The eigenvalues of the Laplacian on domains with small slits

2008/02/19 by Hillairet, Luc, Judge, Chris · 1 citation
#35P99 #58J37 #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.0802.2597

Abstract

We introduce a small slit into a planar domain and study the resulting effect upon the eigenvalues of the Laplacian. In particular, we show that as the length of the slit tends to zero, each real-analytic eigenvalue branch tends to an eigenvalue of the original domain. By combining this with our earlier work (arXiv:math/0703616), we obtain the following application: The generic multiply connected polygon has simple spectrum.

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• Generic failure of uniform separation in planar Dirichlet spectra

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