IS) spin-boson system, the Hamiltonian of which is defined by

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... (1) where 2L1 is the level spacing of two-state spin described by the Pauli matrix 6z and 6±=6x±i6y, w; is the excitation energy of boson with quantum number j, and v = 'LJ;(J..;/w;) · (bj-b;). This is the unitary transformed version offamiliar H =-Ll· 6x +LJ;w;b;+b;+1/2·6z·'LJ;J..;(b;++b;)-LJ;J..]/4w; by unitary operator U=exp[6z·v/2]. The IS means that fi and H are invariant under inversion operation P( 6x, 6y, 6z, b;, bt)P-1 =(6x, -(Jy, -6z, -b;, -b;+), where P=exp[i7r'LJ;b/b;+i(.n/2)(6x-1)]. The spectral density of interaction is assumed to be 'LJ;J../8(w;-w)=2aw·(w/wcY-r Xexp[ -w/wc] where s>O and a is a dimensionless strength. Boson system fiB = 'LJ;w;b; + b; in (1) consists of bosons which displace themselves according to flips of spin. We assume that the boson system is thermodynamically large and white in the sense that cutoff wc')>Ll. We choose the eigenstates of 6z for those of spin. In previous works 1 l,ZJ the author proposed the concept of dynamic compensation, and predicted that the ground state (GS) in the white limit exhibits the transition at ac=1/2 from the delocalized (tunneling) one in a< ac, to the localized one in a>ac for s = 1, the localized one over the whole a except zero for s < 1, and the delocalized one over the whole a for s > 1. The first two results of GS are due to the dynamic compensation of infrared divergence. There is no infrared divergence in s > 1 from the outset. This prediction is different from the paradigmatic description based on the static renormalization of infrared divergence. 3 l, 4 l In this paper we present the rigorous results of the two-branch structure of GS, excited state and their energies, thermodynamics and dynamics, restricting ourselves to the case s=1 in order to save space. The full details and other cases of s will be reported elsewhere. Quantum mechanics Let us first study the eigenvalue equation (fi-E) lfJ"=O. It is immediately proved by substitution that w<RJ and w<LJ defined by Downloaded from https://academic.oup.com/ptp/article-abstract/92/2/465/1889245 by guest on 30 July 2018