Remarks on Periodic Jacobi Matrices on Trees

TL;DR

This paper investigates spectral properties of periodic Jacobi matrices on trees, establishing bounds on spectral gaps and exploring conjectures related to antibound states and specific models.

Contribution

It provides new bounds on spectral gaps for Jacobi matrices on trees and offers insights into antibound states and the rg-model, extending spectral theory in this context.

Findings

Bounds on spectral gaps for Jacobi matrices on trees

Conjectures about antibound states

Observations on the rg-model

Abstract

We look at periodic Jacobi matrices on trees. We provide upper and lower bounds on the gap of such operators analogous to the well known gap in the spectrum of the Laplacian on the upper half-plane with hyperbolic metric. We make some conjectures about antibound states and make an interesting observation for what [3] calls the rg-model.