Spectral theory for the dihedral group

Abstract.

Let $H = -\Delta + V$ be a two-dimensional Schrödinger operator defined on a bounded domain $\Omega \subset {\Bbb R}^2$ with Dirichlet boundary conditions on $\partial \Omega$. Suppose that H commutes with the actions of the dihedral group $\Bbb D_{2n}$, the group of the regular n-gon. We analyze completely the multiplicity of the groundstate eigenvalues associated to the different symmetry subspaces related to the irreducible representations of $\Bbb D_{2n}$. In particular we find that the multiplicities of these groundstate eigenvalues equal the degree of the corresponding irreducible representation. We also obtain an ordering of these eigenvalues. In addition we analyze the qualitative properties of the nodal sets of the corresponding ground state eigenfunctions.