On the Calculus of the Spectrum of the Laplacian for functions in Lens Spaces
Gedeana Pantoja da Silva, Alexandre Casassola Gonçalves, Ferreira da Silva Rafael
TL;DR
The paper addresses the problem of explicitly determining eigenvalue multiplicities for the Laplace-Beltrami operator on lens spaces by linking spectral data to congruence-lattice counts. It introduces a refined combinatorial framework that replaces dependence on $N_{\\mathcal L}(h,l)$ with a direct formula for $N_{\\mathcal L}(k+np)$ involving sums over subsets of indices and the auxiliary counts $\\gamma(U,k+tp)$, weighted by binomial coefficients. The key contribution is the explicit, efficient expression for $N_{\\mathcal L}(k+np)$ and the associated mapping arguments, which facilitate computation of eigenspace dimensions and reveal parity properties for even $p$. This advances spectral geometry of spherical space forms and has potential implications for isospectrality and spectral zeta-function analyses in lens spaces.
Abstract
We improve a specific method to obtain the dimension of the eigenspaces of the Laplace-Beltrami operator on lens spaces and establish some applications related to the explicit description of the dimension of the smallest positive eigenvalue and the parity of the dimensions of the eigenspaces.