Dispersive waves with multiple tunnel effect on a star-shaped network

F. Ali Mehmeti; R. Haller-Dintelmann; V. Régnier

doi:10.3934/dcdss.2013.6.783

Article Contents

2013, Volume 6, Issue 3: 783-791. Doi: 10.3934/dcdss.2013.6.783

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Dispersive waves with multiple tunnel effect on a star-shaped network

1.
Université de Valenciennes et du Hainaut-Cambrésis, LAMAV, FR CNRS 2956, F-59313 Valenciennes
2.
TU Darmstadt, Fachbereich Mathematik, Schloßgartenstraße 7, D-64289 Darmstadt

Received: March 31, 2010

Revised: November 30, 2010

Published: May 2013

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Abstract

We consider the Klein-Gordon equation on a star-shaped network composed of $n$ half-axes connected at their origins. We add a potential which is constant but different on each branch. The corresponding spatial operator is self-adjoint and we state explicit expressions for its resolvent and its resolution of the identity in terms of generalized eigenfunctions. This leads to a generalized Fourier type inversion formula in terms of an expansion in generalized eigenfunctions. This paper is a survey of a longer article, nevertheless the proof of the central formula is indicated.

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Mathematics Subject Classification: Primary 34B45; Secondary 42A38, 47A10, 47A60, 47A70.

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