Effective-Potential Formulation of Schrödinger–Newton Dynamics and the \(m^{11}\) Stiffness Scaling

Abstract

The Schrödinger–Newton (SN) equation is the foundational model of semiclassical gravity. In this work, we develop a Gaussian variational reduction that makes the intrinsic scale-invariant structure of the SN equation analytically explicit, yielding a compact and accurate one-dimensional effective Hamiltonian. This framework maps the phase-space geometry of the SN dynamics and reveals that the dimensionally required \(m^{11}\) stiffness scaling produces a strongly mass-dependent restoring force against wave-packet deformation. Applying this self-gravitational stiffness to levitated optomechanics, we identify a concrete and falsifiable laboratory signature: a mass-dependent, nonlinear upward shift in the mechanical frequency of a trapped nanoparticle.