Spectral stability under removal of small segments

He, Xiang

doi:10.1090/proc/16813

Spectral stability under removal of small segments
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by Xiang He;

Proc. Amer. Math. Soc. 152 (2024), 3499-3507

DOI: https://doi.org/10.1090/proc/16813

Published electronically: June 12, 2024

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Abstract:

In the present paper, we deepen the works of L. Abatangelo, V. Felli, L. Hillairet and C. Léna on the asymptotic estimates of the eigenvalue variation under removal of segments from the domain in $\mathbb {R}^2$. We get a sharp asymptotic estimate when the eigenvalue is simple and the removed segment is tangent to a nodal line of the associated eigenfunction. Moreover, we extend their results to the case when the eigenvalue is not simple.

References

Laura Abatangelo, Virginie Bonnaillie-Noël, Corentin Léna, and Paolo Musolino, Asymptotic behavior of $u$-capacities and singular perturbations for the Dirichlet-Laplacian, ESAIM Control Optim. Calc. Var. 27 (2021), no. suppl., Paper No. S25, 43. MR 4222163, DOI 10.1051/cocv/2020078
Laura Abatangelo, Veronica Felli, Luc Hillairet, and Corentin Léna, Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators, J. Spectr. Theory 9 (2019), no. 2, 379–427. MR 3950657, DOI 10.4171/JST/251
Laura Abatangelo, Corentin Léna, and Paolo Musolino, Ramification of multiple eigenvalues for the Dirichlet-Laplacian in perforated domains, J. Funct. Anal. 283 (2022), no. 12, Paper No. 109718, 50. MR 4489278, DOI 10.1016/j.jfa.2022.109718
Veronica Felli, Alberto Ferrero, and Susanna Terracini, Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 1, 119–174. MR 2735078, DOI 10.4171/JEMS/246
Gilles Courtois, Spectrum of manifolds with holes, J. Funct. Anal. 134 (1995), no. 1, 194–221. MR 1359926, DOI 10.1006/jfan.1995.1142
M. Kac, Probabilistic methods in some problems of scattering theory, Rocky Mountain J. Math. 4 (1974), 511–537. Notes by Jack Macki and Reuben Hersh. MR 510113, DOI 10.1216/RMJ-1974-4-3-511
Jeffrey Rauch and Michael Taylor, Potential and scattering theory on wildly perturbed domains, J. Functional Analysis 18 (1975), 27–59. MR 377303, DOI 10.1016/0022-1236(75)90028-2