Relationship between different types of inverse data for the one-dimensional Schrödinger operator on a half-line

We consider inverse dynamical, spectral, quantum and acoustical scattering problems for the Schrödinger operator on the half line. The goal of the paper is to establish the connections between different types of inverse data for these problems. The central objects which serve as a source for all formulaes are kernels of so-called connecting operators and Krein equations.