The hexagon quantum billiard

A subset of eigenfunctions and eigenvalues for the hexagon quantum billiard are constructed by way of tessellation of the plane and incorporation of s ymmetries of the hexagon. These eigenfunctions are given as a double Fourie r series, obeying C-6 symmetry. A table of the lower lying eigen numbers fo r these states is included. The explicit form for these eigenstates is give n in terms of a sum of six exponentials each of which contains a pair or qu antum numbers and a symmetry integer. Eigenstates so constructed are found to satisfy periodicity of the hexagon array. Contour read-outs of a lower l ying eigenstate reveal in each case hexagonal 6-rold symmetric arrays. Deri ved solutions satisfy either Dirichlet or Neumann boundary conditions and a re irregular in neighborhoods about vertices. This singular property is int rinsic to the hexagon quantum billiard. Dirichlet solutions are valid in th e open neighborhood of the hexagon, due to singular boundary conditions. Fo r integer phase factors, Neumann solutions are valid over the domain of the hexagon. These doubly degenerate eigenstates are identified with the basis of a two-dimensional irreducible representation of the C-6p group. A descr iption is included on the application of these findings to the hexagonal ni tride compounds.