On a transformation of Bohl and its discrete analogue

Fritz Gesztesy's varied and prolific career has produced many transformational contributions to the spectral theory of one-dimensional Schr\"odinger equations. He has often done this by revisiting the insights of great mathematical analysts of the past, connecting them in new ways, and reinventing them in a thoroughly modern context. In this short note we recall and relate some classic transformations that figure among Fritz Gestesy's favorite tools of spectral theory, and indeed thereby make connections among some of his favorite scholars of the past, Bohl, Darboux, and Green. After doing this in the context of one-dimensional Schr\"odinger equations on the line, we obtain some novel analogues for discrete one-dimensional Schr\"odinger equations. \smallskip Dem einzigartigen Fritz gewidmet.