  We consider a class of singular, zero-range perturbations of the Hamiltonian
of a quantum system composed by a test particle and a harmonic oscillators in
dimension one, two and three and we study its spectrum. In facts we give a
detailed characterization of point spectrum and its asymptotic behavior with
respect to the parameters entering the Hamiltonian. We also partially describe
the positive spectrum and scattering properties of the Hamiltonian.
