Dynamical Casimir effect for a massless scalar field between two concentric spherical shells

In this work we consider the dynamical Casimir effect for a massless scalar field -- under Dirichlet boundary conditions -- between two concentric spherical shells. We obtain a general expression for the average number of particle creation, for an arbitrary law of radial motion of the spherical shells, using two distinct methods: by computing the density operator of the system and by calculating the Bogoliubov coefficients. We apply our general expression to breathing modes: when only one of the shells oscillates and when both shells oscillate in or out of phase. We also analyze the number of particle production and compare it with the results for the case of plane geometry.