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arXiv · 1805.03145

Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map

Gregory Berkolaiko·Graham Cox·Jeremy L. Marzuola
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Abstract

It was recently shown that the nodal deficiency of an eigenfunction is encoded in the spectrum of the Dirichlet-to-Neumann operators for the eigenfunction's 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 Schrödinger 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.

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Gregory Berkolaiko, Graham Cox, Jeremy L. Marzuola. 2024-02-13. Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map. https://doi.org/10.1007/s11005-019-01159-x

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