Although the costly experiments performed on gigantic accelerators
did not confirm the predicted presence of supersymmetry (i.e., of
certain correspondence between fermions and bosons) in nature, the
underlying mathematical idea itself survived and found its second
life in the theory as well as in the applications of ordinary linear
differential equations. Its use contributed, first of all, to the
study of solvable Schr\"{o}dinger equations describing a point
particle confined by a one-dimensional shape-invariant potential
(cf. review [2]). The formalism (not unrelated to the theory of
Darboux transformations and non-linear Riccati equations) made the
classification of these potentials transparent. The present paper
contributes to the current effort aimed at the extension of this
classification to the coupled systems of Schr\"{o}dinger equations
(or, if you wish, to the solvable models of confined particles with
spin, etc). In a continued development and partial completion of the
results of their last-year paper [17], the authors base their
classification on the so called superpotentials (= the concept
related, roughly speaking, to the logarithmic derivative of the
ground-state wavefunction in the scalar case). Due attention is paid
to the applicability of these results to the (separable cases of)
motion in more dimensions and to the detailed description of the
solvable shape-invariant models using two coupled equations (where
the authors arrive at a complete classification) and 3 coupled
equations.
MR2847058 Nikitin, Anatoly G.; Karadzhov, Yuri Enhanced
classification of matrix superpotentials. J. Phys. A 44 (2011), no.
44, 445202, 24 pp. 81Qxx