Variational approach to the time-dependent Schr\"odinger-Newton equations

TL;DR

This paper introduces a variational method using Gaussian trial functions to simplify the complex time-dependent Schr"odinger-Newton equations, providing insights into ground states, energies, and long-term dynamics with minimal computational effort.

Contribution

It develops a simplified dynamical system via a variational approach that accurately models key features of the Schr"odinger-Newton equations, offering an efficient alternative to full simulations.

Findings

Model agrees well with full numerical simulations

Provides estimates of ground state density and energy

Captures long-time nonlinear behavior

Abstract

Using a variational approach based on a Lagrangian formulation and Gaussian trial functions, we derive a simple dynamical system that captures the main features of the time-dependent Schr\"odinger-Newton equations. With little analytical or numerical effort, the model furnishes information on the ground state density and energy eigenvalue, the linear frequencies, as well as the nonlinear long-time behaviour. Our results are in good agreement with those obtained through analytical estimates or numerical simulations of the full Schr\"odinger-Newton equations.