
Symmetry in Nonlinear Mathematical Physics  2009
Christiane Quesne (Universite Libre de Bruxelles, Brussels, Belgium)
Solvable rational potentials in supersymmetric quantum mechanics
Abstract:
It has been recently shown by a point canonical
transformation method approach that some (so far unknown) exactly solvable
rational potentials give rise to boundstate solutions to the
Schrödinger equation, which are expressible in terms of Laguerre or
Jacobitype X_{1} exceptional orthogonal polynomials. Here
we use secondorder supersymmetric quantum mechanics to
develop a systematic
construction of potentials of this sort, which turn out to be isospectral
to some wellknown quantum potentials. We discuss in detail
the example of rationallyextended radial oscillator potentials,
related to X_{1}Laguerre polynomials or to other Laguerretype
polynomials. The case of rationallyextended Scarf I potentials is
also briefly sketched in connection with X_{1}Jacobi polynomials or
with generalizations thereof.

