Constraints for quasi-elastic neutron reflections in the range dE~10^(-12)-10^(-9) eV

A restrictive constraint for any quasi-elastic process was obtained in the previously inaccessible energy range dE~10^(-12)-3x10^(-10) eV for reflections of ultracold neutrons from surfaces in the experiement on neutron quantum states in the earth's gravitational field. This could be useful for precision neutron spectrometry experiments and for the verification of extensions of quantum mechanics.