Conformal bridge transformation, $\mathcal{PT}$- and super- symmetry

Supersymmetric extensions of the 1D and 2D Swanson models are investigated by applying the conformal bridge transformation (CBT) to the first order Berry-Keating Hamiltonian multiplied by $i$ and its conformally neutral enlargements. The CBT plays the role of the Dyson map that transforms the models into supersymmetric generalizations of the 1D and 2D harmonic oscillator systems, allowing us to define pseudo-Hermitian conjugation and a suitable inner product. In the 1D case, we construct a $\mathcal{PT}$-invariant supersymmetric model with $N$ subsystems by using the conformal generators of supersymmetric free particle, and identify its complete set of the true bosonic and fermionic integrals of motion. We also investigate an exotic $N=2$ supersymmetric generalization, in which the higher order supercharges generate nonlinear superalgebras. We generalize the construction for the 2D case to obtain the $\mathcal{PT}$-invariant supersymmetric systems that transform into the spin-1/2 Landau problem with and without an additional Aharonov-Bohm flux, where in the latter case, the well-defined integrals of motion appear only when the flux is quantized. We also build a 2D supersymmetric Hamiltonian related to the "exotic rotational invariant harmonic oscillator" system governed by a dynamical parameter $\gamma$. The bosonic and fermionic hidden symmetries for this model are shown to exist for rational values of $\gamma$.