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**Extra resources for Statistical Physics and Dynamical Systems: Rigorous Results**

**Example text**

Asym denotes the totally antisymmetric part (under permutation) of n-fold tensor product of L with itself, and with H~ is generated by E h. = h. , which belongs to + J ~~ J even or odd. or n~(B(h 1 ) •.. B(hn))n~ according as ~"- ... 3) n=o KE+ acting on L. Since K+ does not have eigenvalue 0 K+ and is positive, n (corresponding to n = 0 in the above decomposi- where ~ tion) is the only eigenvector of H~ belonging to 0 . Furthermore, is the restriction of H~ to H~. If (,\,y) H IH (O,:tl), ~ ~ K+ has an (absolutely) continuous spectrum, and hence H~IH~, whose spectrum is obtained by n-fold "convolution" of Spec K+, also has an (absolutely) continuous spectrum.

10). In particular we oband c*'s by taking we will use a basis projection E E = (b). Immediately below, for which annihilation operators B(g) is a linear combination of the original c's We now want to find ground states of evolution as annihilation operator and ACAR c*'s. (relative to the time at) , although this is not exactly the problem we are sol- ving. For this purpose, we need a digression on Fock states. Let E be a basis projection and ~E be a state of ACAR satis- fying ~E(B(h)*B(h)) = (h,Eh).

Publ. RIMS, Kyoto Univ. ~. 384-442 (1970). [2] Araki, H. : On a C*-algebra approach to phase transition in the two dimensional Ising model. Cornrnun. Math. Phys. 2l· 489-503 (1983). : On the XY-rnodel on two-sided infinite chain. Publ. RIMS Kyoto Univ. 20, 277-296 (1984). [4] Araki, H. : Ground states of the XY-rnodel, to be published. [5] Bratteli, 0. : Operator algebras and quantum statistical mechanics II. Springer-Verlag, New York-BerlinHeidelberg, 1981. , Schultz, T. : Two soluble models of an antiferrornagnetic chain.