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Physica Scripta

The Dirac equation with P T /non- P T -symmetric potentials in curved spacetime backgrounds

Özlem Yeşiltaş, Ikram Imane Kouachi

Published Jun 2, 2023• 4 citations
Publisher

Abstract

Abstract In this paper, we have presented the Dirac equation in the frame of position-dependent mass on two dimensional gravitational static background in the presence of   /non-   -symmetric potential interactions. The exponential metric component have formed the reduced Dirac operator into the general supersymmetric model within mass changing with coordinates. We have obtained the eigenvalues of the Dirac operator for the complex Morse and trigonometric complex Scarf-II potentials SL (2, C ) using Lie algebras and the supersymmetric quantum mechanical approaches. Moreover, after obtaining a general Sturm-Liouville type equation using a convenient mapping, the system have become available to be investigated within η - pseudo-Hermiticity. With this context, η operator is found for the examples of complex trigonometric Rosen-Morse potential and complex Morse potentials with real and complex parameters of the initial system and finally the solutions are obtained for each model with the graphics of energy values and probability densities.

DOI
10.1088/1402-4896/acdb05

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Abstract In this paper, we have presented the Dirac equation in the frame of position-dependent mass on two dimensional gravitational static background in the presence of   /non-   -symmetric potential interactions. The exponential metric component have formed the reduced Dirac operator into the general supersymmetric model within mass changing with coordinates. We have obtained the eigenvalues of the Dirac operator for the complex Morse and trigonometric complex Scarf-II potentials SL (2, C ) using Lie algebras and the supersymmetric quantum mechanical approaches. Moreover, after obtaining a general Sturm-Liouville type equation using a convenient mapping, the system have become available to be investigated within η - pseudo-Hermiticity. With this context, η operator is found for the examples of complex trigonometric Rosen-Morse potential and complex Morse potentials with real and complex parameters of the initial system and finally the solutions are obtained for each model with the graphics of energy values and probability densities.
Abstract This paper presents a general method to solve non-hermeticity potentials with PT (Parity-Time) symmetry using two first-order operators against η -weak-pseudo-hermiticity having position-dependent effective mass. It futher suggests a way to apply this method to Dirac equation with spinor wave functions and non-hermetic potentials with real energy spectrum considering position-dependent effective mass. To show the utilities this method was apply to some superpotentials such as Eckart, Scarf-II, Rosen-Mörse-II and Pöchl-Teller. Finally, it was shown that the real potentials can be converted to complex potentials with eigenvalues of a class of η -pseudo-hermeticity.
We present a new procedure to construct the one-dimensional non-Hermitian imaginary potential with a real energy spectrum in the context of the position-dependent effective mass Dirac equation with the vector-coupling scheme in 1+1 dimensions. In the first example, we consider a case for which the mass distribution combines linear and inversely linear forms, the Dirac problem with a PT -symmetric potential is mapped into the exactly solvable Schrödinger-like equation problem with the isotonic oscillator by using the local scaling of the wavefunction. In the second example, we take a mass distribution with smooth step shape, the Dirac problem with a non- PT -symmetric imaginary potential is mapped into the exactly solvable Schrödinger-like equation problem with the Rosen–Morse potential. The real relativistic energy levels and corresponding wavefunctions for the bound states are obtained in terms of the supersymmetric quantum mechanics approach and the function analysis method.
PT\mathcal {PT}PT symmetric Hamiltonian model and Dirac equation in 1+1 dimensions
In this paper, we have introduced a symmetric non-Hermitian Hamiltonian model which is given as where ω and α are real constants, and are first-order differential operators. The Hermitian form of the Hamiltonian is obtained by suitable mappings and it is interrelated to the time-independent one-dimensional Dirac equation in the presence of position-dependent mass. Then, Dirac equation is reduced to a Schrödinger-like equation and two new complex non- symmetric vector potentials are generated. We have obtained a real spectrum for these new complex vector potentials using the shape invariance method. We have searched the real energy values using numerical methods for the specific values of the parameters.
Jia and Dutra ( 2006 J. Phys. A: Math. Gen. 39 11877 ) have considered the one- dimensional non-Hermitian complexified potentials with real spectra in the context of position-dependent mass in the Dirac equation. In their second example, a smooth step shape mass distribution is considered and a non-Hermitian non- PT -symmetric Lorentz vector potential is obtained. They have mapped this problem into an exactly solvable Rosen–Morse Schrödinger model and claimed that the energy spectrum is real. The energy spectrum they have reported is pure imaginary or at best forms an empty set. Their claim on the reality of the energy spectrum is fragile, therefore.
In this paper, the relativistic Dirac equation in one dimension is investigated for a particle in an external electromagnetic field, with the property of position-dependent effective mass approximation (PDEM), in the absence of vector potential. By removing the lower spinor component and combining the pair of equations, a Schrödinger-like equation is obtained for the upper spinor component. Using canonical transformations and introducing two first-order Hermitian and anti-Hermitian differential operators, a formalism for pseudo-hermitic Hamiltonians with parity-time reversal symmetry (PT) has been obtained. Comparing the equation derived from pseudo-Hermitian Hamiltonian with the non-relativistic Schrödinger equation leads to a general formalism for one-dimensional solvable imaginary non-Hermitian potentials with real energy spectra. Also, using this process, the complex potentials of Pöschl-Teller and Scarf II with real energy spectra in Dirac equation with PDEM approximationand PT symmetry have been investigated and their application has been expressed. For some particular parameters we will see the phenomenon of energy-levels crossing. In fact, it means that energy levels disappear from the spectrum. Also, for the mentioned examples, potential figures are drawn.
The bound state solutions of Dirac equation for q-deformed trigonometric Scarf potential with q-deformed trigonometric cotangent and cosecant tensor coupling potential under spin and pseudospin symmetric limits are investigated using Supersymmetric Quantum Mechanics (SUSY QM) method. The new tensor potentials proposed is inspired by superpotential form in SUSY quantum mechanics. The Dirac equations for Scarf potential coupled by new tensor potential in the pseudospin and spin symmetric cases reduce to Schrodinger type equations for shape invariant potential since the proposed new potentials are equivalent to the superpotential of q-deformed trigonometric Scarf potential. The relativistic wave functions are exactly obtained by using SUSY operator method and the relativistic energy equation are exactly obtained by using SUSY method and the idea of shape invariance in the approximation scheme of centrifugal term. The new tensor potential causes the energy degeneracies is omit both for pseudospin and spin symmetric case.