Estimating Scattering Potentials in Inverse Problems with a Non-Causal Volterra Model

In this paper, a finite memory, non-causal Volterra model is proposed to estimate the potential functions in various inverse quantum mechanical problems, where the bound or scattered wave functions are used as inputs of the Volterra system, while the potential is the desired output. Two simple examples are given to show the model capabilities, where in both cases, a really good match is achieved for a very wide range of potential functions. The first example is a simple one-dimensional bound state problem, where the wave function of the first bound state is used as input to determine the model potential. The second example is a one-dimensional scattering problem, where the scattered wave is used as the system input. In both cases, a higher order, non-causal description is needed to be able to give a good estimation to the solution of the inverse problem. The model sensitivity to input perturbations is also examined, showing that the Volterra representation is capable of giving a robust estimate to the underlying dynamical system. The model could be useful in real-life situations, where the scattering potential should be found from measured data, where the precise equations that govern the dynamics of the system are not known.