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Fractal drums and then-dimensional modified Weyl-Berry conjecture

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Abstract

In this paper, we study the spectrum of the Dirichlet Laplacian in a bounded (or, more generally, of finite volume) open set Ω∈R n (n≧1) with fractal boundary ∂Ω of interior Minkowski dimension δ∈(n−1,n]. By means of the technique of tessellation of domains, we give the exact second term of the asymptotic expansion of the “counting function”N(λ) (i.e. the number of positive eigenvalues less than λ) as λ→+∞, which is of the form λδ/2 times a negative, bounded and left-continuous function of λ. This explains the reason why the modified Weyl-Berry conjecture does not hold generally forn≧2. In addition, we also obtain explicit upper and lower bounds on the second term ofN(λ).

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Authors and Affiliations

  1. Department of Mathematics, Wuhan University, 430072, Wuhan, P.R. China

    Chen Hua

  2. Department of Mathematics and Computer Science, University of Dundee, DD1 4HN, Dundee, U.K.

    B. D. Sleeman

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  1. Chen Hua
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  2. B. D. Sleeman
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Communicated by B. Simon

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Hua, C., Sleeman, B.D. Fractal drums and then-dimensional modified Weyl-Berry conjecture. Commun.Math. Phys. 168, 581–607 (1995). https://doi.org/10.1007/BF02101845

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  • DOI: https://doi.org/10.1007/BF02101845

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