Asymptotics of eigenvalues and eigenfunctions of the Dirichlet problem in thin spatial network with nodules

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Abstract

We perform homogenization of a thin spatial network of quantum waveguides with small nodules (the Dirichlet problem for the Laplace operator). In contrast to a problem with the Neumann boundary
condition, the low-frequency but situated far away from the coordinate origin range of the
spectrum of the Dirichlet problem is characterized by localization of the corresponding
eigenfunctions near angular joints of ligaments or at the ligaments themselves in accordance with
distribution of eigenvalues in the discrete spectra (surely non-empty) of model problems
in junctions of semi-infinite cylindrical quantum waveguides of various shapes. The behavior of eigenvalues and eigenfunctions in the mid-frequency range of the spectrum depend crucially on
the phenomenon of threshold resonances in the above-mentioned junctions as well as the relation
between small parameters, namely, the period of distribution of the nodules and their diameter,
comparable in order but bigger than diameter of the ligaments. We consider concrete cases
of rectangular and circular cross-sections and formulate open questions.

About the authors

Sergei Aleksandrovich Nazarov

Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg

Email: srgnazarov@yahoo.co.uk; srgnazarov108@gmail.com
ORCID iD: 0000-0002-8552-1264
Scopus Author ID: 35616414800
ResearcherId: N-3503-2015
Doctor of physico-mathematical sciences, Professor

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