2018/09/28 by С. А. Назаров, Nazarov, Sergei A., Nicolas Popoff +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1809.10963
openalex publication_date 2018/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the Robin Laplacian in the domains \Ω and\n\Ω^\ε, \ε >0, with sharp and blunted cusps,\nrespectively. Assuming that the Robin coefficient a is large enough, the\nspectrum of the problem in \Ω is known to be residual and to cover the\nwhole complex plane, but on the contrary, the spectrum in the Lipschitz domain\n\Ω^\ε is discrete. However, our results reveal the strange\nbehavior of the discrete spectrum as the blunting parameter \ε tends\nto 0: we construct asymptotic forms of the eigenvalues and detect families of\n"hardly movable" and "plummeting" ones. The first type of the eigenvalues do\nnot leave a small neighborhood of a point for any small \ε > 0 while\nthe second ones move at a high rate O(|\ln \ε|) downwards along the\nreal axis \ℝ to -\∞. At the same time, any point \λ \∈\n\ℝ is a "blinking eigenvalue", i.e., it belongs to the spectrum of the\nproblem in \Ω^\ε almost periodically in the |\ln\n\ε|-scale. Besides standard spectral theory, we use the techniques of\ndimension reduction and self-adjoint extensions to obtain these results.\n