New exactly solvable periodic potentials for the Dirac equation

Abstract

Abstract.A new exactly solvable relativistic periodic potential is obtained by the periodic extension of a well-known transparent scalar potential. It is found that the energy band edges are determined by a transcendental equation which is very similar to the corresponding equation for the Dirac Kronig-Penney model. The solutions of the Dirac equation are expressed in terms of elementary functions.

Dirac systems with magnetic field and position dependent mass: Darboux transformations and equivalence with generalized Dirac oscillators

“…We observe that in comparison to its counterpart (39), this equation contains a term proportional to k y because the settings (62) do not comply with the condition (35). Equation ( 63) is exactlysolvable with particular solution…”

“…This is so because the parameter k y must be real-valued due to our definition (12) of the Dirac solution. Let us also point out that the term (35) is determined once the mass m and the potential V have been chosen.…”

“…We observe here that the term proportional to k y has vanished. This is so because our choice of parameters in (37) satisfies (35). The general solution of equation ( 39) can be written as…”

“…In this section we shall apply Darboux transformation to a more general relativistic system, namely, Dirac equation in the presence of a matrix potential [9] [16] [37] [35] and find new matrix potentials for which the Dirac equation remains solvable. More precisely, we consider our initial Dirac equation in the form…”

Bloch Solutions of Periodic Dirac Equations in SPPS Form

“…In this work, we are interested in the same connection in the case of periodic potentials, see e.g. [12]. We here write the Dirac Bloch solutions in Kravchenko form (power series in the spectral parameter) and also the Dirac Hill discriminant in the same form and apply the results to an interesting quasi-exactly solvable periodic potential.…”