Extensions of solvable potentials with finitely many discrete eigenstates

Abstract

We address the problem of rational extensions of six examples of shape-invariant potentials having finitely many discrete eigenstates. The overshoot eigenfunctions are proposed as candidates unique in this group for the virtual state wavefunctions, which are an essential ingredient for multi-indexed and iso-spectral extensions of these potentials. They have exactly the same form as the eigenfunctions but their degrees are much higher than n_{max} so that their energies are lower than the groundstate energy.

22 pages, 3 figures. Typo corrected, comments and references added. To appear in J.Phys.A. arXiv admin note: text overlap with arXiv:1212.6595