Parity-Resolved Spectral Functions Yield Exact Bound-State Energies for 1D Dirac Soft-Core Coulomb Problem

The method uniformizes algebraic branch points via the substitution t = sqrt(x^2 + beta^2) while preserving the Z2 symmetry generated by Pi = sigma_z P_x. Each parity sector is recast as a global connection problem between the regular solution at the origin and the decaying solution at infinity, producing a spectral determinant whose roots give the energies. This approach draws on Braak's quantum Rabi model techniques without equating the resulting function to the G-function. The paper contrasts local quasi-exact solvability, where polynomial solutions satisfy the Dirac equation for specific parameter values, against the global determinant condition that locates all discrete spectrum points.

Comparison with the Bethe-ansatz sector reveals that quasi-exact states serve as independent checks rather than exhaustive solutions, exposing how local analyticity fails to capture the full spectrum when branch points are present. This distinction matters for effective models in graphene nanoribbons and trapped-ion Dirac simulators, where soft-core potentials approximate screened interactions. The construction avoids perturbative expansions, relying instead on exact matching across the complex plane.

Future extensions could incorporate time-dependent driving or higher-dimensional generalizations, testing whether the same uniformization yields closed-form spectral conditions for Dirac-Coulomb problems in two dimensions. Experimental platforms using ultracold atoms or photonic lattices may soon measure the predicted parity-dependent level crossings to within 1 percent accuracy, providing a falsifiable test within two years.