The Point Spectrum Of Periodic Quantum Trees

Abstract

We study the point spectrum of a periodic quantum tree equipped with a Schr\"odinger type differential operator with delta-type vertex conditions, using subsets of the compact graph that generates the tree. We prove analogs of existing discrete results concerning the eigenvalues of such operators (see Aomoto, 1991 and see Banks, Garza-Vargas and Mukherjee, 2022). In particular, we define the density of states measure and find the measure of eigenvalues of the periodic tree. While most results carry over from the discrete case, a notable difference between the continuum and discrete cases is that a \textbf{regular} quantum periodic tree may have eigenvalues. We prove that after an arbitrarily small adjustment of edge lengths, the point spectrum of the universal cover of a compact quantum graph, with at least one cycle and the standard Schr\"odinger operator, is empty.