The effects of the dark energy on the static Schr\"odinger-Newton system -- an Adomian Decomposition Method and Pad\'e approximants based approach
Abstract
The Schr\"odinger-Newton system is a nonlinear system obtained by coupling together the linear Schr\"odinger equation of quantum mechanics with the Poisson equation of Newtonian mechanics. In the present work we will investigate the effects of a cosmological constant (dark energy or vacuum fluctuation) on the Schr\"odinger-Newton system, by modifying the Poisson equation through the addition of a new term. The corresponding Schr\"odinger-Newton- system cannot be solved exactly, and therefore for its study one must resort to either numerical or semianalytical methods. In order to obtain a semianalytical solution of the system we apply the Adomian Decomposition Method, a very powerful method used for solving a large class of nonlinear ordinary and partial differential equations. Moreover, the Adomian series are transformed into rational functions by using the Pad\'e approximants. The semianalytical approximation is compared with the full numerical solution, and the effects of the dark energy on the structure of the Newtonian quantum system are investigated in detail.