Conformal bridge transformation and PT symmetry

Abstract The conformal bridge transformation (CBT) is reviewed in the light of PT symmetry. Originally, CBT was presented as a non-unitary (a complex canonical classical case) that relates two different forms dynamics sense Dirac. Namely, it maps asymptotically free form into harmonically confined associated with mathvariant="fraktur">s mathvariant="fraktur">o ( 2 /> 1 stretchy="false">) ? mathvariant="fraktur">l ? However, non-Hermitian operator i D ^ , where generator dilations, compact Hermitian mathvariant="script">J 0 algebra, can be -symmetric metric. In this work we review applications for one- and two-dimensional systems, well systems on cosmic string background, conformally extended charged particle field Dirac monopole. We also compare unify Darboux transformation. latter used to construct solutions equations KdV hierarchy properties extreme waves. As new result, by using modified relate one-dimensional -regularized mechanics model version de Alfaro, Fubini Furlan system.