We introduce a new theoretical concept for the spectral data defined for Schrödinger operators with complex-valued potentials on the half-line. Then we discuss properties of the corresponding spectral map which assigns the spectral data to a given Schrödinger operator. First, we show that the spectral map is injective, i.e. we have a Borg-Marchenko type theorem within the suggested framework. If time allows, we also present partial results on the image of the spectral map concerning asymptotic properties of the spectral data. The talk is based on a joint work in progress with A. Pushnitski (King's College London) and is motivated by our recent paper on a similar theory developed for non-self-adjoint Jacobi matrices [1].

[1] A. Pushnitski, F. Štampach, An inverse spectral problem for non-self-adjoint Jacobi matrices, Int. Math. Res. Not. 2024 (2024) 6106-6139.