Series solution of the time-dependent Schr\"{o}dinger-Newton equations in the presence of dark energy via the Adomian Decomposition Method

TL;DR

This paper develops a semianalytical series solution for the Schr"{o}dinger-Newton equations with dark energy effects using the Adomian Decomposition Method, providing accurate results efficiently.

Contribution

It introduces the first application of the Adomian Decomposition Method to solve the Schr"{o}dinger-Newton-$ m extLambda$ system with dark energy effects.

Findings

Series solutions are consistent with numerical results.

Dark energy influences the system's behavior significantly.

Method offers quick, accurate solutions with minimal computational resources.

Abstract

The Schr\"{o}dinger-Newton model is a nonlinear system obtained by coupling the linear Schr\"{o}dinger equation of canonical quantum mechanics with the Poisson equation of Newtonian mechanics. In this paper we investigate the effects of dark energy on the time-dependent Schr\"{o}dinger-Newton equations by including a new source term with energy density $\rho_{\Lambda} = \Lambda c^2/(8\pi G)$, where $\Lambda$ is the cosmological constant, in addition to the particle-mass source term $\rho_m = m|\psi|^2$. The resulting Schr\"{o}dinger-Newton-$\Lambda$ (S-N-$\Lambda$) system cannot be solved exactly, in closed form, and one must resort to either numerical or semianalytical (i.e., series) solution methods. We apply the Adomian Decomposition Method, a very powerful method for solving a large class of nonlinear ordinary and partial differential equations, to obtain accurate series solutions of the S-N-$\Lambda$ system, for the first time. The dark energy dominated regime is also investigated in detail. We then compare our results to existing numerical solutions and analytical estimates, and show that they are consistent with previous findings. Finally, we outline the advantages of using the Adomian Decomposition Method, which allows accurate solutions of the S-N-$\Lambda$ system to be obtained quickly, even with minimal computational resources.