On a Lattice Problem for Geodesic Double Differential Forms in the $n$-Dimensional Hyperbolic Space

Abstract

Generalizing a lattice point problem we solve a lattice problem for geodesic double differential forms in the $n$-dimensional hyperbolic space. Thereby we use a properly discontinuous group $\mathfrak{G}$ of isometrics of ${\mathbb{H}}_n$ with compact fundamental domain. We study the relation between lattice sums and the eigenvalue spectrum of the Laplace operator for $p$-forms on the hyperbolic space form ${\mathbb{H}}_n/\mathfrak{G}$. Our approach essentially uses mean value operators for differential forms, their kernel double differential forms and a Landau difference method.