Energy-dependent separable potentials

A method for constructing energy-dependent separable potentials of the form $V=\lambda \mathrm{}\left(E\right)\mathrm{}|v><v|\mathrm{}$ from (complex) elastic scattering phase shifts is presented. The energy-dependent coupling constant $\lambda \mathrm{}\left(E\right)\mathrm{}$ is shown to satisfy a dispersion relation as a consequence of the requirement of multichannel unitarity. It is this knowledge of the analytic properties of $\lambda \mathrm{}\left(E\right)\mathrm{}$ which makes it possible to solve the inverse problem. It is further demonstrated that in some cases the inverse problem can still be solved analytically even when the phase shift changes sign below the inelastic threshold.