Gauge-invariant perturbation expansion in powers of electric charge for the density-of-states of a network model for charged-particle motion in a uniform background magnetic flux density

An explicitly gauge-invariant expansion in powers of e/ħ times the magnetic flux density is formally obtained for the density of states [as characterized by the trace of the resolvent Ĝ = (ω−ĥ )⁻¹] of a charged particle moving on a Hermitian quantum network that is embedded in a Euclidean background that supports a uniform magnetic flux density. The explicit expressions, given here up to third order in the flux density, are also valid for the “local trace” (the trace of P̂_iĜ , where P̂_i is the projector on a network node) and do not appear to have been previously given.