Quasi Kepler's third law for quantum many-body systems

The Kepler's third law is a relation between the period and the energy of two classical particles interacting via a gravitational potential. Recent works showed that this law could be extended, at least approximately, to classical three-body systems, or even many-body classical systems. So, a classical quasi Kepler's third law seems to exist. In this paper, approximate analytical solutions are computed for quantum self-gravitating particles with different masses. The results give strong indications in favor of the existence of a quasi Kepler's third law for such systems. The relevance of the proposal is checked with accurate numerical data for the ground state of self-gravitating identical bosons and with numerical estimations for systems with identical particles plus a different one. Connections between the quantum and classical systems are discussed.