Extensions of solvable potentials with finitely many discrete eigenstates

TL;DR

This paper explores rational extensions of specific shape-invariant potentials with finitely many discrete eigenstates, introducing overshoot eigenfunctions as virtual states for constructing multi-indexed and iso-spectral extensions.

Contribution

It identifies overshoot eigenfunctions as unique virtual states for extending shape-invariant potentials with finitely many discrete eigenstates.

Findings

Overshoot eigenfunctions serve as virtual states for potential extensions.

Extensions preserve shape-invariance and spectral properties.

New classes of exactly solvable potentials are constructed.

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.