Universal scattering with general dispersion relations

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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 $D\ensuremath{\ge}1$ 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 $\ensuremath{\epsilon}(k)=\ifmmode\pm\else\textpm\fi{}|d|{k}^{m}$, where $m\ensuremath{\ge}2$ 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 $E\ensuremath{\rightarrow}0$ 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 $D\ensuremath{\ge}1$, dispersion relations $\ensuremath{\epsilon}(\mathbit{k})={|\mathbit{k}|}^{a}$ for a $D$-dimensional momentum vector $\mathbit{k}$ with any real positive $a$, and separable potential scattering.