Two-dimensional ?inverse scattering problem? for negative energies and generalized-analytic functions. I. Energies below the ground state

Abstract

The two-dimensional inverse problem of reconstructing the general (without the hypothesis of continuous symemetry) Schrodinger operator L = − ∑ α(∂α−iAα)+ u on the basis of data, “collected” from the family of eigenfunctions of one energy level LΨ = e0Ψ, was first considered in 1976 in [1] for the periodic case. From [2] the idea arose of a profound connection of this problem with integrable problems of the theory of solitons in dimension 2+ 1. [3]–[5] are devoted to the development of this approach in the periodic case and later [6, 7] in the rapidly decreasing one. The contemporary stage of research began with [3, 4] in 1984, where, in the periodic case, the group of reductions Z2×Z2 was explicitly found, singling out those data of the inverse problem from which one gets purely potential self-adjoint operators (1)