Bound states and inverse scattering for the Schrödinger equation in one dimension

The one‐dimensional Schrodinger equation is considered when the potential and its first moment are absolutely integrable. When the potential has support contained on the left (right) half‐line, it is uniquely constructed by using only the reflection coefficient from the right (left). The bound state norming constants determine whether the potential has support contained on a half‐line or on the full‐line. The bound state energies and the unique set of norming constants yielding the potential with support contained on the left (right) half‐line are completely determined by the reflection coefficient from the right (left). An explicit example is provided.