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 bound-state solutions to the Schrödinger equation, which are expressible in terms of Laguerre- or Jacobi-type X₁ exceptional orthogonal polynomials. Here we use second-order supersymmetric quantum mechanics to develop a systematic construction of potentials of this sort, which turn out to be isospectral to some well-known quantum potentials. We discuss in detail the example of rationally-extended radial oscillator potentials, related to X₁-Laguerre polynomials or to other Laguerre-type polynomials. The case of rationally-extended Scarf I potentials is also briefly sketched in connection with X₁-Jacobi polynomials or with generalizations thereof.