 01128 B. Helffer, M. HoffmannOstenhof, T. HoffmannOstenhof, N. Nadirashvili
 Spectral Theory for the Dihedral Group.
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Apr 2, 01

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Abstract. Let $H=\Delta+V$ be a twodimensional Schr\"odinger operator defined on
a bounded domain $\Omega \subset \mathbb{R}^2$ with Dirichlet boundary
conditions on $\partial \Omega$. Suppose that $H$ commutes with the
actions of the dihedral group $\mathbb D_{2n}$, the group of the
regular $n$gone.
We analyze completely the multiplicity of the groundstate eigenvalues
associated to the different symmetry subspaces related to the irreducible
representations of $\mathbb D_{2n}$. 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.
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