Spiked potentials and quantum toboggans

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