  The variation of the dimensionless fundamental physical constant mu=m_p/m_e
can be checked through observation of Lyman and Werner lines of molecular
hydrogen in the spectra of distant QSOs. Our intention is to asses the accuracy
of the investigation concerning a possible variation of mu and to provide more
robust results for QSO 0347-383. The goal in mind is to resolve the current
controversy on variation of mu and devise explanations for the different
findings. We achieve this not by another single result but by providing
alternative approaches to the problem. An analysis based on independent data
sets of QSO 0347-383 is put forward and new approaches for some of the steps
involved in the data analysis are introduced. We analyse two independent sets
of observations of the same absorption system and for the first time we apply
corrections for the observed offsets between discrete spectra Drawing on two
independent observations of a single absorption system in QSO 0347-383 our
detailed analysis yields dmu/mu = 15 +/- (9_stat + 6_sys) x 10^{-6} at
z_abs=3.025. Based on the overall goodness-of-fit we estimate the limit of
accuracy to about 300 m/s, consisting of roughly 180 m/s due to the uncertainty
of the fit and about 120 m/s allocated to systematics This work presents
alternative approaches to handle systematics and introduces methods required
for precision analysis of QSO spectra available in the near future.
