Some solutions to the eigenstate equation for the free quantum field Hamiltonian in the Schr\"odinger representation

Using closed positive extensions of the quadratic form in the potential term we provide alternative solutions to the eigenstate equation for the free quantum field Hamiltonian in the Schr\"o\-din\-ger representation. We show that admissible extensions stem from the singular behaviour of the quantum field simultaneously in at least two points, the distance between the latter being limited by the extension parameter. Including position of singularities into dynamical model we provide example of quantum field evolved by the action of the free Hamiltonian interacting with external sources via functional boundary conditions.