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

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 $-\mathrm{\infty }$, a narrow cluster of finitely many eigenvalues tends to $-\mathrm{\infty }$, while the eigenvalues above the cluster remain bounded from below. Certain “rogue” eigenvalues break away from this cluster and tend even faster toward $-\mathrm{\infty }$. 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.