Quantum dynamics of a scalar particle in Schwarzschild spacetime using the generalized Feshbach-Villars transformation

In this paper, we investigate the quantum dynamics of spin-0 scalar particles in a Schwarzschild gravitational field using the Generalized Feshbach-Villars Transformation (GFVT). Starting from the covariant Klein-Gordon equation, we reformulate the system in the two-component FV representation, enabling a consistent separation of positive- and negative-energy sectors and restoring a positive-definite probability density in curved spacetime. In the far-field regime, the GFVT reproduces a Coulomb-like effective potential that leads to a hydrogenic bound spectrum, preserving the symmetry between particle and antiparticle states. Extending the formalism by incorporating a relativistic harmonic oscillator potential, we demonstrate that the resulting radial equation transforms into a biconfluent Heun form, where the square-integrability condition yields a discrete and gravity-independent oscillator spectrum. The Schwarzschild curvature affects only the admissible quantum states through a geometric selection rule. Explicit solutions, probability densities, and visual analyses confirm the self-consistency of the formalism. Our findings show that the GFVT offers a powerful and unified framework for describing relativistic scalar fields under both gravitational and confining potentials, providing deeper insight into quantum phenomena in curved geometries and near strong gravitational sources.