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

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