DOI

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.

Original languageEnglish
JournalProceedings of the Royal Society of Edinburgh Section A: Mathematics
DOIs
StateE-pub ahead of print - 14 Aug 2019

    Scopus subject areas

  • Mathematics(all)

    Research areas

  • cuspidal domain, eigenvalue, Laplace operator, residual spectrum, robin condition, spectral problem

ID: 71562360