Potentials with two shifted sets of equally spaced eigenvalues and their Calogero spectrum

New solvable singular potentials

“…Specifically, the spectrum described in ref. [2] for the harmonic-oscillator like potential is identical to that previously found by us [4], where the discussion focused on real potentials with two sets of equally spaced eigenvalues. In this paper, we extend previous results and also find new real but singular potentials corresponding to the Pöschl-Teller I and Pöschl-Teller II potentials.…”

Algebraic Approach for Shape Invariant Potentials in Klein-Gordon Equation

“…Supersymmetry was first studied in the simplest case of supersymmetric quantum mechanics by Witten [6], Cooper and Freeman [7]. In 1983, the concept of shape invariant potential was introduced by Gendenstein [8], there has been considerable discussion in the literature of this technique and it has been applied to many of the solvable systems in quantum mechanics [7,9]. It is well known that a potential is said to be shape invariant, if its supersymmetry partner potential has the same spatial dependence as the original potential with possibly altered parameters, and the corresponding Hamiltonian can be solved [2,3,7,9,10,11].…”

Shape Invariant Potentials in Second-Order Supersymmetric Quantum Mechanics

“…Additionally, the shape invariant potentials constructed in the framework of 2-SUSY QM are similar to those shape invariant potentials in two steps discussed in the context of standard SUSY QM. That is, there must be an analogous potential algebra underlying the 2-SUSY shape invariant potentials, thus providing an alternative way of getting the energy eigenvalues by algebraic methods [34,46,69,71].…”

“…Having established the general properties for the shape invariant potentials in this class, now let us go back to (3.2). By solving the equation, we are able to construct three different kinds of solution [34,46] as follows.…”

“…A regularization that preserves SUSY and shape invariance needs to be introduced. In effect, the corresponding regularized potentials will exhibit an extra Dirac delta-function like singularity at the origin over the unregularized ones [34,46]. Class 2.…”