The CPT-violating effects on neutron's gravitational bound state

Analytical solutions with effective CPT-violating spin--gravity corrections to the neutron's gravitational bound states are obtained. The helicity-dependent phase evolution due to $\vec{\sigma}\cdot\vec{\tilde{b}}$ and $\vec{\sigma}\cdot\hat{\vec{p}}$ couplings not only leads to spin precession, %shown by the tip of the spin on the Bloch sphere,but also to transition-frequency shifts between different gravitational bound states. Utilizing transition frequencies measured in the qBounce experiment,\cite{GRS} we obtain the rough bound $|\vec{\tilde{b}}|<6.9\times10^{-21}$GeV. Incorporating known systematic errors may lead to more robust and tighter constraints.