A complete set of solutions of the scalar field equation on the background of three-dimensional flat space-time with a single extracted point (impurity) is obtained. The defect is described within the technique of Laplacian self-adjoint extension, in which the impurity is identified with a delta-like singularity. It is shown that in the case of two spatial dimensions, the existence of a single bound state in self-adjoint extensions, which raises no objections at the level of quantum mechanics, corresponds to unphysical states in quantum field theory. The physical states are isolated, and an exact expression for the renormalized Hadamard function is obtained. As an example, the renormalized vacuum expectation value of the field squared ^2 (x) ₑ₄₍ is computed. It is shown that the toy model can be regarded as a first approximation to the extended issue of field effects in locally flat conical spaces, as well as in the cosmic-string background.
Grats et al. (Mon,) studied this question.