The influence of the chameleon field potential on transition frequencies of gravitationally bound quantum states of ultra-cold neutrons

We calculate the chameleon field potential for ultracold neutrons, bouncing on top of one or between two neutron mirrors in the gravitational field of the Earth. For the resulting non--linear equations of motion we give approximate analytical solutions and compare them with exact numerical ones for which we propose the analytical fit. The obtained solutions may be used for the quantitative analysis of contributions of a chameleon field to the transition frequencies of quantum states of ultra-cold neutrons bound in the gravitational field of the Earth.