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\≥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 ϵ(k)=\±|d|km, where m\≥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\→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\&$\#$39;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\≥1, dispersion relations ϵ(k)=|k|a for a D-dimensional momentum vector k with any real positive a, and separable potential scattering.

}, keywords = {FOS: Physical sciences, Mathematical Physics (math-ph), Quantum Physics (quant-ph)}, doi = {https://doi.org/10.1103/PhysRevResearch.4.023014}, url = {https://arxiv.org/abs/2103.09830}, author = {Wang, Yidan and Michael Gullans and Na, Xuesen and Whitsitt, Seth and Alexey V. Gorshkov} }