Discrete Bargmann and Neumann systems and finite-dimensional integrable systems

The nonlinearization approach of eigenvalue problems is equally well applied to the discrete KdV hierarchy. Two kinds of constraints between the potentials and eigenfunctions are suggested, from which the discrete Schrödinger eigenvalue problem, the spatial part of the Lax pairs of the discrete KdV hierarchy, is nonlinearized to be a discrete Bargmann system and a discrete Neumann system, while the nonlinearization of the time part of the Lax pairs leads to two hierarchies of new finite-dimensional completely integrable systems in the Liouville sense. The solutions of the discrete KdV equation are reduced to solving the compatible system of difference equations and ordinary differential equations.