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 $epsilon(k)=pm |d|k^m$, where $mgeq 2$ is an integer. We study impurity scattering problems in which a single-particle in a one-dimensional waveguide scatters off of an inhomogeneous, discrete set of sites locally coupled to the waveguide. For a large class of these problems, we rigorously prove that when there are no bright zero-energy eigenstates, the $S$-matrix evaluated at an energy $Eto 0$ converges to a universal limit that is only dependent on $m$. We also give a generalization of a key index theorem in quantum scattering theory known as Levinsons theorem -- which relates the scattering phases to the number of bound states -- to impurity scattering for these more general dispersion relations.
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