Vacuum Polarization Effects of Pointlike Impurity

We develop precise formulation for the effects of vacuum polarization near a pointlike source with a zero-range ($\delta$-like) potential in three spatial dimensions. There are different ways of introducing $\delta$-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 $- \Delta$ 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 $\langle \phi^{ 2}(x)\rangle_{\rm ren}$ and the renormalized vacuum average of the scalar-field's energy-momentum tensor $\langle T_{\mu\nu}(x)\rangle_{\rm ren}$. Asymptotic cases are discussed in detail.