  Many synthetic quantum systems allow particles to have dispersion relations
that are neither linear nor quadratic functions. Here, we explore
single-particle scattering in general spatial dimension $D\geq 1$ when the
density of states diverges at a specific energy. To illustrate the underlying
principles in an experimentally relevant setting, we focus on waveguide quantum
electrodynamics (QED) problems (i.e. $D=1$) with dispersion relation
$\epsilon(k)=\pm |d|k^m$, where $m\geq 2$ is an integer. For a large class of
these problems for any positive integer $m$, 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 waveguide QED scattering for these more
general dispersion relations. We then extend these results to general integer
dimensions $D \geq 1$, dispersion relations $\epsilon(\boldsymbol{k}) =
|\boldsymbol{k}|^a$ for a $D$-dimensional momentum vector $\boldsymbol{k}$ with
any real positive $a$, and separable potential scattering.
