Interlacing Property of Zeros of Eigenvectors of Schr\"odinger Operators on Trees

TL;DR

This paper proves an interlacing property of zeros of eigenvectors of Schrödinger operators on finite trees, extending Courant's theorem to a new setting with specific vertex conditions.

Contribution

It establishes an analogue of Courant's theorem for eigenfunctions of Schrödinger operators on trees, with conditions on zero vertices' degrees.

Findings

Zeros of eigenvectors exhibit interlacing property on trees.

The result applies to eigenvectors ordered by increasing eigenvalues.

Zeros are constrained by degree conditions at vertices.

Abstract

We prove an analogue for trees of Courant's theorem on the interlacing property of zeros of eigenfunctions of a Schr\"{o}dinger operator. Let $\Gamma$ be a finite tree, and $\mathcal A$ a Schr\"{o}dinger operator on $\Gamma$. If the eigenvectors of $\mathcal A$ are ordered according to increasing eigenvalues, and the vertices corresponding to zero coordinates are of degree at most two, then the zeros of the linear extensions of eigenvectors have the interlacing property.