Authors: Carlos Castro
Exact solutions to the stationary spherically symmetric Newton-Schroedinger equation are proposed in terms of integrals involving $generalized$ Gaussians. The energy eigenvalues are also obtained in terms of these integrals which agree with the numerical results in the literature. A discussion of infinite-derivative-gravity follows which allows to generalize the Newton-Schroedinger equation by $replacing$ the ordinary Poisson equation with a $modified$ non-local Poisson equation associated with infinite-derivative gravity. We proceed to replace the nonlinear Newton-Schroedinger equation for a non-linear quantum-like Bohm-Poisson equation involving Bohm's quantum potential, and where the fundamental quantity is $no$ longer the wave-function $ \Psi$ but the real-valued probability density $ \rho$. Finally, we discuss how the latter equations reflect a $nonlinear$ $feeding$ loop mechanism between matter and geometry which allows us to envisage a ``Schwarzschild atom" as a spherically symmetric probability cloud of matter which curves the geometry, and in turn, the geometry back-reacts on this matter cloud perturbing its initial distribution over the space, which in turn will affect the geometry, and so forth until static equilibrium is reached.
Comments: 14 Pages. submitted to Physics and Astronomy International Journal
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