A global minimum principle is reported for inverse scattering for Schrödinger's equation without bound states. For the one-dimensional problem, the line integral of the potential on the interval $\{-\infty ,x\}$ can be found by minimizing a certain functional of the scattering amplitude and a variational wave field. In three dimensions, the minimum of a closely related functional is shown to yield $-\int d^3\mathbf{y}\nu \left(\mathbf{y}\right)/|\mathbf{x}-\mathbf{y}{|}^2$, where $\nu \left(\mathbf{x}\right)$ is the potential.

• Received 26 June 1996

DOI:https://doi.org/10.1103/PhysRevLett.77.4126