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

We investigate the spectrum of Schrodinger 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 Schrodinger potential, and a set of curves depending on the branching factor, the diameter of the tree, and the Robin parameter.