Quantum Cayley graphs for free groups

Abstract

Dierential operators $\Delta +q$ are considered on metric Cayley graphs of the nitely generated free groups ${\mathbb{F}}_M$. The function $q$ and the graph edge lengths may vary with the $M$ edge types. Using novel methods, a set of $M$ multipliers ${\mu }_m(\lambda )$ depending on the spectral parameter is found. These multipliers are used to construct the resolvent and characterize the spectrum.