Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map
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abstract
2019, Springer Nature B.V. It has been recently shown that the nodal deficiency of an eigenfunction is encoded in the spectrum of the Dirichlet-to-Neumann operators for the eigenfunctions positive and negative nodal domains. While originally derived using symplectic methods, this result can also be understood through the spectral flow for a family of boundary conditions imposed on the nodal set, or, equivalently, a family of operators with delta function potentials supported on the nodal set. In this paper, we explicitly describe this flow for a Schrdinger operator with separable potential on a rectangular domain and determine a mechanism by which lower-energy eigenfunctions do or do not contribute to the nodal deficiency.
