Bragg gap ladder for a linear dispersion

For a scalar one-dimensional dispersion $E(k)=\hslash v|k|$ and a potential of period $L$, a reciprocal vector $nG$, $G=2\pi /L$, connects degenerate right- and left-moving states at $k=nG/2$. Their unperturbed energy is $E_n=n\pi \hslash v/L$. Thus consecutive allowed Bragg thresholds have spacing $\pi \hslash v/L$, whereas shifting momentum by a full reciprocal period changes a linear branch's energy by $2\pi \hslash v/L$. The potential's Fourier coefficients and higher-order couplings determine which gaps actually open and their widths, so a geometric period alone does not imply an exactly energy-periodic spectrum. A graphene pseudospin selection rule can further suppress scattering; the scalar-chain model omits that internal structure.