Eigenvalue counting function for Robin Laplacians on conical domains

TL;DR

This paper analyzes the eigenvalue distribution of Robin Laplacians on conical domains, revealing how the geometry of the boundary influences the accumulation of discrete spectrum near the essential spectrum.

Contribution

It introduces an effective boundary Hamiltonian to describe the eigenvalue accumulation and derives an asymptotic formula involving boundary curvature.

Findings

The discrete spectrum accumulation is governed by an effective boundary operator.

The eigenvalue count near the essential spectrum follows an inverse proportionality to the spectral gap.

The asymptotic behavior depends on the integral of the squared positive geodesic curvature of the boundary.

Abstract

We study the discrete spectrum of the Robin Laplacian $Q^{\Omega}_\alpha$ in $L^2(\Omega)$, \[ u\mapsto -\Delta u, \quad \dfrac{\partial u}{\partial n}=\alpha u \text{ on }\partial\Omega, \] 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 known from previous papers that the bottom of the essential spectrum of $Q^{\Omega}_\alpha$ is $-\alpha^2$ and that the finiteness of the discrete spectrum depends on the geometry of the cross-section. We show that the accumulation of the discrete spectrum of $Q^\Omega_\alpha$ is determined by the discrete spectrum of an effective Hamiltonian defined on the boundary and far from the origin. By studying this model operator, we prove that the number of eigenvalues 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 the cross-section boundary.