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

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.

Figures (29)

• Figure 1: Bound-state energy spectrum of the regularised one-dimensional Coulomb potential in the Feshbach--Villars formalism, for $\delta = 0.05\,m^{-1}$ and three values of the coupling constant $\alpha$. Left panel: particle eigenvalues $E_n^{\rm part}/m$ as a function of the principal quantum number $n$. Right panel: antiparticle eigenvalues $E_n^{\rm anti}/m = -E_n^{\rm part}$, obtained from the charge-conjugation symmetry $E \to -E$, $\alpha \to -\alpha$ of Eq. \ref{['eq:fv_cutoff_master']}. All levels lie inside the mass gap, $|E_n|<m$, and accumulate toward $|E|=m$ as $n$ increases. Deeply localised states (Loudon core states) are excluded from both panels, since they are artefacts of the finite cutoff and disappear in the limit $\delta\to 0^+$ (see text).
• Figure 2: Normalised Coulomb wave functions $\psi_n(x)$ for the one-dimensional problem, with $\alpha = 0.20$ and $m=1$, shown for the four lowest states $n=0,1,2,3$. Each curve is plotted over a sufficiently large interval to capture both its oscillatory structure and asymptotic decay, and all of them are displayed together on a common axis for direct comparison. As $n$ increases, the wave functions extend over progressively larger spatial regions and develop the expected increase in the number of nodes, while their amplitudes become more spread out over the classically allowed region.
• Figure 3: Probability densities $|\psi_n(x)|^2$ for the one-dimensional Coulomb problem with $\alpha = 0.20$ and $m=1$, for the four lowest states: (a) $n=0$, (b) $n=1$, (c) $n=2$, and (d) $n=3$. Each panel is displayed over an adaptive spatial interval chosen to make the full profile visible. The densities become progressively broader as $n$ increases and exhibit the corresponding increase in radial structure, with the number of local maxima growing from the ground state to the excited states. This behaviour reflects the increasingly extended character of the higher Coulomb eigenstates.
• Figure 4: Feshbach--Villars spinor components for the regularised Coulomb potential with $\alpha = 0.20$ and $\delta = 0.05\,m^{-1}$. (a) lowest odd-parity state ($n=0$, odd). (b) lowest even-parity state ($n=0$, even). Solid blue: sum component $\psi_s = \psi_1 + \psi_2$. Dashed red: particle component $\psi_1$. Dotted green: antiparticle component $\psi_2$. The vertical dotted grey lines mark the cutoff boundary $x = \pm\delta$, across which the interior oscillatory solution is matched to the exterior Whittaker function. The particle component dominates throughout, while $\psi_2$ acquires a visible enhancement inside the core owing to the large constant potential $eV_\delta = \alpha/\delta$.
• Figure 5: Ratio $|\psi_2/\psi_1|$ of the antiparticle to particle components as a function of $x > 0$, for the lowest odd state with $\delta = 0.05\,m^{-1}$ and three values of the coupling constant $\alpha$. The vertical dotted line marks the cutoff radius $x = \delta$. For $x > \delta$ the ratio is given analytically by Eq. \ref{['eq:psi2_psi1_ratio']} and decreases monotonically from a maximum at the cutoff boundary toward the asymptotic value $(m-E_0)/(m+E_0)$ in magnitude at large $x$. Larger coupling produces a uniformly higher ratio, reflecting the increasing relativistic character of the more deeply bound states.