A short proof of the Gaillard-Matveev theorem based on shape invariance arguments

We propose a simple alternative proof of the Wronskian representation formula obtained by Gaillard and Matveev for the Darboux-P"oschl-Teller (DPT) potentials. It rests on the use of singular Darboux-B"acklund transformations applied to the free particle system combined to the shape invariance properties of the DPT.