Magnetic Casimir effect of a Lorentz-violating scalar with higher-order derivatives

In this paper, I study the Casimir effect caused by a charged and massive scalar field that breaks Lorentz invariance in a CPT-even, ether-like manner. The breaking of Lorentz invariance is implemented by a constant space-like vector directly coupled to higher order derivatives of the field. I take this vector to be space-like to avoid non-causality problems that could arise with a time-like vector. I examine the two scenarios of the scalar field satisfying either Dirichlet or mixed boundary conditions on a pair of plane parallel plates. I use the zeta function technique to investigate the effect of a constant magnetic field, perpendicular to the plates, on the Casimir energy and pressure. I examine two different directions of the unit vector: parallel and perpendicular to the plates. I fully examine both scenarios for both types of boundary conditions and, in both cases and for both types of boundary conditions, I obtain simple analytic expressions of the Casimir energy and pressure in the three asymptotic limits of strong magnetic field, large mass, and small plate distance.