Variational approach to the time-dependent Schrödinger-Newton equations

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 Schrödinger-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 Schrödinger-Newton equations.

14 pages, 7 figures

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• Models of dark matter halos based on statistical mechanics: I. The classical King model
• Relativistic self-gravitating Bose-Einstein condensates and cold baryons with a stiff equation of state
• Jeans type instability of a complex self-interacting scalar field in general relativity
• Jeans mass-radius relation of self-gravitating Bose-Einstein condensates and typical parameters of the dark matter particle
• The Schrödinger-Newton equations beyond Newton
• Short distance modification of the quantum virial theorem
• Modification of Schrödinger-Newton equation due to braneworld models with minimal length
• Quantum tunneling rate of dilute axion stars close to the maximum mass
• The Schrodinger-Newton System with Self-Field Coupling
• Relativistic effects on the Schrödinger-Newton equation
• Short Distance Modification of a Gravitational System and its Optical Analog
• Series solution of the time-dependent Schrödinger-Newton equations in the presence of dark energy via the Adomian Decomposition Method