| The Asymptotic Maslov Index And Its Applications (2007) | |||||||||||||||
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| . Let N be a 2n-dimensional manifold equipped with a symplectic structure ! and (N ) be the Lagrangian Grassmann bundle over N . Consider a ow t on N that preserves the symplectic structure and a t -invariant connected submanifold . When there exists a continuous section ! (N ), we can associate to any nite, t -invariant measure with support in , a quantity: The Asymptotic Maslov Index, that describes the way Lagrangian planes are asymptotically wrapped in average around the Lagrangian Grassmann bundle. A particular attention is paid to the case when the ow is derived from an optical Hamiltonian and when the invariant measure is the Liouville measure on compact energy levels. The situation when the energy levels are not compact is discussed in an appendix. 1. Introduction 1.1. The Maslov Cocycle. In his book Theorie des perturbations et methodes asymptotiques [15], V. P. Maslov introduced an index of curves relevant in quantum mechanics. V. Arnold [2]... | |||||||||||||||
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