Casimir effect of rough plates under a magnetic field in Hořava-Lifshitz theory

We investigate the Casimir effect for parallel plates within the framework of Hořava-Lifshitz theory in 3 + 1 dimensions, considering the effects of roughness, anisotropic scaling factor, and an uniform constant magnetic field. Quantum fluctuations are induced by an anisotropic charged-scalar quantum field subject to Dirichlet boundary conditions. To incorporate surface roughness, we apply a coordinate transformation to flatten the plates, treating the remaining roughness terms as potential. The spectrum is derived using perturbation theory and regularized with the ζ -function method. As an illustrative example, we consider plates with periodic boundary conditions.