Minimum energy pulse synthesis via the inverse scattering transform

This paper considers a variety of problems in the design of selective RF-pulses. We apply a formula of Zakharov and Manakov to directly relate the energy of an RF-envelope to the magnetization profile and certain auxiliary parameters used in the inverse scattering transform (IST) approach to RF-pulse synthesis. This allows a determination of the minimum possible energy for a given magnetization profile. We give an algorithm to construct both the minimum energy RF-envelope as well as any other envelope that produces a given magnetization profile. This includes an algorithm for solving the Gel'fand-Levitan-Marchenko equations with bound states. The SLR method is analyzed in terms of traditional scattering data, and shown to be a special (singular) case of the IST approach. RF-envelopes are computed for a variety of examples.