  We study solutions of the functional eigenstate equation of a free quantum
field Hamiltonian. Admissible solutions are to have a finite norm and a finite
eigenvalue w.r.t. the norm and eigenvalue of the ground state of the free
theory. We show that in the simple cases of a scalar field and of a vector
field in the Coulomb gauge the admissible eigenstates exist and possess
negative energy. The functionals can be treated as infinite-dimensional
counterparts of the eigenfunctions of the theory of singular perturbations of
differential operators, and can be deployed for construction of the
renormalized states of models with asymptotic freedom.
