Vacuum Polarization Effects of Pointlike Impurity

We develop precise formulation for the effects of vacuum polarization near a pointlike source with a zero-range (δ-like) potential in three spatial dimensions. There are different ways of introducing δ-interaction in the framework of quantum theory. We discuss the approach based on the concept of self-adjoint extensions of densely defined symmetric operators. Within this approach we consider the real massive scalar field in three-dimensional Euclidean space with a single extracted point. Appropriate boundary conditions imposed at this point enable one to consider all self-adjoint extensions of - Δ as operators which can describe a pointlike source with a zero-range potential. In this framework we compute the renormalized vacuum expectation value of the field square φ²(x)ᵣₑₙ and the renormalized vacuum average of the scalar-field's energy-momentum tensor Tμν(x)ᵣₑₙ. Asymptotic cases are discussed in detail.