Quantum toboggans with two branch points

Abstract

In an innovated version of PT −symmetric Quantum Mechanics, wave functions ψ (QT ) (x) describing quantum toboggans (QT) are defined along complex contours of coordinates x(s) which spiral around the branch points x (BP ) . In the first nontrivial case with x (BP ) = ±1, a classification is found in terms of certain "windingis 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.

Rectifiable PT -symmetric Quantum Toboggans with Two Branch Points

“…It will be used here in the form introduced in Ref. [12], where each curve C ( ) (s) has been assumed moving from its "left asymptotics" (where s −1) to a point which lies below one of the branch points x (this move is represented by letter P or symbol R −1 ≡ P ). In this manner we may compose the moves and characterize each contour by a word composed of the sequence of letters selected from the four-letter alphabet R, L, Q and P .…”

Time-Dependent and/or Nonlocal Representations of Hilbert Spaces in Quantum Theory

“…This difficulty becomes almost insurmountable when the wave functions describing quantum toboggans happen to possess two or more branch points (cf. [8] for an illustrative example). For these reasons it is recommended to rectify the tobogganic integration paths via a suitable change of variables in a preparatory step [9].…”

Supersymmetric quantum mechanics living on topologically non-trivial Riemann surfaces

“…Marginally let us note that a set of nice analytic examples of wave functions where M went up to five has been constructed by Sinha and Roy [16] and that the amazing combinatorial complications related to an exhaustive classification of the nonequivalent QT paths already emerge in the first less trivial case with M = 2 [5]. This being said, our final return to the QT SUSY QM is very easy because once we overcome the combinatorial classification barriers and choose any particular member T of the family of conjugations (22) we only have to modify our abovementioned formula (17) and set…”

“…We intend to show how, on an overall background of quantum mechanics, one could interconnect the purely algebraic concept of supersymmetry (SUSY, e.g., review [1]) with the more or less purely analytic concept of quantum toboggans (see refs [2][3][4][5][6] or a compact review paper [7]). …”