Potentials with Identical Scattering Properties Below a Critical Energy

A pair of scattering potentials are called $\alpha$-equivalent if they have identical scattering properties for incident plane waves with wavenumber $k\leq\alpha$ (energy $k^2\leq\alpha^2$.) We use a recently developed multidimensional transfer-matrix formulation of scattering theory to obtain a simple criterion for $\alpha$-equivalence of complex potentials in two and three dimensions.