  The Schr\"odinger-Newton equation is a proposed model to explain the
localization of macroscopic particles by suppressing quantum dispersion with
the particle's own gravitational attraction. On cosmic scales, however, dark
energy also acts repulsively, as witnessed by the accelerating rate of
universal expansion. Here, we introduce the effects of dark energy in the form
of a cosmological constant $\Lambda$, that drives the late-time acceleration of
the Universe, into the Schr\"odinger-Newton approach. We then ask in which
regime dark energy dominates both canonical quantum diffusion and gravitational
self-attraction. It turns out that this happens for sufficiently delocalized
objects with an arbitrary mass and that there exists a minimal delocalization
width of about $67$ m. While extremely macroscopic from a quantum perspective,
the value is in principle accessible to laboratories on Earth. Hence, we
analyze, numerically, how the dynamics of an initially spherical Gaussian wave
packet is modified in the presence of $\Lambda > 0$. A notable feature is the
gravitational collapse of part of the wave packet, in the core region close to
the center of mass, accompanied by the accelerated expansion of the more
distant shell surrounding it. The order of magnitude of the distance separating
collapse from expansion matches analytical estimates of the classical
turnaround radius for a spherically symmetric body in the presence of dark
energy. However, the time required to observe these modifications is
astronomical. They can potentially be measured only in physical systems
simulating a high effective cosmological constant, or, possibly, via their
effects on the inflationary universe.
