New exactly solvable Hamiltonians: shape invariance and self-similarity

Abstract

We discuss in some detail the self-similar potentials of Shabat [Inverse Prob. 8, 303 (1992)] and Spiridonov [Phys. Rev. Lett. 69, 298 (1992)] 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 and (ii) extending the usual concept of shape invariance in one step to a multistep situation. These extensions can be viewed as q deformations of the harmonic oscillator or multisoliton solutions corresponding to the Rosen-Morse potential.