Principal vectors of crystallographic groups and applications

Abstract

For a crystallographic group\(\mathfrak{G}\) acting on ann-dimensional Euclidean space we consider the\(\mathfrak{G}\)-invariant linear elliptic differential operatorP with constant coefficients and to it the\(\mathfrak{G}\)-automorphic eigenvalue problemP [ω] + μψ = 0. N(λ) is the number of all eigenvaluesμ smaller than or equal to the “frequency bound”λq (q: order ofP). Earlier we found the asymptotic estimationN(λ) ∼ c0 · λn + c1 · λn−1 (c0,c1: certain volumina). Furthermore,N(λ) was interpreted as the number of so-called principal classes of principal lattice vectors within a convex domain. In this paper we demonstrate these results for the casen = 2 for two representative crystallographic groups\(\mathfrak{G}\) and the assigned lattices. Above all we demonstrate a counting method for an exact estimation ofN(λ) if a is not too big. In an analogous way we can treat all the 230 space groups of crystallography. It will be seen that these applications are brought about by the so-called principal vectors of these lattices.