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

TL;DR

This paper investigates the spectral behavior of the Robin Laplacian in domains with cusps, revealing phenomena like plummeting eigenvalues and blinking eigenvalues as the domain's cusp is blunted.

Contribution

It provides asymptotic descriptions of eigenvalues and uncovers novel spectral phenomena in cusp domains with Robin boundary conditions, using advanced analytical techniques.

Findings

Eigenvalues split into 'hardly movable' and 'plummeting' types.

'Plummeting' eigenvalues descend rapidly to -infinity as the cusp is blunted.

Any real number becomes a 'blinking eigenvalue' periodically in the blunting parameter.

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.