Universal scattering with general dispersion relations

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 underlying principles an experimentally relevant setting, focus on waveguide electrodynamics (QED) problems (i.e., $D=1$) with relation $\ensuremath{\epsilon}(k)=\ifmmode\pm\else\textpm\fi{}|d|{k}^{m}$, where $m\ensuremath{\ge}2$ is integer. For large class these for any positive integer $m$, rigorously prove there no bright zero-energy eigenstates, $S$ matrix evaluated energy $E\ensuremath{\rightarrow}0$ converges universal limit only dependent $m$. We also give generalization key index theorem theory known as Levinson's theorem---which relates phases number bound states---to QED more relations. then extend results dimensions $D\ensuremath{\ge}1$, $\ensuremath{\epsilon}(\mathbit{k})={|\mathbit{k}|}^{a}$ $D$-dimensional momentum vector $\mathbit{k}$ real $a$, and separable potential scattering.