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”λ ^q (q: order ofP). Earlier we found the asymptotic estimationN(λ) ∼ c₀ · λⁿ + c₁ · λⁿ⁻¹ (c ₀,c ₁: 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.