Renormalization of bosonic quadratic Hamiltonians involving rank-one perturbations

Abstract

We study the renormalization of a bosonic quadratic Hamiltonian with an ultraviolet divergence. The Hamiltonian is composed of the sum of a free part and the square of the smeared field operator. We explicitly diagonalize the Hamiltonian via Bogoliubov transformations, thus simplifying its definition as a self-adjoint operator. Depending on the field operator's smearing, we discuss different renormalizations, either of the energy alone, or the energy and coupling constant together.