The One-Dimensional Modified Weyl-Berry Conjecture: An Elementary Approach

Abstract

Let $\varOmega$ be a bounded domain in $\mathbb {R}^n$ with boundary $\delta \varOmega$ and consider the eigenvalue problem

$$ -\varDelta u=\lambda u, $$

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

$$ N(\lambda ):=\#\{(0<)\lambda _k<\lambda \}, $$

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

$$ N(\lambda )=\pi ^{-1}\left| \varOmega \right| _1\lambda ^{\frac{1}{2}}+\mathscr {O}(\lambda ^{\frac{d}{2}}), $$

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.