On the Casimir Effect with δ-Like Potentials, and a Recent Paper by K. Ziemian (Ann. Henri Poincaré, 2021)

The local and global aspects of the Casimir effect for a scalar field in the presence of a point-like impurity were treated in our papers (Fermi and Pizzocchero in Symmetry 10(2):38, 2018; Fermi in Mod. Phys. Lett. A 35(03):2040008, 2020), using the zeta regularization method. A paper by Ziemian, recently published in (Ann. Henri Poincaré 22:1751–1781, 2021), discusses the Casimir effect for a scalar field in presence of one or two, extended or point-like impurities, using the Herdegen–Stopa approach. Ziemian claims that his result for the energy density with a single point-like impurity differs from that derived in Fermi and Pizzocchero (2018), ascribing the mismatch to a basic conceptual discrepancy. In the present work, we show that the formula in Ziemian (2021) for the energy density in the presence of a point-like impurity coincides (upon amending minor computational errors) with the formula of Fermi and Pizzocchero (2018) for the same quantity. In order to make our discussion self-contained, in the present paper we also survey some basic facts related to Fermi and Pizzocchero (2018), Fermi (2020), Ziemian (2021). This survey includes Schroedinger’s operators with point-like interactions, as described in the celebrated book (Solvable models in quantum mechanics, Springer, New York, 1988) by Albeverio et al., the zeta regularization method for a canonically quantized scalar field and the implementation of point-like interactions in this setting.