TY - JOUR

T1 - Plummeting and blinking eigenvalues of the Robin Laplacian in a cuspidal domain

AU - Nazarov, Sergei A.

AU - Popoff, Nicolas

AU - Taskinen, Jari

N1 - Publisher Copyright: © Royal Society of Edinburgh 2019. Copyright: Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2019/8/14

Y1 - 2019/8/14

N2 - 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 to -∞. At the same time, any point λ 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.

AB - 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 to -∞. At the same time, any point λ 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.

KW - cuspidal domain

KW - eigenvalue

KW - Laplace operator

KW - residual spectrum

KW - robin condition

KW - spectral problem

UR - http://www.scopus.com/inward/record.url?scp=85070688415&partnerID=8YFLogxK

U2 - 10.1017/prm.2019.48

DO - 10.1017/prm.2019.48

M3 - Article

AN - SCOPUS:85070688415

JO - Royal Society of Edinburgh - Proceedings A

JF - Royal Society of Edinburgh - Proceedings A

SN - 0308-2105

ER -