Two-dimensional ``inverse scattering problem'' for negative energies and generalized-analytic functions. I: Energies below the ground state (Q1120080)

Let \(L=-\Delta +u(x)\), \(x\in {\mathbb{R}}^ 2\), be an operator where the function u(x) is periodic or short-range. It is assumed that certain information about the solutions of the equation \(L\psi =\epsilon_ 0\psi\) with some fixed \(\epsilon_ 0\) is given. The problem of reconstruction of u(x) is considered. In particular, conditions ensuring that \(\epsilon_ 0\) is smaller than the lower bound of L are found.