Construction of theSMatrix from Its Left-Hand-Cut Discontinuity, When the Latter is Asymptotically Unbounded

A new procedure is proposed for constructing the partial-wave $S$ matrix, given its left-hand-cut discontinuity $\lambda \mathrm{}\left(m\right)\mathrm{}$ ($m=-2\mathrm{ik}$, where $k$ is the c.m. momentum). The advantage over the $\frac{N}{D}$ method is that this procedure works even when $\lambda \mathrm{}\left(m\right)\mathrm{}$ is unbounded as $m\to \infty$ (provided that it oscillates in a suitable way); in fact, the new technique works even when $\lambda \mathrm{}\left(m\right)\mathrm{}$ is not bounded by any finite power of $m$, so that not only does the usual $\frac{N}{D}$ decomposition break down, but there exists no partial-wave dispersion relation with a finite number of subtractions. An equivalent potential is introduced that provides considerable insight into the nature of elementary-particle, bound-state, ghost, and Castillejo-Dalitz-Dyson poles. The method is generalized to include inelastic effects.