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

• Wronskian representation of TDPT and Bessel potentials.
• Simple proof of Gaillard–Matveev theorem.
• Regular and singular extensions of the constant potential.
• Darboux transformations based on excited states.
• Confluent case and Wronskian Rayleigh formula.

We propose a simple alternative proof of the Wronskian representation formula obtained by Gaillard and Matveev for the trigonometric Darboux–Pöschl–Teller (TDPT) potentials. It rests on the use of singular Darboux–Bäcklund transformations applied to the free particle system combined to the shape invariance properties of the TDPT.