Dirac equation in minimal length quantum mechanics with energy- dependent harmonic potential

Bound states of the two-dimensional Dirac equation for an energy-dependent hyperbolic Scarf potential

“…Hence, if either condition (32) or (33) are satisfied, then the corresponding solution in (29) vanishes at both infinities. Furthermore, the derivative of (29) with respect to x shows the same behavior, implying that the remaining component (16) forming our solution spinor (14) vanishes at both infinities. For the sake of brevity we omit to show this rigorously, as it would require a similar series of considerations as done above for the function (29).…”

“…where we assume without restriction that k y = 0. Once the first solution component ψ 1 has been found from the Schrödinger-type equation (15), the remaining counterpart ψ 2 is generated by means of (16). These two functions are then substituted into (14) in order to obtain the solution spinor of the stationary Dirac equation.…”

“…Note that we included the parameter n as an index in order to indicate the bound-state character. In order to construct the solution to the Dirac equation (3), we calculate ψ 2,n by means of (16). After that, we can calculate the norm of these bound-state solutions by means of (17).…”

“…These two functions are then substituted into (14) in order to obtain the solution spinor of the stationary Dirac equation. Before we apply the results of this section to a specific model, we rewrite the orthogonality relation (11) and norm (12) in terms of the solutions to the system ( 15), (16). To this end, we introduce two pairs of functions ψ 1,m , ψ 2,m and ψ 1,n , ψ 2,n that are solutions to the latter system for E = E m and E = E n , respectively.…”

“…In this note we show that a similar type of modification is also necessary in the relativistic context if the potential is energy-dependent. There is very few literature on the topic, particular systems were studied for example in [13] [15], [16]. We consider here a particular case of the two-dimensional, massless Dirac equation for an external scalar potential that we assume to depend on the energy and on a single spatial variable.…”