Pauli–Villars regularization for some models with singular perturbations

We show how Pauli–Villars regularization works in the construction of renormalized Hamiltonian for two examplars of quantum systems with singular pertuabations. The systems are the scalar 3-dimensional particle interacting with $\delta$-potential and the infrared extensions of the quadratic forms of the gaussian functional of the ground state in the quantum field theory.