Quantum toboggans: models exhibiting a multisheeted PT-symmetry

A generalization of the concept of PT-symmetric Hamiltonians H = p2 + V (x) is presented. For the usual analytic potentials V (x) (with singularities) and for the recently widely accepted 'PT-symmetric' asymptotic boundary conditions for wave functions ψ(x) (selected inside a pair of complex wedges generalizing the usual x → ±∞ asymptotics), non-equivalent quantum toboggans are defined as integrated along topologically different paths C of coordinates x ε Cl.