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

13 mins 27 secs, 50.36 MB, iPod Video 480x270, 59.94 fps, 44100 Hz, 511.19 kbits/sec

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Description:	Chernyshev, V (BMSTU) Tuesday 27 July 2010, 17:30-17.45

Created:

2010-07-29 08:46

Collection:

Analysis on Graphs and its Applications

Publisher:

Isaac Newton Institute

Chernyshev, V

Language:

eng (English)

Credits:

Author:

Chernyshev, V


Abstract:	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.

Abstract:

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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