Wave scattering in one dimension with absorption

Wave scattering is analyzed in a one-dimensional nonconservative medium governed by the generalized Schrödinger equation dc/dx1kc5@ ikP(x) 1Q(x)#c , where P(x) and Q(x) are real, integrable potentials with finite first moments. Various properties of the scattering solutions are obtained. The corresponding scattering matrix is analyzed, and its small-k and large-k asymptotics are established. The bound states, which correspond to the poles of the transmission coefficient in the upper-half complex plane, are studied in detail. When the medium is not purely absorptive, i.e., unless P(x)<0, it is shown that there may be bound states at complex energies, degenerate bound states, and singularities of the transmission coefficient imbedded in the continuous spectrum. Some explicit examples are provided illustrating the theory. © 1998 American Institute of Physics. @S0022-2488~98!01503-5#