Analytical Solutions of One-Dimensional ($1\mathcal{D}$) Potentials for Spin-0 Particles via the Feshbach-Villars Formalism

• URL: http://arxiv.org/abs/2603.25375v1
• Date: Thu, 26 Mar 2026 12:24:29 GMT
• Title: Analytical Solutions of One-Dimensional ($1\mathcal{D}$) Potentials for Spin-0 Particles via the Feshbach-Villars Formalism
• Abstract summary: We present a unified study of the one-dimensional Feshbach--Villars equation for spin-0 particles.<n>We analyse its solutions for Coulomb, power-exponential, Cornell, Pschl--Teller, and Woods--Saxon interactions.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present a unified analytical and numerical study of the one-dimensional Feshbach--Villars (FV) equation for spin-0 particles in the presence of several representative external potentials. Starting from the FV formulation of the Klein--Gordon equation, we derive the corresponding one-dimensional master equation and analyse its solutions for Coulomb, power-exponential, Cornell, Pöschl--Teller, and Woods--Saxon interactions. For the singular Coulomb and Cornell cases, a Loudon-type cutoff regularisation is implemented on the full line, allowing a mathematically controlled treatment of the origin and an explicit classification of the states by parity. The Coulomb problem exhibits the expected near-degenerate even--odd structure in the cutoff limit, while the Cornell potential combines short-distance Coulomb behaviour with long-distance confinement and produces a finite set of bound states for fixed parameters. The power-exponential potential with $p=1$ is reduced to a Whittaker-type equation and yields an intrinsically relativistic spectrum with no standard Schrödinger bound-state limit in the parameter regime considered. For the smooth short-range Pöschl--Teller and Woods-Saxon potentials, the FV formalism reveals, respectively, the effects of definite parity and spatial asymmetry on the spectrum, wave functions, and particle--antiparticle mixing. In all cases, we reconstruct the full FV spinor, analyse the associated charge density, and compare the relativistic behaviour with the corresponding non-relativistic expectations whenever such a limit exists. The results provide a coherent set of analytical and numerical benchmarks for relativistic scalar bound states in one dimension.

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