Quantum toboggans with two branch points

In an innovated version of PTPT-symmetric quantum mechanics, wave functions ψ(QT)(x)ψ(QT)(x) describing quantum toboggans (QT) are defined along complex contours of coordinates x(s)x(s) which spiral around the branch points x(BP)x(BP). In the first nontrivial case with x(BP)=±1x(BP)=±1, a classification is found in terms of certain “winding descriptors” ϱ. For some  ϱ0ϱ0, a mapping x(ϱ)(s)→y(0)(s)x(ϱ)(s)→y(0)(s) is presented which rectifies the contours and which enables us to extend, to our QTs, the standard proofs of the reality/observability of the energy spectrum.