Spectral asymptotics of the Laplacian on Platonic solids graphs

Abstract

We investigate the high-energy eigenvalue asymptotics quantum graphs consisting of the vertices and edges of the five Platonic solids considering two different types of the vertex coupling. One is the standard $\delta$-condition, the other is the preferred-orientation one introduced in [ET18]. The aim is to provide another illustration of the fact that the asymptotic properties of the latter coupling are determined by the vertex parity by showing that the octahedron graph differs in this respect from the other four for which the edges at high energies effectively disconnect and the spectrum approaches the one of the Dirichlet Laplacian on an interval.

arXiv comment25 pages, 10 figures

References in corpus (3)

• Convergence of spectra of graph-like thin manifolds
• Non-Weyl asymptotics for quantum graphs with general coupling conditions
• Quantum graphs with vertices of a preferred orientation

Cited by in corpus (6)

• Topological bulk-edge effects in quantum graph transport
• Quantum graphs: self-adjoint, and yet exhibiting a nontrivial $\mathcal{P}\mathcal{T}$-symmetry
• Kagome network with vertex coupling of a preferred orientation
• Magnetic ring chains with vertex coupling of a preferred orientation
• Magnetic square lattice with vertex coupling of a preferred orientation
• Topologically induced spectral behavior: the example of quantum graphs