The eigenvalues of the Laplacian on domains with small slits

Details The eigenvalues of the Laplacian on domains with small slits The eigenvalues of the Laplacian on domains with small slits

2008-02-19

arxiv.org/abs/0802.2597v1

Mathematics

Spectral Theory

29 pages, 3 figures

Scientific paper

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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