Bound states of the one-dimensional Dirac equation for scalar and vector double square-well potentials

Abstract

We have analytically studied bound states of the one-dimensional Dirac equation for scalar and vector double square-well potentials (DSPs), by using the transfer-matrix method. Detailed numerical calculations of the eigenvalue, wave function and density probability have been performed for the three cases: (1) vector DSP only, (2) scalar DSP only, and (3) scalar and vector DSPs with equal magnitudes. We discuss the difference and similarity among results of the cases (1)- (3)

Fermions in the background of mixed vector–scalar–pseudoscalar square potentials

“…The square potentials, wells and barriers, are models widely used in low-dimensional systems such as the quantum dots [24], Dirac fermions in graphene [25], electrons in semiconductor heterostructures [26] and theoretical studies [27,28]. Besides these applications, the well and barrier potentials are extremely examples used as toy-models in textbooks (for example, see [7] ), further increasing its importance in quantum mechanics.…”

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].…”

Low-regularity integrators for nonlinear Dirac equations

“…Equation (1.1) occurs as a core part of many related models, such as the Gross-Neveu system [33], the Dirac-Klein-Gordon system [16,53], the Maxwell-Dirac system [18,32] and the Chern-Simons-Dirac system [10]. Due to these applications, the numerical solutions of the NDE (1.1) are widely interested in computational physics [4,11,24,32,35,40,45], especially the bound states, the solitary waves and the interaction dynamics of solitons [1,29,55]. These existing works in computational physics or applied mathematics mainly assume that the initial data of NDE is smooth enough for theoretical and/or numerical studies.…”