  In this paper we investigate, within the standard model extension framework,
the influence of Lorentz- and CPT-violating terms on gravitational quantum
states of ultracold neutrons. Using a semiclassical wave packet, we derive the
effective nonrelativistic Hamiltonian which describes the neutrons vertical
motion by averaging the contributions from the perpendicular coordinates to the
free falling axis. We compute the physical implications of the Lorentz- and
CPT-violating terms on the spectra. The comparison of our results with those
obtained in the GRANIT experiment leads to an upper bound for the
symmetries-violation $c_{\mu\nu} ^{n}$ coefficients. We find that ultracold
neutrons are sensitive to the $a _{i} ^{n}$ and $e _{i} ^{n}$ coefficients,
which thus far are unbounded by experiments in the neutron sector. We propose
two additional problems involving ultracold neutrons which could be relevant
for improving our current bounds; namely, gravity-resonance-spectroscopy and
neutron whispering gallery wave.
