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Statistical properties of semiclasssical solutions of the non-stationary Schrödinger equation on metric graphs

Presented by: 
V Chernyshev [BMSTU]
Tuesday 27th July 2010 - 17:30 to 17:45
INI Seminar Room 1
The talk is devoted to the development of the semiclassical theory on quantum graphs. For the non-stationary Schrödinger equation, propagation of the Gaussian packets initially localized in one point on an edge of the graph is described. Emphasis is placed on statistics behavior of asymptotic solutions with increasing time. It is proven that determination of the number of quantum packets on the graph is associated with a well-known number-theoretical problem of counting the number of integer points in an expanding polyhedron. An explicit formula for the leading term of the asymptotics is presented. It is proven that for almost all incommensurable passing times Gaussian packets are distributed asymptotically uniformly in the time of passage of edges on a finite compact graph. Distribution of the energy on infinite regular trees is also studied. The presentation is based on the joint work with A.I. Shafarevich.
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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons