Dynamical Casimir effect with Robin boundary conditions in a three dimensional open cavity

We consider a massless scalar field in 1+1 dimensions inside a cavity composed by a fixed plate, which imposes on the field a Robin BC, and an oscillating one, which imposes on the field a Dirichlet BC. Assuming that the plate moves for a finite time interval, and considering parametric resonance, we compute the total number of created particles inside the cavity. We generalize our results to the case of two parallel plates in 3+1 dimensions.