Vacuum and thermal fluctuations of a scalar field with point interactions

We investigate the vacuum and thermal fluctuations of a neutral massless scalar field living in Minkowski spacetime and interacting with a finite number of point-like obstacles, modelled by zero-range potentials. The system is described rigorously in terms of self-adjoint realizations of the Laplacian, under assumptions ensuring the absence of instabilities. Using the relative zeta-function technique, we determine the renormalized connected partition function and derive explicit expressions for the thermodynamic observables, characterizing both their low- and high-temperature behaviors. Furthermore, we derive of a convergent Born series expansion for the Casimir energy, which identifies multiple-scattering processes as the mechanism underlying vacuum forces. The latter decompose into pairwise contributions directed along the lines joining the obstacles, with intensities depending non-locally on the full configuration. We also present some numerical results for identical obstacles, indicating that the Casimir forces are always attractive in this context.