Vacuum Polarization in the Point Impurity Background

The vacuum polarization effect of a massive scalar field \(\phi(x)\) near the point \(\delta\)-like source is considered. The corresponding interaction is introduced within a technique of self-adjoint extension of the Laplace operator. This method has been widely discussed within the framework of quantum mechanics. We propose to use it to investigate vacuum field effects. This approach allows computing the renormalized Hadamard function and the renormalized vacuum energy density \(\langle T_{00}(x)\rangle_{\textrm{ren}}\) for massive real scalar field with minimal curvature coupling. The dependence of the vacuum polarization effect upon the fields’s mass is analyzed.