2026/07/28 by Vincent Boulard
Mathematics · #math.AP #math.SP #msc:35B20 #msc:35P15 #msc:35P20 #msc:54E52
arxiv created 2026/07/28 · arxiv updated 2026/07/30
Can a bounded planar domain have a simple Dirichlet spectrum with uniformly separated consecutive eigenvalues? Dimension two is critical: Weyl's law permits both uniform separation and arbitrarily small gaps. We prove that uniform separation is nevertheless exceptional in a natural rough-domain setting, even after multiplicities are removed. Let D⊂ℝ2 be a bounded domain and, for ℓ≥ 1, let C_ℓ(D) be the space of nonempty connected open sets Ω⊂ D such that D∖Ω has at most ℓ connected components, endowed with the complementary-Hausdorff topology. We prove that C_ℓ(D) is completely metrizable and Baire, and that smooth domains are dense in it. If 0<ν1(Ω)<ν2(Ω)<⋯ are the distinct Dirichlet eigenvalues, our main result states that \Ω\inC_ℓ(D):infm≥ 1(νm+1(Ω)-νm(Ω))=0\ is residual. This statement requires no simplicity assumption. Combining it with our transfer of Micheletti's classical generic-simplicity theorem to C_ℓ(D) shows that a generic domain has simple spectrum and consecutive gaps with zero lower limit. The proof uses Šverák's planar spectral continuity theorem and a local surgery that implants an arbitrarily high pair of close consecutive distinct eigenvalues.