English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We study the discrete spectrum of the Robin Laplacian $Q^{\Omega}_\alpha$ in$L^2(\Omega)$, \[ u\mapsto -\Delta u, \quad \dfrac{\partial u}{\partialn}=\alpha u \text{ on }\partial\Omega, \] where $\Omega\subset \mathbb{R}^{3}$is a conical domain with a regular cross-section $\Theta\subset \mathbb{S}^2$,$n$ is the outer unit normal, and $\alpha>0$ is a fixed constant. It is knownfrom previous papers that the bottom of the essential spectrum of$Q^{\Omega}_\alpha$ is $-\alpha^2$ and that the finiteness of the discretespectrum depends on the geometry of the cross-section. We show that theaccumulation of the discrete spectrum of $Q^\Omega_\alpha$ is determined by thediscrete spectrum of an effective Hamiltonian defined on the boundary and farfrom the origin. By studying this model operator, we prove that the number ofeigenvalues of $Q^{\Omega}_\alpha$ in $(-\infty,-\alpha^2-\lambda)$, with$\lambda>0$, behaves for $\lambda\to0$ as \[ \dfrac{\alpha^2}{8\pi \lambda}\int_{\partial\Theta} \kappa_+(s)^2d s +o\left(\frac{1}{\lambda}\right), \]where $\kappa_+$ is the positive part of the geodesic curvature of thecross-section boundary.

Eigenvalue counting function for Robin Laplacians on conical domains