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arXiv · 2006.12377

Spectra of Regular Quantum Trees: Rogue Eigenvalues and Dependence on Vertex Condition

Zhaoxia W. Hess·Stephen P. Shipman
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Abstract

We investigate the spectrum of Schr\"odinger operators on finite regular metric trees through a relation to orthogonal polynomials that provides a graphical perspective. As the Robin vertex parameter tends to $-\infty$, a narrow cluster of finitely many eigenvalues tends to $-\infty$, while the eigenvalues above the cluster remain bounded from below. Certain "rogue" eigenvalues break away from this cluster and tend even faster toward $-\infty$. The spectrum can be visualized as the intersection points of two objects in the plane--a spiral curve depending on the Schr\"odinger potential, and a set of curves depending on the branching factor, the diameter of the tree, and the Robin parameter.

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Zhaoxia W. Hess, Stephen P. Shipman. 2020-06-22. Spectra of Regular Quantum Trees: Rogue Eigenvalues and Dependence on Vertex Condition. https://doi.org/10.1007/s00023-021-01035-2

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