The eigenvalues of the Laplacian on domains with small slits

Hillairet, Luc; Judge, Chris

doi:10.1090/S0002-9947-2010-04943-8

The eigenvalues of the Laplacian on domains with small slits
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by Luc Hillairet and Chris Judge PDF

Trans. Amer. Math. Soc. 362 (2010), 6231-6259 Request permission

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 (2009), we obtain the following application: The generic multiply connected polygon has a simple spectrum.

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