Near‐exact supersymmetric partner potentials: Construction and exploitation

Abstract

Precise supersymmetric (SUSY) partner potentials can be generated only for exactly solvable problems of the stationary Schrö dinger equation. This is a severe restriction, as most problems are not amenable to exact solutions. We employ here a linear variational strategy to explicitly construct approximate SUSY partners of a few common, not exactly solvable potentials and subsequently examine their properties to explore the advantages in practical implementation. The efficacy of our proposed scheme is commendable. We demonstrate that, for symmetric potentials, the constructed partners may be so good that the overall recipe has the nicety of generating the whole eigenspectrum by employing only half of the full Hilbert space functions. A similar strategy is shown to work for the odd states too, with proper boundary conditions. Pilot calculations involve a number of low-lying states of some mixed oscillator and doublewell potentials. Analysis of the results reveals a few interesting features of the problem of construction of approximate SUSY partners and their practical use. Particularly, we identify places where the operator-level approximations are involved and how far they affect the bounding properties of energies that are obtained as eigenvalues of a matrix diagonalization problem associated with linear variations.

Approximate analytical versus numerical solutions of Schrödinger equation under molecular Hua potential

“…They also provide us with a deeper understanding of the physics of the problems. The most common analytical techniques include supersymmetry quantum mechanics (SUSYQM),10–14 Nikiforov–Uvarov,15, 16 point canonical transformation,17 Ansatz solution,18–19 Pekeris‐type approximation,20 Lie algebraic methods,21 Taylor expansion,22 and so forth. Here, we follow both SUSYQM and finite difference methods to obtain reliable solutions.…”