Computing nodal deficiency with a refined Dirichlet-to-Neumann map

TL;DR

This paper introduces a refined Dirichlet-to-Neumann map to better understand the spectral properties and nodal deficiencies of Laplacian eigenfunctions, applicable to complex partitions and eigenvalues.

Contribution

It provides an improved framework for analyzing nodal deficiency using a refined Dirichlet-to-Neumann map, extending previous results to more general settings.

Findings

Enhanced bounds on nodal deficiency for degenerate eigenfunctions

Framework applicable to non-bipartite and non-smooth partitions

Generalized approach for spectral minimal partition studies

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 partitions of generic Laplacian eigenfunctions.