Interlacing Property of Zeros of Eigenvectors of Schrödinger Operators on Trees

Details Interlacing Property of Zeros of Eigenvectors of Schrödinger Operators on Trees Interlacing Property of Zeros of Eigenvectors of Schrödinger Operators on Trees

2009-04-08

arxiv.org/abs/0904.1336v1

Mathematics

Combinatorics

7 pages, 1 figure

Scientific paper

We prove an analogue for trees of Courant's theorem on nodes 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 eigenvalues of $\mathcal A$ are all simple, and ordered according to increasing eigenvalues, then the $n$-th eigenvector has exactly $n$ nodal domains, and the zeros of eigenvectors have the interlacing property.

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