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Eigenvalues on a domain with discrete rotational symmetry. (English) Zbl 0629.58024

The author describes the structure of eigenfunctions of the Laplace operator in a planar domain under deformations that preserve symmetry under a discrete rotation. There are two types of eigenfunctions, namely symmetric and asymmetric. The main theorem shows that generically, symmetric eigenfunctions are simple and asymmetric eigenfunctions are of multiplicity two. This is a partial answer to a conjecture of Arnol’d concerning the codimension of large multiplicity eigenspaces in the space of domains preserving symmetry.
Reviewer: M.Puta

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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