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arXiv · 1601.06225

Simplicity of eigenvalues and non-vanishing of eigenfunctions of a quantum graph

Gregory Berkolaiko·Wen Liu
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Abstract

We prove that after an arbitrarily small adjustment of edge lengths, the spectrum of a compact quantum graph with $δ$-type vertex conditions can be simple. We also show that the eigenfunctions, with the exception of those living entirely on a looping edge, can be made to be non-vanishing on all vertices of the graph. As an application of the above result, we establish that the secular manifold (also called "determinant manifold") of a large family of graphs has exactly two smooth connected components.

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Gregory Berkolaiko, Wen Liu. 2016-06-24. Simplicity of eigenvalues and non-vanishing of eigenfunctions of a quantum graph. https://arxiv.org/abs/1601.06225

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