Combinatorial aspects of the Darboux transformation. (Q2781701)

The present article undertakes a formal, combinatorial analysis of the Darboux transformation for the one-dimensional Schrödinger equation. To this end the article introduces a new combinatorial technique: the spectral residue sequence. This is an infinite sequence of numbers that represents each given potential function, and is related in a nonlinear, invertible manner to the power-series coefficients of the latter. The spectral residue sequence possesses a number of remarkable properties; one of these is the fact that exact-solvability manifests as the factorizability of the spectral residues with respect to the potential coupling constants. Another property is the fact that a Darboux transformation of the potential function -- a nonlinear mapping -- corresponds to a simple negation of the spectral residues. This fact gives a useful criterion for establishing whether a given potential is shape invariant, and permits, in principle, the creation of new shape-invariant potentials.NEWLINENEWLINEFor the entire collection see [Zbl 0974.00040].