Stability of eigenvalues of quantum graphs with respect to magnetic perturbation and the nodal count of the eigenfunctions

TL;DR

This paper establishes a relationship between the zeros of eigenfunctions and the stability of eigenvalues under magnetic perturbations on quantum graphs, extending the magnetic nodal theorem.

Contribution

It introduces an analogue of the magnetic nodal theorem for quantum graphs, linking eigenfunction zeros to eigenvalue stability under magnetic perturbations.

Findings

Eigenfunction zeros relate to the Morse index of eigenvalues.

The Morse index at zero magnetic field equals the number of zeros minus (n-1).

Provides a new understanding of spectral stability in quantum graphs.

Abstract

We prove an analogue of the magnetic nodal theorem on quantum graphs: the number of zeros $\phi$ of the $n$-th eigenfunction of the Schr\"odinger operator on a quantum graph is related to the stability of the $n$-th eigenvalue of the perturbation of the operator by magnetic potential. More precisely, we consider the $n$-th eigenvalue as a function of the magnetic perturbation and show that its Morse index at zero magnetic field is equal to $\phi - (n-1)$.