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The Casimir effect for a scalar field in presence of delta-type potentialshas been investigated for a long time in the case of surface delta functions,modelling semi-transparent boundaries. More recently Albeverio, Cacciapuoti,Cognola, Spreafico and Zerbini [9,10,51] have considered some configurationsinvolving delta-type potentials concentrated at points of $\mathbb{R}^3$; inparticular, the case with an isolated point singularity at the origin can beformulated as a field theory on $\mathbb{R}^3\setminus \{\mathbf{0}\}$, withself-adjoint boundary conditions at the origin for the Laplacian. However, theabove authors have discussed only global aspects of the Casimir effect,focusing their attention on the vacuum expectation value (VEV) of the totalenergy. In the present paper we analyze the local Casimir effect with a pointdelta-type potential, computing the renormalized VEV of the stress-energytensor at any point of $\mathbb{R}^3\setminus \{\mathbf{0}\}$; to this purposewe follow the zeta regularization approach, in the formulation already employedfor different configurations in previous works of ours (see [29-31] andreferences therein).

Local Casimir Effect for a Scalar Field in Presence of a Point Impurity