Generic failure of uniform separation in planar Dirichlet spectra

Abstract

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 $D\subset {\mathbb{R}}^2$ be a bounded domain and, for $\ell \ge 1$, let ${\mathcal{C}}_{\ell }(D)$ be the space of nonempty connected open sets $\Omega \subset D$ such that $\overline{D}\setminus \Omega$ has at most $\ell$ connected components, endowed with the complementary-Hausdorff topology. We prove that ${\mathcal{C}}_{\ell }(D)$ is completely metrizable and Baire, and that smooth domains are dense in it. If $0<{\nu }_1(\Omega )<{\nu }_2(\Omega )<\cdots$ are the distinct Dirichlet eigenvalues, our main result states that \[ \left\{Ω\in\mathcal{C}_\ell(D):\inf_{m\geq 1}\bigl(ν_{m+1}(Ω)-ν_m(Ω)\bigr)=0\right\} \] is residual. This statement requires no simplicity assumption. Combining it with our transfer of Micheletti's classical generic-simplicity theorem to ${\mathcal{C}}_{\ell }(D)$ 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.