Supersymmetric quantum mechanics is constructed in a new non-Hermitian representation. Firstly, the map between the partner operators $H^{ (\pm)}$ is chosen antilinear. Secondly, both these components of a super-Hamiltonian ${ \cal H}$ are defined along certain topologically nontrivial complex curves $r^{ (\pm)}(x)$ which spread over several Riemann sheets of the wave function. The non-uniqueness of our choice of the map ${ \cal T}$ between "tobogganic" partner curves $r^{ (+)}(x)$ and $r^{ (-)}(x)$ is emphasized.
Categories: math-ph math.MP quant-ph