The effects of the dark energy on the static Schr\"{o}dinger-Newton system -- an Adomian Decomposition Method and Pad\'{e} approximants based approach

TL;DR

This paper investigates how dark energy influences the Schr"{o}dinger-Newton system by applying the Adomian Decomposition Method and Padé approximants to obtain semianalytical solutions, comparing them with numerical results.

Contribution

It introduces a semianalytical approach combining Adomian Decomposition and Padé approximants to study the Schr"{o}dinger-Newton-$ ext{Lambda}$ system affected by dark energy.

Findings

Dark energy modifies the structure of the quantum system.

Semianalytical solutions closely match numerical results.

The approach effectively handles nonlinear quantum-gravitational equations.

Abstract

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.