Rectifiable PT-symmetric Quantum Toboggans with Two Branch Points

Certain complex-contour (a.k.a. quantum-toboggan) generalizations of Schroedinger's bound-state problem are reviewed and studied in detail. Our key message is that the practical numerical solution of these atypical eigenvalue problems may perceivably be facilitated via an appropriate complex change of variables which maps their multi-sheeted complex domain of definition to a suitable single-sheeted complex plane.


Introduction
One-dimensional Schrödinger equation for bound states − 2 2m d 2 dx 2 ψ n (x) + V (x) ψ n (x) = E n ψ n (x) , ψ n (x) ∈ L 2 (R) (1) belongs among the most friendly phenomenological models in quantum mechanics [1]. For virtually all of the reasonable phenomenological confining potentials V (x) the numerical treatment of this eigenvalue problem remains entirely routine.
During certain recent numerical experiments [2] it became clear that many standard (e.g., Runge-Kutta [3]) computational methods may still encounter new challenges when one follows the advice by Bender and Turbiner [4], by Buslaev and Grecchi [5], by Bender et al [6] or by Znojil [7] and when one replaces the most common real line of coordinates x ∈ R in ordinary differential Eq. (1) by some less trivial complex contour of x ∈ C(s) which may be conveniently parametrized, whenever necessary, by a suitable real pseudocoordinate s ∈ R, Temporarily, the scepticism has been suppressed by Weideman [8] who showed that many standard numerical algorithms may be reconfirmed to lead to reliable results even for many specific analytic samples of complex interactions V (x) giving real spectra via Eq. (2).
Unfortunately, the scepticism reemerged when we proposed, in Ref. [7], to study the so called quantum toboggans characterized by the relaxation of the most common tacit assumption that the above-mentioned integration contours C(s) must always lie just inside a single complex plane R 0 equipped by suitable cuts. Subsequently, the reemergence of certain numerical difficulties accompanying the evaluation of the spectra of quantum toboggans has been reported by Bíla [9] and by Wessels [10].
Their empirical detection of the presence of instabilities in their numerical results may be recognized as one of the key motivations of our present considerations. 2 Illustrative tobogganic Schrödinger equations

Assumptions
Whenever the complex integration contour C(s) used in Eq. (2) becomes topologically nontrivial (cf. Figures 1 -4 for illustration), it may be interpreted as connecting several sheets of the Riemann surface R (multisheeted) supporting the general solution ψ (general) (x) of the underlying complex ordinary differential equation. It is well known that these solutions ψ (general) (x) are non-unique (i.e., two-parametric -cf. [9]). From the point of view of physics this means that they may be restricted by some suitable (i.e., typically, asymptotic [4,5]) boundary conditions (cf. also Ref. [7]). In what follows we shall assume that (A1) these general solutions ψ (general) (x) live on unbounded contours called "tobogganic", with the name coined and with the details explained in Ref. [7]; (A2) our particular choice of the tobogganic contours will be specified by certain multiindex ̺ so that C (tobogganic) (s) ≡ C (̺) (s); (A3) for the sake of brevity our attention may be restricted to the tobogganic models where the multiindices ̺ are nontrivial but still not too complicated. For this reason we shall study just the subclass of the tobogganic models containing, typically, potentials with two strong singularities inducing branch points in the wave functions.
In this manner we shall have to deal with the two branch points x In the language of mathematics the obvious topological structure of the corresponding multi-sheeted Riemann surface R (multisheeted) will be "punctured" at x (BP ) (±) = ±1. In the vicinity of these two "spikes" we shall assume the generic, "logarithmic" [11] structure of R (multisheeted) .

Winding descriptors ̺
The multiindex ̺ will be called "winding descriptor" in what follows. 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 (BP ) (±) = ±1. During the further increase of s one simply selects one of the following four alternative possibilities: • one moves counterclockwise around the left branch point x (BP ) (−) (this move is represented by the first letter L in the "word" ̺), • one moves counterclockwise around the right branch point x (BP ) (+) (this move is represented by letter R), • one moves clockwise around the left branch point x • one moves clockwise around the right branch point x (BP ) (+) (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 At N = 0 we may assign the empty symbol ̺ = ∅ or ̺ = 0 to the one-parametric family of the straight lines of Ref. [5], Thus, one encounters precisely four possible arrangements of the descriptor, viz., in the first nontrivial case. In the more complicated cases where N > 1 it makes sense to re-express the requirement of PT −symmetry in the form of the stringdecomposition ̺ = Ω Ω T where the superscript T marks an ad hoc transposition, i.e., the reverse reading accompanied by the L ↔ R interchange of symbols. Thus, besides the illustrative Eq. (7) we may immediately complement the first nontrivial list etc. The four "missing" words LL −1 , L −1 L , RR −1 and R −1 R had to be omitted as trivial here because they cancel each other when interpreted as windings [12].

Formula
The core of our present message lies in the idea that the non-tobogganic straight lines (6) may be mapped on their specific (called "rectifiable") tobogganic descendants.
For this purpose one may use the following closed-form recipe of Ref. [12], where one defines This formula guarantees the PT symmetry of the resulting contour as well as the stability of the position of our pair of the branch points. Another consequence of this choice is that the negative imaginary axis of z = −i|z| is mapped upon itself.
Some purely numerical features of the mapping (10) may be also checked via the freely available software of Ref. [13]. On this empirical basis we shall demand that the exponent κ will be chosen here as an odd positive integer, κ = 2M + 1, M = 1, 2, . . .. In this case the asymptotics of the resulting nontrivial tobogganic contours (with M = 0) will still parallel the κ = 1 real line C (0) (s) in the leadingorder approximation.
These relations numbered by m = 0, ±1, . . . , M may further be simplified via the two known elementary trigonometric real and non-negative constants A and B such that In terms of these constants we separate Eq. (12) into it real and imaginary parts yielding the pair of relations As long as ε > 0 we may restrict our attention to the non-negative s and eliminate s = B/(2 ε). The remaining quadratic equation finally leads to the following unique solution of the problem, This formula perfectly confirms the validity and precision of our illustrative graphical constructions.

Samples of countours of complex coordinates
For the most elementary toboggans characterized by the single branching point the winding descriptor ̺ becomes trivial because it is being formed by the words in a one-letter alphabet. This means that all the information about windings degenerates just to the length of the word ̺ represented by an (arbitrary) integer [14]. Obviously, these models would be too trivial from our present point of view.
In an opposite direction one could also contemplate tobogganic models where a larger number of branch points would have to be taken into account. An interesting series of exactly solvable models of this form may be found, e.g., in Ref. [15].  Once we fix the distance ε of the complex line C (0) from the real line R we may still vary the odd integers κ. Vice versa, even at the smallest κ = 3 the recipe enables us to generate certain mutually non-equivalent tobogganic contours C (̺) (s) in the ε−dependent manner. This confirms the existence of discontinuities. Their emergence and form are best illustrated by the pair of Figures 3 and 4. We may conclude that in general one has to deal here with the very high sensitivity of the results to the precision of the numerical input or to the precision of computer arithmetics. This confirms the expectations expressed in our older paper [12] where we emphasized that the descriptor ̺ is not necessarily easilly inferred from a nontrivial, detailed analysis of the mapping M.

Rectifiable tobogganic contours with κ ≥ 5
Once we select the next odd integer κ = 5 in Eq. (10) the study of the knot-shaped structure of the resulting integration contours C (̺) (s) becomes even more involved because in the generic case sampled by Figure 6 the size of the internal loops proves unexpectedly small in comparison. As a consequence, their very existence may in principle escape our attention. Thus, one might even mistakenly perceive the curve of Figure 6 as an inessential deformation of the curves in Figures 1 or 2. Naturally, not all of the features of our toboganic integration contours will change during transition from κ = 3 to κ = 5. In particular, the partial parallelism between Their M ≤ 6 sample is listed here in Table 1.
On this basis we may summarize that at a generic κ the variation (i.e., in all of our examples, the growth) of the shift ε makes certain subspirals of contours C (̺ ) larger and moving closer and closer to each other. In this context our Table 1

Conclusions
We confirmed the viability of an innovated, "tobogganic" version of PT −symmetric Quantum Mechanics of bound states in models where the general solutions of the underlying ordinary differential Schrödinger equation exhibit two branch-point singularities located, conveniently, at x (BP ) = ±1.
In particular we clarified that many topologically complicated complex integrations contours which spiral around the branch points x (BP ) in various ways may be rectified. This means that one can apply an elementary change of variables z(s) → x(s) and replace the complicated original tobogganic quantum bound-state problem by an equivalent simplified differential equation defined along the straight line of complex pseudocoordinates z = s − iε.
In detail a few illustrative rectifications have been described where we succeeded in an assignment of the different winding descriptors ̺ to the tobogganic contours controlled solely by the variation of the "initial" complex shift ε. An interesting supplementary result of our present considerations may be also seen in the constructive demonstration of feasibility of an explicit description of these transitions between topologically non-equivalent quantum toboggans characterized by non-equivalent winding descriptors ̺. Still, the full understanding of these structures remains to be an open problem recommended to a deeper analysis in the nearest future.
In summary we have to emphasize that our present rectification-mediated reconstruction of the ordinary-differential-equation representation of quantum toboggans could be perceived as an important step towards their rigorous mathematical analysis and, in particular, towards the extension of the existing rigorous proofs of the reality/observability of the energy spectra to these promising innovative phenomenological models.