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 ϵ(k)=\±|d|km, where m\≥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\→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 impurity scattering for these more general dispersion relations.

}, doi = {https://doi.org/10.48550/arXiv.2103.09830}, url = {https://arxiv.org/abs/2103.09830}, author = {Yidan Wang and Michael Gullans and Xuesen Na and Alexey V. Gorshkov} }