Exact solutions of the newton-schrödinger equation, infinite derivative gravity and schwarzschild atoms

Abstract

Exact solutions to the stationary spherically symmetric Newton-Schrödinger 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 generalizing the Newton-Schrödinger 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-Schrödinger 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 but the real-valued probability density . 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.