Krein-Adler transformations for shape-invariant potentials and pseudo   virtual states
  For eleven examples of one-dimensional quantum mechanics with shape-invariant potentials, the Darboux-Crum transformations in terms of multiple pseudo virtual state wavefunctions are shown to be equivalent to Krein-Adler transformations deleting multiple eigenstates with shifted parameters. These are based upon infinitely many polynomial Wronskian identities of classical orthogonal polynomials, i.e. the Hermite, Laguerre and Jacobi polynomials, which constitute the main part of the eigenfunctions of various quantum mechanical systems with shape-invariant potentials. 
Satoru Odake and Ryu Sasaki
math-ph hep-th math.CA math.MP nlin.SI quant-ph