  Within the self-gravitating Bose-Einstein condensate (BEC) model of dark
matter (DM), we argue that the axionlike self-interaction of ultralight bosons
ensures the existence of both rarefied and dense phases in the DM halo core of
(dwarf) galaxies. In fact, this stems from two independent solutions of the
Gross-Pitaevskii equation corresponding to the same model parameters. For a
small number of particles, this structure disappears along with the
gravitational interaction, and the Gross-Pitaevskii equation reduces to the
stationary sine-Gordon equation, the one-dimensional antikink solution of which
mimics a single-phase DM radial distribution in the halo core. Quantum
mechanically, this solution corresponds to a zero-energy bound state of two
particles in a closed scattering channel formed by the domain-wall potential
with a finite asymptotics. To produce a two-particle composite with low
positive energy and a finite lifetime, we appeal to the resonant transition of
one asymptotically free particle of a pair from an open channel (with a model
scattering potential) to the closed channel. Using the Feshbach resonance
concept, the problem of two-channel quantum mechanics is solved in the presence
of a small external influence which couples the two channels, and an analytical
solution is obtained in the first approximation. Analyzing the dependence of
scattering data on interaction parameters, we reveal a long-lived two-particle
composite (dimer) possessing a lifetime of millions of years. This result is
rather surprising and supposes important implications of dimers' being involved
in forming large DM structures. It is shown that the dimers' appearance is
related with the regime of infinite scattering length due to resonance. The
revealed dependence of the DM scattering length $a$ on the parameters of
interactions can theoretically justify variation of $a$ in the DM dominated
galaxies.

Astrophysics of Galaxies;General Relativity and Quantum Cosmology;High Energy Physics - Theory;Quantum Physics