  We consider the problem of a fixed impurity coupled to a small number $N$ of
non-interacting bosons. We focus on impurity-boson interactions that are
mediated by a closed-channel molecule, as is the case for tuneable interatomic
interactions in cold-atom experiments. We show that this two-channel model can
be mapped to a boson model with effective boson-boson repulsion, which enables
us to solve the three-body ($N=2$) problem analytically and determine the
trimer energy for impurity-boson scattering lengths $a>0$. By analysing the
atom-dimer scattering amplitude, we find a critical scattering length $a^*$ at
which the atom-dimer scattering length diverges and the trimer merges into the
dimer continuum. We furthermore calculate the tetramer energy exactly for $a>0$
and show that the tetramer also merges with the continuum at $a^*$. Indeed,
since the critical point $a^*$ formally resembles the unitary point $1/a = 0$,
we find that all higher-body bound states (involving the impurity and $N>1$
bosons) emerge and disappear at both of these points. We show that the behavior
at these 'multi-body resonances' is universal, since it occurs for any model
with an effective three-body repulsion involving the impurity. Thus, we see
that the fixed-impurity problem is strongly affected by a three-body parameter
even in the absence of the Efimov effect.
