Inverse Problems in Multidimensions

The direct and inverse problems associated with off-diagonal N × N matrix-valued potentials in Rn + 1 are studied rigorously. For $N = 2$ and $n = 1$, they are related to the Davey–Stewartson II equation. The components of the potentials are assumed to be rapidly decreasing Schwartz functions. Under small norm conditions, both the direct and the inverse problems areshown to be uniquely solvable. The inverse data are characterized by certain nonlinear equations. Furthermore, the general problem of reconstructing the potential is reduced to the reconstruction of 2 × 2 potentials.