Generalization of the Concept of Invariance of Differential Equations. Results of Applications to Some Schrödinger Equations

We have found that differential equations can be form invariant under a larger class of infinitesimal transformations than those considered by Lie and Ovsjannikov. We give a generalization of the concept of point transformation. It is necessary for the systematic determination of the generators of continuous invariance groups of, e.g., the partial differential equations of physics. Applying it to Schr\"odinger's equation, time-dependent constants of the motion are found systematically, as illustrated here for the hydrogen atom.