Klassifikation av polygoner med trigonometriska egenfunktioner till Laplaceoperatorn under Dirichletrandvillkor

Abstract

We consider the eigenstructure of the Laplace operator on triangles with the angles (60°, 60°,60°); (30°, 60°,90°) och (45°, 45°,90°). Using the earlier work by M. Práger (1998) and M. A. Pinsky (1980) we find eigenfunctions of the Laplace operator with Dirichlet boundary conditions. We show completeness of eigenfunctions in L2 for each triangle. Moreover, we present a result by Brian J. McCartin (2008) that classifies which polygons have a complete set of trigonometric eigenfunctions. These polygons are the triangles mentioned above, the rectangle and the square. We connect McCartins result to symmetries of lattices, crystals and Weyl groups. In 1980 Pierre H. Bérard studied the connection between different types of eigenfunctions and symmetries and proved that all alcoves of Weyl groups have trigonometric eigenfunctions. We point out the fact that in R2 the converse is also true. That is, all polygons with a complete set of trigonometric eigenfunctions are alcoves.