Bernard Helffer (Nantes Université)
in collaboration with
G. Berkolaiko, G. Cox, and M. Persson Sundqvist
Abstract
Recent work of the authors and their collaborators has uncovered fundamental connections between the Dirichlet-to-Neumann map, the spectral flow of a certain family of self-adjoint operators, and the nodal deficiency of a Laplacian eigenfunction (or an analogous deficiency associated to a non-bipartite equipartition). Using a more refined construction of the Dirichlet-to-Neumann map, we strengthen all of these results, in particular getting improved bounds on the nodal deficiency of degenerate eigenfunctions. Our framework is very general, allowing for non-bipartite partitions, non-simple eigenvalues, and non-smooth nodal sets. Consequently, our results can be used in the general study of spectral minimal partitions, not just nodal partitions of generic Laplacian eigenfunctions.
