No abstract

By imposing special compatible similarity constraints on a class of integrable partial q-difference equations of KdV-type we derive a hierarchy of seconddegree ordinary q-difference equations. The lowest (non-trivial) member of this hierarchy is a second-order second-degree equation which can be considered as an analogue of equations in the class studied by Chazy. We present corresponding isomonodromic deformation problems and discuss the relation between this class of difference equations and other equations of Painlevé type.

Abstract.The path integral approach to the quantization of one degree-of-freedom Newtonian particles is considered within the discrete time-slicing approach, as in Feynman's original development. In the time-slicing approximation the quantum mechanical evolution will generally not have any stationary states. We look for conditions on the potential energy term such that the quantum mechanical evolution may possess stationary states without having to perform a continuum limit. When the stationary states are postulated to be solutions of a second-order ordinary differential equation (ODE) eigenvalue problem it is found that the potential is required to be a solution of a particular first-order ODE. Similarly, when the stationary states are postulated to be solutions of a second-order ordinary difference equation (O∆E) eigenvalue problem the potential is required to be a solution of a particular first-order O∆E. The classical limits (which are at times very nontrivial) are integrable maps.

The discrete equations of motion for the quantum mappings of KdV type are given in terms of the Sklyanin variables (which are also known as quantum separated variables). Both temporal (discrete-time) evolutions and spatial (along the lattice at a constant time-level) evolutions are considered. In the classical limit, the temporal equations reduce to the (classical) discrete Dubrovin equations as given in a previous publication (Nijhoff F W 2000 Chaos, Solitons and Fractals 11 19-28). The reconstruction of the original dynamical variables in terms of the Sklyanin variables is also achieved.