Monochromatic Reconstruction Algorithms for Two-dimensional Multi-channel Inverse Problems

We consider two inverse problems for the multi-channel two-dimensional Schrödinger equation at fixed positive energy, i.e., the equation −Δψ+V (x)ψ=Eψ at fixed positive E, where V is a matrix-valued potential. The first is the Gel’fand inverse problem on a bounded domain D at fixed energy and the second is the inverse fixed-energy scattering problem on the whole plane $\mathbb{R^{2}}$. We present in this paper two algorithms which give efficient approximate solutions to these problems: in particular, in both cases, we show that the potential V is reconstructed with Lipschitz stability by these algorithms up to O(E^−(m−2)/2) in the uniform norm as $E \rightarrow + \infty$, under the assumptions that V is m-times differentiable in L¹, for m≥3, and has sufficient boundary decay.