Spectral flow for pair compatible equipartitions

TL;DR

This paper extends a spectral flow method to analyze spectral equipartitions satisfying pair compatibility, broadening its applicability beyond nodal partitions to include spectral minimal partitions and others.

Contribution

It generalizes the spectral flow approach for nodal deficiency analysis to more general pair compatible spectral equipartitions, including minimal partitions.

Findings

Spectral flow approach can be extended to general equipartitions.

Multiple methods for Dirichlet-to-Neumann operators are discussed.

The framework applies to nodal and spectral minimal partitions.

Abstract

We show that a recent spectral flow approach proposed by Berkolaiko-Cox-Marzuola for analyzing the nodal deficiency of the nodal partition associated to an eigenfunction can be extended to more general partitions. To be more precise, we work with spectral equipartitions that satisfy a pair compatible condition. Nodal partitions and spectral minimal partitions are examples of such partitions. Along the way, we discuss different approaches to the Dirichlet-to-Neumann operators: via Aharonov-Bohm operators, via a double covering argument, and via a slitting of the domain.