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

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