  We study the Hamiltonian for a system of N identical bosons interacting with
an impurity, i.e., a different particle, via zero-range forces in dimension
three. It is well known that, following the standard approach, one obtains the
Ter-Martirosyan Skornyakov Hamiltonian which is unbounded from below. In order
to avoid such instability problem, we introduce a three-body force acting at
short distances. The effect of this force is to reduce to zero the strength of
the zero-range interaction between two particles, i.e., the impurity and a
boson, when another boson approaches the common position of the first two
particles. We show that the Hamiltonian defined with such regularized
interaction is self-adjoint and bounded from below if the strength of the
three-body force is sufficiently large. The method of the proof is based on a
careful analysis of the corresponding quadratic form.
