Infrared extensions of the quadratic form of the ground state of scalar field theory

We extend the quadratic forms of the Gaussian functionals of the free quantum scalar field theory to the set of functions decreasing in the infinity as $ |\vec{x}|^{-1} $. We use the momentum-space representation (after the Fourrier transform) and as the scalar product we take the product generated by the quadratic form of the Laplace operator (potential term of the quantum Hamiltonian).