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Erschienen in:
2019 | OriginalPaper | Buchkapitel
The One-Dimensional Modified Weyl-Berry Conjecture: An Elementary Approach
verfasst von : Roland J. Etienne
Erschienen in: Mathematics, Informatics, and Their Applications in Natural Sciences and Engineering
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Abstract
Let \(\varOmega \) be a bounded domain in \(\mathbb {R}^n\) with boundary \(\delta \varOmega \) and consider the eigenvalue problem with Dirichlet boundary conditions, i.e. \(u\left| _{\delta \varOmega }\right. =0\). Its set of eigenvalues, \(0<\lambda _1\le \lambda _2\le \cdots \le \lambda _k\le \cdots \)—each eigenvalue being repeated according to (algebraic) multiplicity—is countable and the eigenvalue counting function may be defined as for a given positive \(\lambda \). The modified Weyl-Berry conjecture for the asymptotics of the eigenvalues of the Laplacian on bounded open subsets of the line (fractal strings) then states that with \(\left| \varOmega \right| _1\) being the one-dimensional Lebesgue measure of \(\varOmega \) and \(d\in [0,1]\) the Minkowski dimension of the boundary. Based upon a matrix representation of the Laplacian, it will be shown how to obtain some of the key results on the one-dimensional modified Weyl-Berry conjecture through elementary methods.
$$ -\varDelta u=\lambda u, $$
$$ N(\lambda ):=\#\{(0<)\lambda _k<\lambda \}, $$
$$ N(\lambda )=\pi ^{-1}\left| \varOmega \right| _1\lambda ^{\frac{1}{2}}+\mathscr {O}(\lambda ^{\frac{d}{2}}), $$