Rational recursion operators for integrable differential-difference equations

In this paper we introduce the concept of pre-Hamiltonian pairs of difference operators, demonstrate their connections with Nijenhuis operators and give a criteria for the existence of weakly nonlocal inverse recursion operators for differential {diference equations. We begin with a rigorous setup of the problem in terms of the skew field of rational (pseudo{diference) operators over a difference field with a zero characteristic subfield of constants and the principal ideal ring of matrix rational (pseudo{diference) operators. In particular, we give a criteria for a rational operator to be weakly nonlocal. A difference operator is called pre-Hamiltonian, if its image is a Lie subalgebra with respect to the Lie bracket on the difference field. Two pre-Hamiltonian operators form a pre-Hamiltonian pair if any linear combination of them is preHamiltonian. Then we show that a pre-Hamiltonian pair naturally leads to a Nijenhuis operator, and a Nijenhuis operator can be represented in terms of a pre-Hamiltonian pair. This provides a systematic method to check whether a rational operator is Nijenhuis. As an application, we construct a hamiltonian pair and thus a Nijenhuis recursion operator for the diferential{difference equation recently discovered by Adler & Postnikov. The Nijenhuis operator obtained is not weakly nonlocal. We prove that it generates an infinite hierarchy of local commuting symmetries. We also illustrate our theory on the well-known examples including the Toda, the Ablowitz{Ladik and the Kaup{Newell differential difference equations.