Analytical and Numerical Bound-State Analysis of One-Dimensional Time-like Vector Potentials in the Feshbach–Villars Formalism

We analyse one-dimensional bound states of spin-zero particles coupled to external time-like vector potentials within the Feshbach–Villars representation. The study is organised around a practical criterion of physical admissibility that distinguishes genuine bound states from scattering solutions and finite-box artefacts by combining asymptotic decay, parity, matching conditions, conserved Feshbach–Villars charge, node counting, and numerical-domain convergence. The regularised Coulomb interaction is treated through full-line Loudon matching, which clarifies the cutoff dependence of the regular odd–even pairs and separates them from the isolated core branch. Symmetric power-exponential, Pöschl–Teller, and localised Woods–Saxon wells are analysed by parity-resolved inward shooting with bounded residual functions and explicit convergence tests. The pure-vector Cornell interaction is excluded because its large-distance solutions remain oscillatory, whereas a one-sided Woods–Saxon step is shown not to support a non-trivial square-integrable state on the full line under simultaneous decay conditions. A signed search also follows a Pöschl–Teller branch through zero energy and confirms a negative-energy continuation with positive integrated Feshbach–Villars charge. The comparison among the four models separates spatial localisation, asymptotic component mixing, and integrated component content, showing that these diagnostics need not follow the same trend. The resulting framework provides reproducible benchmarks for relativistic scalar bound-state calculations.