  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 $m\geq 2$ 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 $E\to 0$ 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 Levinson's theorem -- which relates the scattering phases to
the number of bound states -- to impurity scattering for these more general
dispersion relations.
