PRINCIPAL VECTORS OF CRYSTALLOGRAPHIC GROUPS AND APPLICATIONS

For a crystallographic group G acting on an n-dimensional Euclidean sp ace we consider the G-invariant linear elliptic differential operator P with constant coefficients and to it the G-automorphic eigenvalue pr oblem P[psi] + mupsi = 0. N(lambda) is the number of all eigenvalues m u smaller than or equal to the ''frequency bound'' lambda(q) (q: order of P). Earlier we found the asymptotic estimation N(lambda) approxima tely c0 . lambda(n) + c1 . lambda(n-1) (c0, c1: certain volumina). Fur thermore, N(lambda) was interpreted as the number of so-called princip al classes of principal lattice vectors within a convex domain. In thi s paper we demonstrate these results for the case n = 2 for two repres entative crystallographic groups G and the assigned lattices. Above al l we demonstrate a counting method for an exact estimation of N(lambda ) if lambda is not too big. In an analogous way we can treat all the 2 30 space groups of crystallography. It will be seen that these applica tions are brought about by the so-called principal vectors of these la ttices.