Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2018
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| _version_ | 1866910329207783424 |
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| author | Berkolaiko, Gregory Cox, Graham Marzuola, Jeremy L. |
| author_facet | Berkolaiko, Gregory Cox, Graham Marzuola, Jeremy L. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1805_03145 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map Berkolaiko, Gregory Cox, Graham Marzuola, Jeremy L. Analysis of PDEs Spectral Theory 35P05 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. |
| title | Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map |
| topic | Analysis of PDEs Spectral Theory 35P05 |
| url | https://arxiv.org/abs/1805.03145 |