Hyperconfluent third-order supersymmetric quantum mechanics

Abstract

The hyperconfluent third-order supersymmetric quantum mechanics, in which all the factorization energies tend to a common value, is analyzed. It will be shown that the final potential as well can be achieved by applying consecutively a confluent second-order and a first-order SUSY transformations, both with the same factorization energy. The technique will be applied to the free particle and the Coulomb potential.

Recursive representation of Wronskians in confluent supersymmetric quantum mechanics

“…This coincides precisely with the result found in [13]. Besides providing a representation of the Wronskian on its left side, our identity (21) can be used to determine SUSY-transformed potentials (12) without the need to calculate the transformation functions u j , j ≥ 1.…”

“…We are therefore interested in choosing the transformation function such that our Wronskian remains free of singularities. In case of second-and third-order confluent SUSY transformations this was achieved by means of constructing a closed-form expression of the Wronskian [13]. In the following we will generalize this construction to arbitrary-order confluent SUSY transformations with the purpose to derive regularity conditions for the potentials resulting from such transformations.…”

Confluent Supersymmetric Partners of Quantum Systems Emerging from the Spheroidal Equation

“…For further details, particularly on spectral design, we refer the reader to [11,22,23] and references therein. We start out by considering an initial equation that has the general form…”

“…In the next step we substitute the series form (10), (11) for the solutions of the spheroidal Equation (8) into Equation (30). Note that due to the settings (21), the characteristic values of A will translate into discrete spectral values of E, denoted by E j , j a nonnegative integer.…”

“…Recent works on the confluent algorithm include its application to the inverted oscillator [7] and to the Dirac equation for pseudoscalar potentials [8]. Further results concern its equivalence with the zero-mode supersymmetry scheme [9], a representation using a differential formula [10,11] and higher-order confluent SUSY transformations [12]. The purpose of the present note is to apply the confluent SUSY algorithm to models that are governed by the spheroidal equation [13], a particular case of the confluent Heun class [14].…”