Spectra of Quantum Trees and Orthogonal Polynomials

Abstract

We investigate the spectrum of regular quantum-graph trees, where the edges are endowed with a Schr\"odinger operator with self-adjoint Robin vertex conditions. It is known that, for large eigenvalues, the Robin spectrum approaches the Neumann spectrum. In this research, we compute the lower Robin spectrum. The spectrum can be obtained from the roots of a sequence of orthogonal polynomials involving two variables. As the length of the quantum tree increases, the spectrum approaches a band-gap structure. We find that the lowest band tends to minus infinity as the Robin parameter increases, whereas the rest of the bands remain positive. Unexpectedly, we find that two groups of isolated negative eigenvalues separate from the bottom of the lowest band. These eigenvalues are computed as they depend asymptotically on the Robin parameter. Our analysis invokes the interlacing property of orthogonal polynomials.