Universality in one-dimensional scattering with general dispersion relations

Many synthetic quantum systems allow particles to have dispersion relations that are neither linear nor quadratic functions. Here, we explore single-particle scattering in one dimension when the dispersion relation is any analytic function. We show that, when the density of states diverges at energy E₀, the S-matrix evaluated at an energy E->E₀ converges to a universal limit that is only dependent on the rate of divergence of the density of states at E₀. This behavior is independent of the nature of the interactions, a feature that we illustrate by considering two distinct scattering problems: a single-particle in a one dimensional waveguide (i) scattering off of a localized potential(“potential” scattering) and (ii) scattering off of an inhomogeneous, discrete set of sites locally coupled to the waveguide (“impurity” scattering). We also give a generalization of a key result in quantum scattering theory known as Levinson’s theorem—which relates the scattering phases to the number of bound states—to these more general dispersion relations