  We study the corrections induced by the theory of non-commutativity, in both
space-space and space-time versions, on the spectrum of hydrogen-like atoms.
For this, we use the relativistic theory of two-particle systems to take into
account the effects of the reduced mass, and we use perturbation methods to
study the effects of non-commutativity.We apply our study to the muon hydrogen
with the aim to solve the puzzle of proton radius [R. Pohl et al., Nature 466,
213 (2010) and A. Antognini et al., Science 339, 417 (2013)]. The shifts in the
spectrum are found more noticeable in muon H (muH) than in electron H (eH)
because the corrections depend on the mass to the third power; This explains
the discrepancy between muH and eH results. In space-space non-commutativity,
the parameter required to resolve the puzzle Theta(ss) (0.35GeV)-2, exceeds the
limit obtained for this parameter from various studies on eH Lamb shift. For
space-time non-commutativity, the value Theta(st) (14.3GeV)-2 has been obtained
and it is in agreement with the limit determined by Lamb shift spectroscopy in
eH. We have also found that this value fills the gap between theory and
experiment in the case of muD and improves the agreement between theoretical
and experimental values in the case of hydrogen-deuterium isotope shift.
