Recovery of singularities from amplitude information

Abstract

Let V(x) denote a potential in the one-dimensional Schrödinger equation, without bound states, for which (1) V(x) = 0 for x<0 and (2) V(x) is piecewise continuous with adequate decay as x goes to infinity. We are interested in the problem of determining V given the reflectivity r(k) = |R(k)|, where R is the usual left-hand reflection coefficient. For very special classes of potentials it is known that r determines V uniquely. Here we show that under much more general (although still restrictive) assumptions, the location and magnitude of discontinuities of V can be determined from r. The nature of the restrictions is related to the behavior of R(k) for k in the upper half of the complex plane. © 1997 American Institute of Physics.