The Point Spectrum Of Periodic Quantum Trees

TL;DR

This paper investigates the eigenvalues of periodic quantum trees with Schrödinger operators, establishing analogs of discrete results, defining the density of states, and showing conditions under which the point spectrum is empty after small adjustments.

Contribution

It extends discrete spectral results to quantum trees, introduces the density of states measure, and analyzes the impact of edge length adjustments on the spectrum.

Findings

Regular quantum periodic trees can have eigenvalues.

The density of states measure is defined for these trees.

Small adjustments to edge lengths can eliminate the point spectrum.

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.