Spherical n-lunes: billiards and eigenvalues

We characterise the periodic orbits of geodesic billiards on spherical lunes on $\mathbb{S}^{n}$. In the case of angle openings of the form $π/p$ for positive integer $p$ we fully determine their Dirichlet and Neumann spectra. We then show that lunes with an angle opening smaller than $π$ which is not a rational multiple of $π$, or those with an angle opening of the form $π/p$ for $p$ larger than one satisfy Pólya's conjecture eventually, independently of whether the corresponding geodesic billiards satisfy the nonperiodicity condition or not. For lunes with an angle opening $π/p$ we further provide a two-term asymptotic formula for the eigenvalues based on sharp upper and lower bounds, together with a corresponding two-term counting function established using the geoesic billiards approach. Finally,we give an explicit bound on $p$ in terms of the dimension ensuring the corresponding lunes satisfy Pólya's conjecture for all eigenvalues.