Self-adjointness of the Yukawa₂ Hamiltonian

We prove that the Hamiltonian H(g) for the Yukawa₂ interaction with a spatial cutoff is self-adjoint. We define H(g) as the limit of operators H(g, κ) with an ultraviolet cutoff. The ultraviolet cutoff makes the mass renormalization constant δm²(g, κ) and the vacuum self-energy E(g, κ) finite, but these constants both diverge logarithmically as κ → ∞. We choose the renormalization constants required by perturbation theory. As κ → ∞, the resolvents of the self-adjoint operators H(g, κ) converge in norm to the resolvent of the self-adjoint operator H(g). In addition H(g) has a vacuum vector.