Possible evidence for a variable fine structure constant from QSO absorption lines: systematic errors

Comparison of quasar absorption spectra with laboratory spectra allow us to probe possible variations in the fundamental constants over cosmological time-scales. In a companion paper we present an analysis of Keck/HIRES spectra and report possible evidence suggesting that the fine structure constant, alpha, may have been smaller in the past: da/a = (-0.72 +/- 0.18) * 10^{-5} over the redshift range 0.5 < z < 3.5. In this paper we describe a comprehensive investigation into possible systematic effects. Most of these do not significantly influence our results. When we correct for those which do produce a significant systematic effect in the data, the deviation of da/a from zero becomes more significant. We are lead increasingly to the interpretation that alpha was slightly smaller in the past.