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

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.