Geometric Spectral Threshold at a Linear Velocity Zero in One-Dimensional Dirac Systems

Title: Numerical code for the square-root ladder at a smoothed linear velocity zero — companion to "Geometric Spectral Threshold at a Linear Velocity Zero in One-Dimensional Dirac Systems" Description: Python scripts supporting the numerical validation of the near-horizon square-root ladder in C. A. S. Almeida, Physica Scripta (2026) [DOI]. The code diagonalizes a one-dimensional Dirac Hamiltonian with domain-wall mass m(x) = m₀ tanh(x/ℓ) and smoothed linear velocity v(x) = κ√(x² + δ²), which has a floor v₀ = κδ. It uses a Hermitian mimetic finite-volume discretization on a staggered grid, uniform in the core and geometrically stretched in the tails. The low-lying spectrum is obtained with sparse shift-invert Lanczos (scipy.sparse.linalg.eigsh). Units: ħ = 1. ladder.py: checks E_n = √(2n m₁v₀) and the ratios E_n/E₁ → √n. convergence.py: grid-refinement study; shows that convergence is controlled by ℓ_osc/h, with ℓ_osc = √(v₀/m₁). validity_sweep.py: residual deviation versus the validity parameter W = m₁v₀/κ² at converged resolution. two_corrections.py: separates the two finite-size corrections, one from the outer velocity region and one from mass saturation. In the wide-wall limit it recovers W·(E₁/E₁^exact − 1) ≈ 0.12. Running.txt: reference output of all scripts. Requirements: Python ≥ 3.10, NumPy, SciPy.Usage: python3 <script>.pyLicense: [MIT]