  The ground-state properties of attractive bosons trapped in a ring lattice
including a single attractive potential well with an adjustable depth are
investigated. The energy spectrum is reconstructed both in the
strong-interaction limit and in the superfluid regime within the Bogoliubov
picture. The analytical results thus obtained are compared with those found
numerically from the exact Hamiltonian, in order to identify the regions in the
parameter space where this picture is effective. The single potential
introduced is the simplest way to break the translational symmetry and to
observe, through a completely analytical approach, how the absence of symmetry
affects the properties of the low-excited eigenstates of the system. This model
gives a first insight into the properties of systems including more complex
potentials.
