Additional numerical and graphical evidence to support some Conjectures on discrete Schr\"odinger operators with a more general long range condition

TL;DR

This paper provides additional numerical and graphical evidence supporting conjectures related to discrete Schrödinger operators with long-range conditions, focusing on specific parameters in two-dimensional cases.

Contribution

It offers new numerical data and insights into the sets of threshold energies for certain conjectures in the context of discrete Schrödinger operators.

Findings

More evidence for $oldsymbol{ heta}_{ ext{threshold}}$ sets at $oldsymbol{ heta}=3,4$ in 2D.

Additional threshold energies identified for $oldsymbol{ heta}=3$ in 2D.

Enhanced understanding of the structure of $oldsymbol{ heta}_{ ext{sets}}$ in specific cases.

Abstract

This document contains additional numerical and graphical evidence to support some of the conjectures mentioned in \cite{GM3}. We give more evidence for $\kappa=3,4$ in dimension 2. As mentioned in that article we still don't quite understand the sets $\boldsymbol{\mu}_{\kappa}(\Delta)$ and $\boldsymbol{\Theta}_{\kappa}(\Delta)$ on $(0,1/2)$ for $\kappa=3$ in dimension 2. Here we give a bunch more threshold energies for $\kappa = 3$ in dimension 2.