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[en] We propose experiments with atomic hydrogen gas at ultra-low temperatures T < 100μK when the thermal energy of atoms is comparable with the changes of their potential energy in the Earth gravity field. At these conditions we suggest implementing a gravitational spectroscopy for studies of quantum properties of ultra-cold atomic hydrogen and its interactions with matter and gravity, similar to experiments with ultra-cold neutrons (Nesvizhevsky et al. Nature 415, 297 2002). A magnetic trap used for reaching the Bose-Einstein Condensation (Fried et al. Phys. Rev. Lett. 81, 3811 1998) can be used for cooling a large number of H atoms below 1 mK. Evaporative cooling over the trap barrier allows effective cooling of the vertical degree of freedom of the trapped atoms. Releasing these ultra-slow atoms from the trap onto the cold surface of superfluid helium will allow studies of quantum bounces and stationary gravitational states of H atoms in the potential well created by this surface and the field of Earth gravity. Experimental study of properties of gravitational quantum states of hydrogen and quantum reflection of ultracold hydrogen from surface would be of major importance for designing similar experiments with antihydrogen, which are currently prepared in CERN.