Symmetry classification of differential equations and reduction techniques

The symmetry classification of differential equations containing arbitrary functions can be a source of several interesting results. We study two particular but significant examples: a nonlinear ODE and a linear PDE (the 1-dimensional Schr\"odinger equation). We provide first of all a necessary, but very restrictive, simple condition involving the arbitrary functions in order that the given equation does admit Lie symmetries. In the first example, we show that some symmetry appears only if a precise numerical relation between the involved parameters is satisfied. In the case of Schr\"odinger equation, we see that only for a very limited class of potential functions some symmetry is admitted, and that the Lie generators of these symmetries are precisely recursion operators and are related to the Dirac step up - step down operators, well known in Quantum Mechanics. In connection with all these symmetries, we also discuss the important problem of the reduction of the differential equations, in both the different contexts of ODE's and of PDE's.