On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions

TL;DR

This paper links the Courant nodal deficiency in quantum graphs to the Morse index of an energy functional, revealing its role as a measure of instability in graph partitions and providing a new interpretative framework.

Contribution

It introduces a novel interpretation of the Courant nodal deficiency as the Morse index of an energy functional on graph partitions, connecting spectral properties to stability analysis.

Findings

Nodal deficiency equals the Morse index at a critical point.

Critical partitions correspond to eigenfunctions of quantum graphs.

The framework reveals the instability directions in graph partitions.

Abstract

Courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of Laplacian-type operators. In particular, it holds for generic eigenfunctions of quantum graph. The theorem stipulates that, after ordering the eigenvalues as a non decreasing sequence, the number of nodal domains $\nu_n$ of the $n$-th eigenfunction satisfies $n\ge \nu_n$. Here, we provide a new interpretation for the Courant nodal deficiency $d_n = n-\nu_n$ in the case of quantum graphs. It equals the Morse index --- at a critical point --- of an energy functional on a suitably defined space of graph partitions. Thus, the nodal deficiency assumes a previously unknown and profound meaning --- it is the number of unstable directions in the vicinity of the critical point corresponding to the $n$-th eigenfunction. To demonstrate this connection, the space of graph partitions and the energy functional are defined and the corresponding critical partitions are studied in detail.