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 symmetries of the hexagon. These eigenfunctions are given as a double Fourier series, obeying C6 symmetry. A table of the lower lying eigen numbers for these states is included. The explicit form for these eigenstates is given in terms of a sum of six exponentials each of which contains a pair of quantum numbers and a symmetry integer. Eigenstates so constructed are found to satisfy periodicity of the hexagon array. Contour read-outs of a lower lying eigenstate reveal in each case hexagonal 6-fold symmetric arrays. Derived solutions satisfy either Dirichlet or Neumann boundary conditions and are irregular in neighborhoods about vertices. This singular property is intrinsic to the hexagon quantum billiard. Dirichlet solutions are valid in the open neighborhood of the hexagon, due to singular boundary conditions. For 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 C6v group. A description is included on the application of these findings to the hexagonal nitride compounds.