Eigenstates of the free quantum field Hamiltonian with special boundary conditions

We study solutions to the eigenstate equation for the Hamiltonian of the free quantum field in the Schrödinger representation with modified functional boundary conditions and reformulate the task in terms of extensions of the quadratic form in the Hamiltonian's potential term. We provide examples of Gaussian solutions corresponding to singular potentials of the power-1/2 Riesz type and show that domains of admissible extensions include functions with antisymmetric singular behavior in at minimum two points, the distance between the latter being restricted by the extension parameter.