Two discrete forms of the inverse scattering problem

Abstract

The inverse problem for a class of quantum mechanical systems with discrete spectra is studied. First, for the case of a particle moving in a large sphere and interacting with a separable potential, it is shown that the potential can be found from the analytic properties of the eigenvalues, if they are given as the roots of a transcendental equation. The same technique is also applied for the construction of the form factor from the eigenfunctions of a two channel Wigner–Weisskopf model, when the system is enclosed in a large sphere. Then a completely different method of inversion is developed for the determination of the matrix elements of a tri–diagonal shell-model Hamiltonian. In this case the input data are the energy eigenvalues and the reduced widths. This method utilizes the continued.J-fraction expansion of a quantity which is analogous to the R-matrix of the nuclear reaction theory. The same technique is also used to investigate the exact inversion of the scattering problem, when the phase shifts are calculated by solving the finite-difference approximate form of the Schrödinger equation.