Counterexamples to the modified Weyl–Berry conjecture on fractal drums

Abstract

Let Ω be a non-empty open set in ℝ n with finite ‘volume’ ( n -dimensional Lebesgue measure). Let be the Laplacian operator. Consider the eigenvalue problem (with Dirichlet boundary conditions): where λ ∈ ℝ and u is a non-zero member of (the closure in the Sobolev space H 1 (Ω) of the set of smooth functions with compact support contained in Ω). It is well known that the values of λ∈ℝ for which (1·1) has a non-zero solution are positive and form a discrete set. Moreover, for each λ, the associated eigenspace is finite dimensional. Let the spectrum of (1·1) be denoted where 0 < λ 1 ≤ λ 2 ≤ … and where the multiplicity of each λ in the sequence is the dimension of the associated eigenspace. Let