Singularly perturbed hamiltonians of a quantum Rayleigh gas defined as quadratic forms

We consider a class of singular, zero-range perturbations of the quantum Hamiltonian of a system consisting of a test particle and N harmonic oscillators (Rayleigh gas). Using the theory of quadratic forms we construct the self-adjoint and bounded from below perturbed Hamiltonian and we give representation formulas for the resolvent and the unitary group. The one-dimensional, two-dimensional and three-dimensional cases are discussed