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Spectral asymptotics of the Dirichlet Laplacian on fat graphs

Presented by: 
D Grieser [Carl von Ossietzky, Oldenburg]
Date: 
Thursday 12th April 2007 - 14:00 to 15:00
Venue: 
INI Seminar Room 1
Session Chair: 
N Datta and V Kendon
Abstract: 

We investigate the behavior of the eigenvalues of the Laplacian, or a similar operator, on a family of Riemannian manifolds with boundary, called fat graphs, obtained by associating to the edges of a given finite graph cross-sectional Riemannian manifolds with boundary, and also to the vertices certain Riemannian manifolds with boundary, glueing them according to the graph structure, and scaling them by a factor of $\varepsilon$ while keeping the lengths of the edges fixed. The simplest model of this is the $\varepsilon$-neighborhood of a the graph embedded with straight edges in $R^n$.

We determine the asymptotics of the eigenvalues with various boundary conditions as $\varepsilon\to 0$ in terms of combinatorial and scattering data.

Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons