  The Schr\"{o}dinger-Newton system is a nonlinear system obtained by coupling
together the linear Schr\"{o}dinger 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\"{o}dinger-Newton system, by modifying the Poisson
equation through the addition of a new term. The corresponding
Schr\"{o}dinger-Newton-$\Lambda$ 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.
