  The formation of bound states involving multiple particles underlies many
interesting quantum physical phenomena, such as Efimov physics or
superconductivity. In this work we show the existence of an infinite number of
such states for some boson impurity models. They describe free bosons coupled
to an impurity and include some of the most representative models in quantum
optics. We also propose a family of wavefunctions to describe the bound states
and verify that it accurately characterizes all parameter regimes by comparing
its predictions with exact numerical calculations for a one-dimensional
tight-binding Hamiltonian. For that model, we also analyze the nature of the
bound states by studying the scaling relations of physical quantities such as
the ground state energy and localization length, and find a non-analytical
behavior as a function of the coupling strength. Finally, we discuss how to
test our theoretical predictions in experimental platforms such as photonic
crystal structures and cold atoms in optical lattices.
