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The Schr\"odinger-Newton equation is a proposed model to explain thelocalization of macroscopic particles by suppressing quantum dispersion withthe particle's own gravitational attraction. On cosmic scales, however, darkenergy also acts repulsively, as witnessed by the accelerating rate ofuniversal expansion. Here, we introduce the effects of dark energy in the formof a cosmological constant $\Lambda$, that drives the late-time acceleration ofthe Universe, into the Schr\"odinger-Newton approach. We then ask in whichregime dark energy dominates both canonical quantum diffusion and gravitationalself-attraction. It turns out that this happens for sufficiently delocalizedobjects with an arbitrary mass and that there exists a minimal delocalizationwidth of about $67$ m. While extremely macroscopic from a quantum perspective,the value is in principle accessible to laboratories on Earth. Hence, weanalyze, numerically, how the dynamics of an initially spherical Gaussian wavepacket is modified in the presence of $\Lambda > 0$. A notable feature is thegravitational collapse of part of the wave packet, in the core region close tothe center of mass, accompanied by the accelerated expansion of the moredistant shell surrounding it. The order of magnitude of the distance separatingcollapse from expansion matches analytical estimates of the classicalturnaround radius for a spherically symmetric body in the presence of darkenergy. However, the time required to observe these modifications isastronomical. They can potentially be measured only in physical systemssimulating a high effective cosmological constant, or, possibly, via theireffects on the inflationary universe.

Dark energy effects in the Schr\"odinger-Newton approach