Series solution of the time-dependent Schrödinger-Newton equations in the presence of dark energy via the Adomian Decomposition Method

Abstract

The Schrödinger-Newton model is a nonlinear system obtained by coupling the linear Schrödinger 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 Schrödinger-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 Schrödinger-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.

arXiv commentExpanded version of the published work: Symmetry 2023, 15(2), 372. 20 pages, 1 table, 8 figures

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• A Brief Introduction to the Adomian Decomposition Method, with Applications in Astronomy and Astrophysics
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