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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

An example of a two-term asymptotics for the ``counting function'' of a fractal drum


Authors: Jacqueline Fleckinger-Pellé and Dmitri G. Vassiliev
Journal: Trans. Amer. Math. Soc. 337 (1993), 99-116
MSC: Primary 58G18; Secondary 58G25
MathSciNet review: 1176086
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Abstract: In this paper we study the spectrum of the Dirichlet Laplacian in a bounded domain $ \Omega \subset {\mathbb{R}^n}$ with fractal boundary $ \partial \Omega $. We construct an open set $ \mathcal{Q}$ for which we can effectively compute the second term of the asymptotics of the "counting function" $ N(\lambda ,\mathcal{Q})$, the number of eigenvalues less than $ \lambda $. In this example, contrary to the M. V. Berry conjecture, the second asymptotic term is proportional to a periodic function of In $ \lambda $, not to a constant. We also establish some properties of the $ \zeta $-function of this problem. We obtain asymptotic inequalities for more general domains and in particular for a connected open set $ \mathcal{O}$ derived from $ \mathcal{Q}$. Analogous periodic functions still appear in our inequalities. These results have been announced in $ [{\text{FV}}]$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1176086-7
PII: S 0002-9947(1993)1176086-7
Article copyright: © Copyright 1993 American Mathematical Society