Magnetic interpretation of the nodal defect on graphs

TL;DR

This paper provides a natural proof linking the Courant nodal defect of a Schrödinger operator on a finite graph to a Morse index, using magnetic field deformations, building on recent and classical results.

Contribution

It offers a new, natural proof of a recent result connecting the nodal defect to Morse index via magnetic deformations, inspired by classical work.

Findings

Nodal defect interpreted as Morse index

Magnetic deformations relate to spectral properties

Provides a natural proof of a recent theorem

Abstract

In this note, we present a natural proof of a recent and surprising result of Gregory Berkolaiko (arXiv 1110.5373) interpreting the "Courant nodal defect" of a Schr\"odinger operator on a finite graph as a Morse index associated to the deformations of the operator by switching on a magnetic field. This proof is inspired by a nice paper of Miroslav Fiedler published in 1975.