Remarks on Quantum Differential Operators

In the course of writing the book [9] and various papers [10, 11, 12, 13, 14, 15, 16] we encountered many q-differential equations but were frustrated by a lack of understanding about natural forms for such equations. One has operators of the type qKP or qKdV for example but even there, expressing the resulting equations (even via Hirota type equations or in bilinear form) seemed curiously difficult. On the other hand Laplace operators, wave operators, heat operators, and Schrödinger operators have been written down in various forms and their symmetries studied (see e.g. [5, 6, 7, 9, 30, 46, 59, 60]). Also various operators associated with q-special functions have been isolated and studied (see e.g. [9, 36, 37, 49, 64]). However when we started deriving nonlinear differential equations from zero curvature conditions on a quantum plane for example (following the procedure of [27, 28]) we were puzzled about their meaning, their solvability and their relation to qKP for example. Thus it seems appropriate to partially survey the area of q-differential operators and isolate the more significant species while looking also for techniques of solvability. One expects of course that meaningful equations will involve quantum groups (QG) in some way. We discuss various examples and techniques relative to q-differential equations and give some new results along with some expository material. The one dimensional linear situation involving q-special functions seems to be fairly well understood but the multidimensional nonlinear situation leaves still many directions for further study.