The confluent supersymmetry algorithm for Dirac equations with pseudoscalar potentials

Abstract

We introduce the confluent version of the quantum-mechanical supersymmetry (SUSY) formalism for the Dirac equation with a pseudoscalar potential. Application of the formalism to spectral problems is discussed, regularity conditions for the transformed potentials are derived, and normalizability of the transformed solutions is established. Our findings extend and complement former results [21].

Spectrally isomorphic Dirac systems: Graphene in an electromagnetic field

“…The presented unitary transformation can set the mass to a constant and induce an inhomogeneous scalar potential. It can provide the new isospectral partners of the recently discussed Dirac systems with reflection-less potential [27], [28] or inhomogeneous magnetic fields [29], [30]. It is worth noticing that a different approach to construction of solvable systems described by Dirac equation was developed in [31].…”

Confluent Supersymmetric Partners of Quantum Systems Emerging from the Spheroidal Equation

“…Note that although h can become imaginary, it enters in the spheroidal equation and in the recursion Equations (12)- (15) for the series coefficients merely as the real-valued h 2 , therefore preventing the spectral values from becoming complex. After substituting both Equation (20) and the settings (21) into the boundary-value problem (8), (9), we obtain after some elementary manipulations…”

“…We will now generate another quantum model from the spheroidal Equation (8). Since the process of construction follows similar steps as in the previous section, we will not give details for all our calculations.…”

“…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].…”

One dimensional Dirac-Moshinsky oscillator-like system and isospectral partners

“…This tool has been applied to a large variety of quantum interactions such as the harmonic oscillator [15], the hydrogen atom [12,21], the Pöschl-Teller potentials [6], among many others. Moreover, it has been possible to extend the SUSY formalism so it can be used to find solutions of different kind of equations like the FokkerPlanck equation [22,24] [5,7,8,10,14,18,20].The purpose of this work is to apply the SUSY technique to a one dimensional Dirac Moshinsky oscillator-like system [17,23] to obtain new isospectral families using the supersymmetric quantum mechanic formalism. In particular we will focus on two versions of this technique: the first-order SUSY quantum mechanics will be used to generate a one-parameter family of isospectral systems, this part is closely related to Bogdan Mielnik's seminal article on isospectral Hamiltonians to the harmonic oscillator, and second, a three parametric family will be generated through the confluent supersymmetry algorithm.…”