Eigenvalue counting function for Robin Laplacians on conical domains

Abstract

We study the discrete spectrum of the Robin Laplacian $Q_{\alpha }^{\Omega }$ in $L^2(\Omega )$, \[ u\mapsto -Δu, \quad \dfrac{\partial u}{\partial n}=αu \text{ on }\partialΩ, \] 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 $Q_{\alpha }^{\Omega }$ 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 $Q_{\alpha }^{\Omega }$ 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 $Q_{\alpha }^{\Omega }$ in $(-\infty ,-{\alpha }^2-\lambda )$, with $\lambda >0$, behaves for $\lambda \to 0$ as \[ \dfrac{α^2}{8πλ} \int_{\partialΘ} κ_+(s)^2d s +o\left(\frac{1}λ\right), \] where ${\kappa }_{+}$ is the positive part of the geodesic curvature of the cross-section boundary.

21 pages

References in corpus (5)

• Schrödinger operators with $\delta$-interactions supported on conical surfaces
• Spectral Properties of a Magnetic Quantum Hamiltonian on a Strip
• Discrete spectrum of interactions concentrated near conical surfaces
• Weyl formulae for the Robin Laplacian in the semiclassical limit
• Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere

Cited by in corpus (5)

• Eigenvalues of Robin Laplacians in infinite sectors
• Dirichlet spectrum of the Fichera layer
• Discrete spectrum of interactions concentrated near conical surfaces
• Attractive conical surfaces create infinitely many bound states
• Spectral asymptotics for $\delta$-interactions on sharp cones