Landscape of scattering universality with general dispersion relations

Universality in physics describes how disparate systems can exhibit identical low-energy behavior. Here, we reveal a rich landscape of new universal scattering phenomena governed by the interplay between an interaction and a system's density of states. We investigate one-dimensional scattering with dispersion relations $ε(k)=|k|^m$ and $ε(k)=\operatorname{sgn}(k)|k|^m$ for any real $m>1$. For emitter scattering and separable potentials, including contact potentials, we classify the zero-energy $S$-matrix limits. These universal $S$-matrix fixed points depend solely on the dispersion exponent $m$ and a finite set of integers characterizing the low-momentum interaction structure. This classification extends far beyond the standard quadratic-dispersion case ($m=2$). We further derive a generalized Levinson's theorem relating the total winding of the $S$-matrix determinant to the number of bound states. Finally, for contact potentials and analytic monomial dispersions, we analyze the leading low-energy expansion of the scattering phase near these fixed points and extract the corresponding Wigner-delay scaling. Our findings are directly relevant to synthetic quantum systems, where engineered dispersion relations in atomic arrays and photonic crystals offer a platform to explore these universal behaviors.