The Schrodinger-Newton System with Self-Field Coupling

We study the Schrodinger-Newton system of equations with the addition of gravitational field energy sourcing - such additional nonlinearity is to be expected from a theory of gravity (like general relativity), and its appearance in this simplified scalar setting (one of Einstein's precursors to general relativity) leads to significant changes in the spectrum of the self-gravitating theory. Using an iterative technique, we compare the mass dependence of the ground state energies of both Schrodinger-Newton and the new, self-sourced system and find that they are dramatically different. The Bohr method approach from old quantization provides a qualitative description of the difference, which comes from the additional nonlinearity introduced in the self-sourced case. In addition to comparison of ground state energies, we calculate the transition energy between the ground state and first excited state to compare emission frequencies between Schrodinger-Newton and the self-coupled scalar case.