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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta$$\end{document}-like) potential in three spatial dimensions. There are different ways of introducing delta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta$$\end{document}-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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$- \Delta$$\end{document} 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 <phi 2(x)> ren\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle \phi <^>{{\hspace{1.0pt}}2}(x)\rangle _{\textrm{ren}}$$\end{document} and the renormalized vacuum average of the scalar-field's energy-momentum tensor < T mu nu(x)> ren\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle T_{\mu \nu }(x)\rangle _{\textrm{ren}}$$\end{document}. Asymptotic cases are discussed in detail.