Intertwining of exactly solvable Dirac equations with one-dimensional potentials

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Abstract
The method of intertwining is used to construct transformations between one-dimensional electric potentials or one-dimensional external scalar fields for which the Dirac equation is exactly solvable. The transformations are analogous to the Darboux transformations between Schrödinger potentials. It is shown that a class of exactly solvable Dirac potentials corresponds to soliton solutions of the modified Korteweg–deVries (MKdV) equation, just as certain Schrödinger potentials are solitons of the Korteweg–deVries equation. It is also shown that the intertwining transformations are related to Bäcklund transformations for MKdV. The structure of the intertwining relations is shown to be described by an N=4 superalgebra, generalizing supersymmetric quantum mechanics to the Dirac case.