Quantum Cayley graphs for free groups

TL;DR

This paper develops a spectral analysis framework for Schrödinger operators on metric Cayley graphs of free groups with variable potentials and edge lengths, introducing new multipliers to characterize the spectrum.

Contribution

It introduces novel methods to construct multipliers depending on the spectral parameter, enabling spectrum characterization of Schrödinger operators on these graphs.

Findings

Derived a set of M multipliers depending on the spectral parameter

Constructed the resolvent for Schrödinger operators on Cayley graphs

Characterized the spectrum of the operators using the multipliers

Abstract

Differential operators of Schrodinger type are considered on metric Cayley graphs of free groups with a minimal set of M generators. The potential and edge lengths may vary with the M edge types. Using novel methods, a set of M multipliers depending on the spectral parameter is found. These multipliers are used to construct the resolvent and characterize the spectrum