The Darboux Method: Supersymmetric Partner Potentials

Abstract

Even if the two partner potentials, V(x, m)and V(x, m–1), of the form [k ²(x, m)∓ k’(x, m) + L(m)] of eqs. (7) and (8) of the last chapter do not have the same shape, it may still be possible to say something about the eigenvalue spectrum of the partner potential if the eigenvalues of one of the potentials are known. This will be true whether or not the potentials are functions of a parameter, m. This has been known since 1882 through the work of G. Darboux; (Comptes Rendus Acad. de Sci. (Paris) 94(1882)1456). This 19^th century work has only recently been rediscovered by quantum theorists in connection with work in particle physics on supersymmetry. Hence, the partner potentials are known as supersymmetric partner potentials.