New exactly solvable Hamiltonians: Shape invariance and self-similarity

Abstract

We discuss in some detail the self-similar potentials of Shabat and Spiridonov which are reflectionless and have an infinite number of bound states. We demonstrate that these self-similar potentials are in fact shape invariant potentials within the formalism of supersymmetric quantum mechanics. In particular, using a scaling ansatz for the change of parameters, we obtain a large class of new, reflectionless, shape invariant potentials of which the Shabat-Spiridonov ones are a special case.These new potentials can be viewed as q-deformations of the single soliton solution corresponding to the Rosen-Morse potential. Explicit expressions for the energy eigenvalues, eigenfunctions and transmission coefficients for these potentials are obtained. We show that these potentials can also be obtained numerically. Included as an intriguing case is a shape invariant double well potential whose supersymmetric partner potential is only a single well. Our class of exactly solvable Hamiltonians is further enlarged by examining two new directions: (i) changes of parameters which are different from the previously studied cases of translation and scaling; (ii) extending the usual concept of shape invariance in one step to a multi-step situation. These extensions can be viewed as q-deformations of the harmonic oscillator or multi-soliton solutions corresponding to the Rosen-Morse potential.(

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

“…was studied and shown to produce new classes of shape invariant potentials [3,4], which included self-similar potentials [12] as a special case. The above mentioned change of parameters are shown in Fig.…”

“…1 have not been discussed in the literature. We will obtain simple, explicit potentials using an approach motivated by recent developments [3,4] in obtaining new shape invariant potentials [5] in supersymmetric quantum mechanics [2,6]. These advances have been made using novel choices of the function f appearing in the change of parameters a 1 = f (a 0 ) in the shape invariance condition.…”

Parasupersymmetry in quantum mechanics

“…We shall see later that in this way we will be able to go much beyond the factorization method and obtain a huge class of new solvable potentials [59].…”

“…Recently, a new class of SIPs has been discovered which involves a scaling of parameters [58]. These new potentials as well as multi-step SIPs [59] have been studied, and their connection with self-similar potentials as well as with q-deformations has been explored [60,61,62,63].…”

Point Canonical Transformation versus Deformed Shape Invariance for Position-Dependent Mass Schrödinger Equations

“…For such a purpose, we have used an approach inspired by a branch of supersymmetric quantum mechanics [17,18,19], whose development dates back to that of quantum groups and q-algebras [20,21,22,23,24,25,26]. It consists in considering those one-dimensional potentials that are translationally shape invariant for a constant mass and in deforming the corresponding shape invariance condition in such a way that it remains solvable.…”