Researchers have explored the Casimir effect for a massive scalar field confined between two parallel plates, introducing an effective mass that varies with position. This approach allows for the study of the interaction between a scalar background and the field, yielding exact normal modes by solving the Klein-Gordon equation. Surprisingly, the resulting transverse energy spectrum exhibits a Landau-like structure, despite the absence of an external magnetic field, suggesting an unexpected analogy between systems.

Quantization of the field allows for the calculation of vacuum energy using generalized zeta-function regularization and a renormalization procedure. The renormalized vacuum energy separates into a Landau-like contribution and an additional term induced by the spatial dependence of the effective mass. It has been shown analytically and numerically that both contributions are exponentially suppressed in the strong-coupling regime. In the opposite limit, the Landau-like contribution smoothly reproduces the standard vacuum energy for a confined massive scalar field, while the additional term becomes singular due to the restricted domain of validity of the exact spectrum.

Except in the vicinity of this singular limit, the vacuum energy is dominated by the Landau-like sector. These results establish a direct connection between position-dependent effective masses and boundary-induced quantum vacuum phenomena. This exactly solvable framework opens new avenues for investigating the Casimir effect in spatially inhomogeneous relativistic systems, offering a theoretical tool for a better understanding of quantum forces in complex environments.