Berry Phase in the Gravitational Quantum Well and the Seiberg-Witten map

We explicitly compute the geometrical Berry phase for the noncommutative gravitational quantum well for different SW maps. We find that they lead to different partial contributions to the Berry phase. For the most general map we obtain that $\mathrm{\Delta }\gamma (S)\sim {\eta }^3$, in a segment S of the path in the configuration space where $\sqrt{\eta }$ is the fundamental momentum scale for the noncommutative gravitational quantum well. For the full closed path, we find, through an explicit computation, that $\gamma (C)=0$. This result is consistent with the fact that physical properties are independent of the SW map and shows that these maps do not introduce degeneracies or level crossing in the noncommutative extensions of the gravitational quantum well.