Abstract
We discuss conceptual problems of zero-range potentials in the problem of vacuum polarization of real-valued massive scalar field. In this way, we develop precise formulation for the effects of vacuum polarization near a pointlike source with a spherically-symmetric \(\delta\) -like potential in three spatial dimensions. In computation, we use the approach based on the concept of self-adjoint extensions (SAE) of densely defined symmetric operators, and compare it with delta-potential introduced heuristically. It turns out that the bare field-strength coupling should be infinitesimally small and has to be renormalized. The renormalized coupling \(\lambda_{\textrm{ren}}\) is related directly with the self-adjoint parameter. In this framework we compute the renormalized vacuum expectation value of the field squared \(\langle\phi^{2}(x)\rangle_{\textrm{ren}}\) . Asymptotic cases are discussed in detail.
