By Stieglitz J.
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For results locating the essential spectrum of - ~ (the points of the spectrum which are either accumulation points of the spectrum or eigenvalues of infinite multiplicity), see for example [85,86), which also contain references discussing the location of the greatest lower bound of the spectrum -~. For such results and results on the nature of the spectrum of the Laplacian on a simply connected complete Riemannian symmetric space of non compact type, see [87J. For complete Riemannian manifolds for which - ~ has pure point spectrum, see [88).
G. Gallavotti, "The elements of MechaniCS," SpringerVerlag, New York (1982). D. Birkhoff, "Dynamical Systems," Providence, Rhode Island (1966) . Moser, Nachr. Acad. Wiss. Gottingen, Math. Phys. Kl, lla, no. 1 (1962), 1-20. H. Rlissmann, Preprint (1983). M. Herman, Unpublished Result. J. M. Greene, J. Math. Phys. 20 1183 (1979). F. Doveil, M. Escande, J. St~ Phys. 26, 257 (1981). Aubry, Y. " To appear in Physica D (1983), and J. Phys. C (1983). Mather, "Existence of quasi-periodic orbits in the twist homeomorphisms of the annulus, " Preprint Princeton (1981).
Leading to the conclusion In particular, if E E K y(E) = that probably the states are extended. This is actually the case. l EPn (E) - Rn Pn- l(E) (19) n' Thanks to (4) we get immediately = Pn P2·n (E) (E 2 - A) (20) leading to the relation: (using (5)). P (Fk(E)) p If E (21) K ,namely in the spectrum, we get: € = k l/J(p2 ) some (22) (J Therefore as E(£) is choosen randomly in the spectrum with the probability given by ~, the value of I/J at points p2k are choosen randomly. Choosing p = 1 , we can prove that lim inf' 11J!