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

Abstract

Abstract. Two different exactly solvable systems are constructed using the supersymmetric quantum mechanics formalism and a pseudoscalar one-dimensional version of the DiracMoshinsky oscillator as a departing system. One system is built using a first-order SUSY transformation. The second is obtained through the confluent supersymmetry algorithm. The two of them are explicitly designed to have the same spectrum as the departing system and pseudoscalar potentials.Keywords: Dirac equation, Dirac-Moshinsky oscillator, pseudoscalar potential, confluent SUSY algorithm 1. Introduction Supersymmetric quantum mechanics (SUSY) is a tool that allows to find new exactly solvable stationary Schrödinger equations. Moreover, the SUSY formalism provides a method to obtain, in principle, a quantum system with a desired spectrum. To apply it we need a solvable initial system and at least one of its solutions, as a result a new system, often referred as SUSY partner, and its solutions are generated [9,11,16]. 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. This paper is organized as follows. Section 2 gives a brief review of the first-order and the confluent case of supersymmetric quantum mechanics. In section 3 the confluent supersymmetry algorithm for Dirac equations is described. Section 4 is devoted to present and apply the SUSY

The eigenvalue problem of one-dimensional Dirac operator

“…The one-dimensional Dirac equation is a nice playground for designing analytical solutions in some non-trivial environments. Here are some examples: Bound states in doublewell potentials [39], pseudo-scalar potential barrier [40], construction of transparent potentials [41,42], quadratic plus inversely quadratic potential [43], Kratzer potential [44], Wood-Saxon potential and effective mass problem [45], hyperbolic tangent potential [46] and Dirac-Moshinsky oscillator [47], just to mention several examples. Among innovative, non-standard, applications one should mention a work by Correa and Jakubsky on the description of optical systems, in the coupled mode theory of the Bragg gratings using 1D Dirac equation with a non-Hermitian Hamiltonian [48].…”