Ultraviolet Renormalisation of a quantum field toy model I

We consider a class of Hamiltonians describing a fermion field coupled to a boson field. The interaction kernels are assumed bounded in the fermionic momentum variable and decaying like $|q|^{-p}$ for large boson momenta $q$. A realistic physical case would be $p=\frac12$. We impose a spatial cutoff and UV-renormalise the resulting Hamiltonian by subtracting its ground state energy. Our renormalisation procedure works for the physically realistic value $p=\frac12$ in spatial dimensions $1$ and $2$ and for $p>\frac34$ in spatial dimension $d=3$.