  We analyze a dynamics of ultracold neutrons (UCNs) caused by interactions
violating Lorentz invariance within the Standard Model Extension (SME)
(Colladay and Kostelecky, Phys. Rev. D55, 6760 (1997) and Kostelecky, Phys.
Rev. D69, 105009 (2004)). We use the effective non-relativistic potential for
interactions violating Lorentz invariance derived by Kostelecky and Lane (J.
Math. Phys. 40, 6245 (1999)) and calculate contributions of these interactions
to the transition frequencies of transitions between quantum gravitational
states of UCNs bouncing in the gravitational field of the Earth. Using the
experimental sensitivity of qBounce experiments we make some estimates of upper
bounds of parameters of Lorentz invariance violation in the neutron sector of
the SME which can serve as a theoretical basis for an experimental analysis. We
show that an experimental analysis of transition frequencies of transitions
between quantum gravitational states of unpolarized and polarized UCNs should
allow to place some new constraints in comparison to the results adduced by
Kostelecky and Russell in Rev. Mod. Phys. 83, 11 (2011); edition 2019, arXiv:
0801.0287v12 [hep-ph].
