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Plummeting and blinking eigenvalues of the Robin Laplacian in a cuspidal domain

Published online by Cambridge University Press:  14 August 2019

Sergei A. Nazarov ,
Nicolas Popoff  and
Jari Taskinen
Sergei A. Nazarov
Affiliation:
Saint-Petersburg State University, Universitetskaya nab., 7–9, St. Petersburg, 199034Russia (s.nazarov@spbu.ru; srgnazarov@yahoo.co.uk)
Nicolas Popoff
Affiliation:
Institut de Mathématiques de Bordeaux UMR 5251, Université de Bordeaux, 351 cours dela Libération – F 33 405 Talence, France (nicolas.popoff@u-bordeaux.fr)
Jari Taskinen
Affiliation:
Department of Mathematics and Statistics, P.O.Box 68, University of Helsinki, 00014 Helsinki, Finland (jari.taskinen@helsinki.fi)
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Abstract

We consider the Robin Laplacian in the domains Ω and Ωε, ε > 0, with sharp and blunted cusps, respectively. Assuming that the Robin coefficient a is large enough, the spectrum of the problem in Ω is known to be residual and to cover the whole complex plane, but on the contrary, the spectrum in the Lipschitz domain Ωε is discrete. However, our results reveal the strange behaviour of the discrete spectrum as the blunting parameter ε tends to 0: we construct asymptotic forms of the eigenvalues and detect families of ‘hardly movable’ and ‘plummeting’ ones. The first type of the eigenvalues do not leave a small neighbourhood of a point for any small ε > 0 while the second ones move at a high rate O(| ln ε|) downwards along the real axis ℝ to −∞. At the same time, any point λ ∈ ℝ is a ‘blinking eigenvalue’, i.e., it belongs to the spectrum of the problem in Ωε almost periodically in the | ln ε|-scale. Besides standard spectral theory, we use the techniques of dimension reduction and self-adjoint extensions to obtain these results.

Keywords

Laplace operatorrobin conditionspectral problemcuspidal domaineigenvalueresidual spectrum
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 2871 - 2893
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Copyright
Copyright © 2019 The Royal Society of Edinburgh

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