Computing Nodal Deficiency with a Refined Dirichlet-to-Neumann Map
(2022) In Journal of Geometric Analysis 32(10).- 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 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, the results can be used in the general study of spectral minimal partitions, not just nodal... (More)
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 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, the results can be used in the general study of spectral minimal partitions, not just nodal partitions of generic Laplacian eigenfunctions.
(Less)
- author
- Berkolaiko, G. ; Cox, G. ; Helffer, B. and Sundqvist, M. P. LU
- organization
- publishing date
- 2022-10
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Dirichlet-to-Neumann operators, Minimal partitions, Nodal deficiency, Spectral flow
- in
- Journal of Geometric Analysis
- volume
- 32
- issue
- 10
- article number
- 246
- publisher
- Springer
- external identifiers
-
- scopus:85134602110
- ISSN
- 1050-6926
- DOI
- 10.1007/s12220-022-00984-2
- language
- English
- LU publication?
- yes
- id
- 77a5dc8b-081e-4075-8dcc-9476df0fb791
- date added to LUP
- 2022-09-05 14:32:13
- date last changed
- 2022-09-05 14:32:13
@article{77a5dc8b-081e-4075-8dcc-9476df0fb791,
abstract = {{<p>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 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, the results can be used in the general study of spectral minimal partitions, not just nodal partitions of generic Laplacian eigenfunctions.</p>}},
author = {{Berkolaiko, G. and Cox, G. and Helffer, B. and Sundqvist, M. P.}},
issn = {{1050-6926}},
keywords = {{Dirichlet-to-Neumann operators; Minimal partitions; Nodal deficiency; Spectral flow}},
language = {{eng}},
number = {{10}},
publisher = {{Springer}},
series = {{Journal of Geometric Analysis}},
title = {{Computing Nodal Deficiency with a Refined Dirichlet-to-Neumann Map}},
url = {{http://dx.doi.org/10.1007/s12220-022-00984-2}},
doi = {{10.1007/s12220-022-00984-2}},
volume = {{32}},
year = {{2022}},
}