Confluent Crum-Darboux transformations in Dirac Hamiltonians with   $PT$-symmetric Bragg gratings
  We consider optical systems where propagation of light can be described by a Dirac-like equation with $PT$-symmetric Hamiltonian. In order to construct exactly solvable configurations, we extend the confluent Crum-Darboux transformation for the one-dimensional Dirac equation. The properties of the associated intertwining operators are discussed and the explicit form for higher-order transformations is presented. We utilize the results to derive a multi-parametric class of exactly solvable systems where the balanced gain and loss represented by the $PT$-symmetric refractive index can imply localization of the electric field in the material. 
Francisco Correa, Vit Jakubsky
hep-th math-ph math.MP nlin.SI physics.optics quant-ph