Phase equivalent potentials, Complex coordinates and Supersymmetric Quantum Mechanics

Supersymmetric Quantum Mechanics may be used to construct reflectionless potentials and phase-equivalent potentials. The exactly solvable case of the $\lambda sech^2$ potential is used to show that for certain values of the strength $\lambda$ the phase-equivalent singular potential arising from the elimination of all the boundstates is identical to the original potential evaluated at a point shifted in the complex cordinate space. This equivalence has the consequence that certain general relations valid for reflectionless potentials and phase-equivalent potentials lead to hitherto unknown identities satisfied by the Associated Legendre functions. This exactly solvable probelm is used to demonstrate some aspects of scattering theory.