Construction of solutions of the classical field equation for a massless Klein-Gordon field coupled to a static source

In this paper, we consider a system of a massless Klein-Gordon field coupled to a static source. The total Hamiltonian is a self-adjoint operator on a boson Fock space. We consider annihilation operators in the Heisenberg picture and define a sesquilinear form. Under infrared regularity conditions, it is proven that the sesquilinear form is a solution of the classical field equation.