Sturm-Hurwitz Theorem for quantum graphs

TL;DR

This paper establishes bounds on the zeros of linear combinations of Schrödinger eigenfunctions on quantum graphs, revealing differences from classical interval and manifold cases, especially in tree graphs.

Contribution

It provides the first bounds for zero counts on quantum graphs and demonstrates that tree graphs can differ from the interval in nodal properties.

Findings

Bounds for zero counts on quantum graphs are established.

Tree graphs can have different nodal counts than the interval.

Examples show sharpness and non-trivial lower bounds.

Abstract

We prove upper and lower bounds for the number of zeroes of linear combinations of Schr\"odinger eigenfunctions on metric (quantum) graphs. These bounds are distinct from both the interval and manifolds. We complement these bounds by giving non-trivial examples for the lower bound as well as sharp examples for the upper bound. In particular, we show that even tree graphs differ from the interval with respect to the nodal count of linear combinations of eigenfunctions. This stands in distinction to previous results which show that all tree graphs have to same eigenfunction nodal count as the interval.