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- W2892541995 abstract "We consider the Robin Laplacian in the domains $Omega$ and $Omega^varepsilon$, $varepsilon >0$, with sharp and blunted cusps, respectively. Assuming that the Robin coefficient $a$ is large enough, the spectrum of the problem in $Omega$ is known to be residual and to cover the whole complex plane, but on the contrary, the spectrum in the Lipschitz domain $Omega^varepsilon$ is discrete. However, our results reveal the strange behavior of the discrete spectrum as the blunting parameter $varepsilon$ 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 neighborhood of a point for any small $varepsilon > 0$ while the second ones move at a high rate $O(|ln varepsilon|)$ downwards along the real axis $mathbb{R}$ to $ -infty$. At the same time, any point $lambda in mathbb{R}$ is a blinking eigenvalue, i.e., it belongs to the spectrum of the problem in $Omega^varepsilon$ almost periodically in the $|ln varepsilon|$-scale. Besides standard spectral theory, we use the techniques of dimension reduction and self-adjoint extensions to obtain these results." @default.
- W2892541995 created "2018-10-05" @default.
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- W2892541995 date "2018-09-28" @default.
- W2892541995 modified "2023-09-27" @default.
- W2892541995 title "Plummeting and blinking eigenvalues of the Robin Laplacian in a cuspidal domain" @default.
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