Trapped modes in thin and infinite ladder like domains: existence and asymptotic analysis

Bérangère Delourme; Sonia Fliss; Patrick Joly; Elizaveta Vasilevskaya

Trapped modes in thin and infinite ladder like domains: existence and asymptotic analysis

Bérangère Delourme ¹ Sonia Fliss ² Patrick Joly ² Elizaveta Vasilevskaya ¹

1 LAGA

LAGA - Laboratoire Analyse, Géométrie et Applications

2 POEMS - Propagation des Ondes : Étude Mathématique et Simulation

Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231

Abstract : We are interested in a 2D propagation medium which is a localized perturbation of a reference homogeneous periodic medium. This reference medium is a "thick graph", namely a thin structure (the thinness being characterized by a small parameter $\varepsilon$ > 0) whose limit (when $\varepsilon$ tends to 0) is a periodic graph. The perturbation consists in changing only the geometry of the reference medium by modifying the thickness of one of the lines of the reference medium. The question we investigate is whether such a geometrical perturbation is able to produce localized eigenmodes. We have investigated this question when the propagation model is the scalar Helmholtz equation with Neumann boundary conditions . This amounts to solving an eigenvalue problem for the Laplace operator in an unbounded domain. We use a standard approach of analysis that consists in (1) find a formal limit of the eigenvalue problem when the small parameter tends to 0, here the formal limit is an eigenvalue problem for a second order differential operator along a graph; (2) proceed to an explicit calculation of the spectrum of the limit operator; (3) deduce the existence of eigenvalues as soon as the thickness of the ladder is small enough and construct an asymptotic expansion of the eigenvalues with respect to the small parameter. Following this approach, we prove the existence of localized modes provided that the geometrical perturbation consists in diminishing the width of one rung. Using the matched asymptotic expansion method, we obtain an asymptotic expansion at any order of the eigenvalues, which can be used for instance to compute a numerical approximations of these eigenvalues and associated eigenvectors.

Keywords : matched asymptotic expansion method spectral theory periodic media quantum graphs

Document type :

Reports

Domain :

Mathematics [math] / Analysis of PDEs [math.AP]

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https://hal.inria.fr/hal-01287023
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