By Bernard Souillard (auth.), Giulio Casati (eds.)
Six years in the past, in June 1977, the 1st overseas convention on chaos in classical dynamical structures happened the following in Como. For the 1st time, physicists, mathematicians, biologists, chemists, economists, and others obtained jointly to debate the relevance of the hot development in nonlinear classical dynamics for his or her personal study box. instantly after, e-book of "Nonlinear technology Abstracts" begun, which, in flip, resulted in the Physica D magazine and to a fast bring up of the learn job within the complete sector with the construction of various "Nonlinear facilities" worldwide. in the course of those years nice growth has been made in figuring out the qualitative habit of classical dynamical structures and now we will have fun with the gorgeous complexity and diversity in their movement. in the meantime, increasingly more scientists started to wonder if and the way such attractive constructions may persist in quantum movement. certainly, normally integrable platforms were formerly con sidered by way of Quantum Mechanics and consequently the matter is open the best way to describe the qualitative habit of structures whose classical restrict is non-integrable. the current assembly used to be geared up in view of the truth that scientists operating in several fields - mathematicians, theoretical physicists, good country physicists, nuclear physicists, chemists and others - had universal difficulties. furthermore, we felt that it was once essential to make clear a few primary questions about the logical foundation for the dialogue together with the very definition of chaos in Quantum Mechanics.
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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!