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

TL;DR

This paper analyzes the spectrum of Schrödinger operators on finite regular metric trees, revealing how eigenvalues behave as the Robin vertex parameter varies, including the emergence of rogue eigenvalues that diverge rapidly.

Contribution

It introduces a graphical framework linking the spectrum to orthogonal polynomials and describes the asymptotic behavior of eigenvalues, including rogue eigenvalues, as the Robin parameter tends to .

Findings

Eigenvalues tend to  as the Robin parameter .

Rogue eigenvalues diverge faster than the main cluster as the Robin parameter .

Spectrum visualization as intersections of spiral and other curves depends on tree parameters.

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.