Spiked potentials and quantum toboggans

Even if the motion of a quantum (quasi-)particle proceeds along a left-right-symmetric (PT-symmetric) curved path in complex plane, the spectrum of bound states may remain physical, i.e., real and bounded below). We propose a generalization. Firstly, we show how the topologically less trivial (tobogganic) contours may be allowed to live on several sheets of a Riemann surface. Secondly, the specification of a scattering regime is formulated for such a class of models.