  In this work, the CPT-violating (CPTV) interactions on neutrons'
gravitational bound state are studied. With simple analytical solutions, we
provide a preliminary investigation on the Lorentz-violation (LV) induced spin
precession due to the $\vec{\sigma}\cdot\vec{\tilde{b}}(1+gz)$ and
$\bar{b}/m_{_I}\vec{\sigma}\cdot\hat{\vec{p}}$ couplings, where
$\vec{\tilde{b}}$ and $\bar{b}$ represent LV coefficients. The
helicity-dependent couplings can induce unusual phase evolutions with position
and momentum dependence. As $\vec{\tilde{b}}$ varies with time due to the
Earth's motion, the spin polarization also shows a sidereal time dependence,
and it may be enhanced with time for ultra-stable polarized state of neutrons.
The inseparability of the spin-momentum coupling of the $\bar{b}$-term can also
lead to motional dependent polarization state. With the precisely measured
transition frequency between different gravitational bound states, we get a
rough bound $|\vec{\tilde{b}}|<3.9\times10^{-3}$GeV for unpolarized neutrons.
If the spin-flip transition frequency can reach comparable precision in the
future, the bound can be improved to the level of $10^{-24}$GeV. The test of
weak equivalence principle with polarized atom may also improve it
significantly.
