An index theorem for Schr\"odinger operators on metric graphs

TL;DR

This paper establishes a connection between the spectral flow of Schrödinger operators on metric graphs and the Maslov index, providing a new way to analyze eigenvalue variations through symplectic geometry.

Contribution

It introduces a novel index theorem linking spectral flow and Maslov index for Schrödinger operators on metric graphs, and derives an Hadamard-type formula for eigenvalue derivatives.

Findings

Spectral flow equals the Maslov index for these operators.

Derived an Hadamard-type formula for eigenvalue derivatives.

Provided a geometric interpretation of vertex conditions.

Abstract

We show that the spectral flow of a one-parameter family of Schr\"odinger operators on a metric graph is equal to the Maslov index of a path of Lagrangian subspaces describing the vertex conditions. In addition, we derive an Hadamard-type formula for the derivatives of the eigenvalue curves via the Maslov crossing form.