  As-yet undiscovered light bosons may constitute all or part of the dark
matter (DM) of our Universe, and are expected to have (weak) self-interactions.
We show that the quartic self-interactions generically induce the capture of
dark matter from the surrounding halo by external gravitational potentials such
as those of stars, including the Sun. This leads to the subsequent formation of
dark matter bound states supported by such external potentials, resembling
gravitational atoms (e.g. a solar halo around our own Sun). Their growth is
governed by the ratio $\xi_{\rm foc} \equiv \lambda_{\rm dB}/R_\star$ between
the de Broglie wavelength of the incoming DM waves, $\lambda_{\rm dB}$, and the
radius of the ground state $R_\star$. For $\xi_{\rm foc}\lesssim 1$, the
gravitational atom grows to an (underdense) steady state that balances the
capture of particles and the inverse (stripping) process. For $\xi_{\rm
foc}\gtrsim 1$, a significant gravitational-focusing effect leads to
exponential accumulation of mass from the galactic DM halo into the
gravitational atom. For instance, a dark matter axion with mass of the order of
$10^{-14}$ eV and decay constant between $10^{7}$ and $10^8$ GeV would form a
dense halo around the Sun on a timescale comparable to the lifetime of the
Solar System, leading to a local DM density at the position of the Earth
$\mathcal{O}(10^4)$ times larger than that expected in the standard halo model.
For attractive self-interactions, after its formation, the gravitational atom
is destabilized at a large density, which leads to its collapse; this is likely
to be accompanied by emission of relativistic bosons (a `Bosenova').
