Generic failure of uniform separation in planar Dirichlet spectra

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 $\ell\geq 1$, 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 \[ \left\{\Omega\in\mathcal{C}_\ell(D):\inf_{m\geq 1}\bigl(\nu_{m+1}(\Omega)-\nu_m(\Omega)\bigr)=0\right\} \] 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.