{'fraction_non_alphanumeric': 0.04417212881162724, 'fraction_numerical': 0.012895411798233114, 'mean_word_length': 4.459743290548425, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Solvable simulation of a double-well problem in PT symmetric quantum mechanics

Within quantum mechanics which works with parity-pseudo-Hermitian Hamiltonians we study the tunneling in a symmetric double well formed by two delta functions with complex conjugate strengths. The model is exactly solvable and exhibits several interesting features. Besides an amazingly robust absence of any PT symmetry breaking, we observe a quasi-degeneracy of the levels which occurs all over the energy range including the high-energy domain. This pattern is interpreted as a manifestation of certain"quantum beats".


Introduction
In current textbooks, many phenomena (like, e.g., vibrational spectra) and methods (like, e.g., perturbation expansions) of quantum mechanics are best illustrated by the one-dimensional Schrödinger equation for bound states in a real and symmetric In such a setting, the transition of the so called PT −symmetric quantum mechanics (say, in its form proposed by Bender and Boettcher [1]) may most easily be illustrated by an inclusion of an asymmetric and purely imaginary additional potential in the same equation, The exactly solvable examples abound [2,3]. As their most elementary and transparent sample we may recollect the harmonic oscillator in D dimensions [4] with Here, the centrifugal barrier is regularized and proportional to ℓ = (D − 3)/2 + m in the m − th partial wave.
The studies of models (2) also often involve much more complicated potentials, which are not exactly solvable. An encouragement is provided by their more immediate applicability as well as by the efficiency of many semiclassical [5], perturbative [6] and/or purely numerical techniques [7]. In the pioneering paper, Caliceti et al [8] paid attention to the very unusual imaginary cubic PT −symmetric anharmonicity. Within sophisticated perturbation theory, she (together with her co-authors) succeeded in showing that in spite of the non-Hermiticity of the Hamiltonian, the spectrum of energies may remain real.
Buslaev and Grecchi [9] returned to the more standard, quartic PT −symmetric anharmonic oscillator with a similar motivation stemming from field theory [10]. They were the first who demonstrated that the bound-state energies of a non-Hermitian, PT −symmetric model (4) may coincide with the spectrum of a certain Hermitian double-well problem. Their sophisticated and explicit, Fourier-type equivalence transformation between the Hermitian and non-Hermitian partners may be now better understood in the language of A. Mostafazadeh [11].
Let us note that the presence of the centrifugal term in eq. (4) played an important technical role in ref. [9], while a similar term has been absent in all the original studies of the imaginary cubic forces [5,8]. Curiously enough, only a re-introduction of this term in the generalized model seems to have opened the way towards the rigorous proof of the reality of the spectrum of the imaginary cubic oscillators [12]. In the light of this proof, the special regular case of eq. (5) with the vanishing ℓ(ℓ + 1) = 0 is not exceptional at all.
On the contrary, the inclusion of a strong repulsion ℓ(ℓ + 1) ≫ 1 in eq. (5) seems instructive and productive in having paved the way towards the recent clarification of the applicability of the current 1/ℓ technique in the PT −symmetric context in ref. [13]. Thus, we shall assume that ℓ is large in all the potentials of the type (3), (4) or (5). At this point, we get quite close to the subject of our present short communication since the implementation of the 1/ℓ technique proved entirely different in the Hermitian and PT −symmetric models, and we intend to re-analyze the latter case via certain simplified models.

Double wells
All three PT −symmetric potentials (3) -(5) offer a very good testing ground for the comparison of the Hermitian and PT −symmetric calculations. Firstly, they may serve as an elementary illustration of the 1/ℓ technique in the Hermitian case because their Hermitian versions emerge simply in the limit c → 0. Secondly, one immediately notices that such a step is not mathematically trivial since their centrifugal barriers become strongly singular. Fortunately, this does not play any role at all because simultaneously, we have to "shrink" the full real axis of the coordinates to the mere positive "radial" half-axis [4]. A fully rigorous discussion of this point is available in ref. [9] and enables us to re-interpret our differential Schrödinger equation (1) with the real and constrained coordinates x = x Hermitian = x (c=0) > 0 as the standard central bound-state problem on half-axis. In such a context, the perturbative 1/ℓ recipe (as reviewed, e.g., in paper [14] and/or in many other references which are quoted therein) is virtually trivial. One simply imagines that the potential V (x) possesses an absolute minimum at some real and positive coordinate R > 0 { for example, in eq. (3) we get the unique minimum at R = [ℓ(ℓ + 1)/B] 1/4 , etc }. In the vicinity of this minimum, the shape of the potential V (x) may be approximated by an appropriate harmonic oscillator well, At this point one discovers that the quality of the latter "zero-order" approximation increases quickly with the growth of ℓ. Moreover, the expansion of the binding energies and wave functions in the powers of 1/R remains feasible and exhibits very good convergence properties in practice [14].
After a return to the non-Hermitian eq. (1) with x ∈ (−∞, ∞), an application of the same idea requires a much more careful analysis. As long as all our sample potentials V (x) remain smooth, analytic and confining for all the shift parameters c = 0, our wave functions ψ(x) stay analytic and normalizable all over the lower halfplane of the complex plane of x [1]. At the same time, the manifest non-Hermiticity of our sample Hamiltonians at c > 0 leads to the necessity of a re-interpretation of their eigenstates. In the spirit of our remarks [15,16] (cf. also the later reviews [11,17]), one must replace the concepts of Hermiticity by pseudo-Hermiticity while the unitarity becomes weakened to pseudo-unitarity. As a consequence, the probabilistic interpretation of PT −symmetric quantum mechanics acquires the forms which find their immediate guidance in the relativistic quantum mechanics, with one of the most popular examples provided by the Feshbach-Villars pseudo-Hermitian re-formulation of the Klein-Gordon equation [18].
The net consequence of all these introductory remarks is that within the framework of our above exemplification of the PT −symmetric quantum mechanics we are free to work with any complex coordinate R of the minimum of V (x). As long as we assumed that ℓ(ℓ + 1) ≫ 1 is large, we discover that there exist many minima R generating, in principle, many alternative large−ℓ expansions. For example, eq. (3) with c > 0 gives the minimum of V (x) whenever  (4) and (5) (both with B = 0 for simplicity) gives the respective rules R 6 = ℓ(ℓ + 1)/(2D) and i R 5 = 2ℓ(ℓ + 1)/(3C) so that, generically, one has to deal with several pairs of the complex minima such that where we only know that |R| ≫ 1. In such a setting, we may choose the shift of the axis in such a way that c ≡ θ, without causing any change in the spectrum of course.
The fully exhaustive discussion of the particular cubic example (5) may be found in ref. [13]. For the simpler quartic oscillator (4), the same large−ℓ construction would lead to the vanishing and non-vanishing θ at the positive and negative couplings D > 0 (occurring, e.g., in refs. [19]) and D < 0 (chosen, e.g., in refs. [9,20]), respectively. Thus, in contrast to the Hermitian case where the real position of the absolute minimum of V (x) is unique in the majority of cases of practical interest, the regularization mediated by the shifts c = 0 leads, in accordance with the pattern (6), to the most frequent occurrence of the two symmetric minima in V (x) at once.

Now we may conclude our introductory considerations by a declaration that any
PT −symmetric double well (6) with the complex (and complex-conjugate) strengths Ω 2 (±) has not been found solvable in our preceding paper [13], and that it offered the main motivation for our forthcoming considerations, therefore.

The model
We shall try to simulate the effects of the general double attraction in the following schematic square-well-like model x ∈ (−1, 1) where, in the spirit of refs. [3], the forces are reduced to the mere PT −symmetric pair of the delta functions at a distance measured by the variable a ∈ (0, 1). This means that we plan to solve eq. (1) with the boundary conditions by the standard matching technique as described in textbooks [21]. Under the usual PT −symmetric normalization convention [16] ψ this enables us to denote E = κ 2 in the obvious ansatz for the wave functions on the three sub-intervals of the whole domain in question, Moreover, as long as we have the appropriate insertions in eq. (8) lead to the four-by-four matrix set of equations Here, µ(ω) = cos κ(1 − a) − ω 2 κ −1 sin κ(1 − a) and ν(η) = η κ −1 sin κ(1 − a) are mere abbreviations.
We may conclude that the matching conditions may be satisfied if and only if the secular determinant F (κ) vanishes in eq. (12). This condition has the following form Its main merit is its compact form, not quite expected in the light of the previous complicated matching formula (12). By its structure it resembles the textbook solution of the simple square well so that, in this sense, it is a mere implicit representation of the spectrum itself. In fact, one might even believe that an explicit construction of this spectrum might be possible (say, in the form of an infinite power series in some of the parameters) but due to the utterly elementary character of eq. (13), one might still deduce the majority of the relevant features of the spectrum from this implicit definition. In order to demonstrate this expectation in detail, the roots of eq. (13) will be studied here via the graphs of the function F (κ). In this sense, we shall determine and display the various qualitative features spectrum of the energies E n = κ 2 n in a series of pictures, via their presentation in the form of the sequences of the nodal zeros κ n of the function F (κ).

Discussion
The results of our study are sampled in Figures 1 -7 at the different sets of the couplings. In all of these cases we observed, first of all, that the influence of the changes of η is not too relevant so that we have fixed, in all our pictures, η = 20 for the sake of definiteness.
In Figure 1 we simulated the system of the two weakly attractive wells which lie far from each other. We see that the choice of a = 0.95 and ω = 3/2 still leads to the mere very weak perturbation of the very well known spectrum of the ω = 0 energies in square well. The function F (κ) oscillates very regularly and its overall amplitude, although not quite constant, exhibits just a small slow variation and remains almost independent of our choice of the interval of κ (the Figure takes κ = √ E ∈ (0, 15)).
In Figure 2 we see that for the same distance of the wells, the growth of their attractive strength up to ω = 15000 leads to the significant growth of the average magnitude of the amplitude of F (κ) and to the emergence of an "envelope" (i.e., an auxiliary curve F av (κ) which connects the maxima or minima) with much slower oscillations. In addition we discover an overall asymptotic decrease of F av (κ) and certain less apparent periodic structures when we try to connect some of the "subdominant" local maxima/minima of F (κ). Even a not too significant shortening of their distance (to a = 0.85 and a = 0.65, respectively) shows that the period of the envelopes becomes shorter and, in the latter case, one can see a clear competition between several parallel envelopes which start to overlap each other.
In order to reveal the latter pattern clearly, we omitted the low lying levels. This enabled us to see that the interplay between different periods remains fairly regular.
Step by step, it introduces a much more complicated pattern to the energy spectrum, anyhow. The separate levels behave in a more and more chaotic manner, especially when we return to the low-lying part of the spectrum in a way illustrated in Figure   5 where we kept a = 0.65 and weakened ω = 150.
The remarkable feature of all this pattern is that our numerical experiments never revealed a merger of the two levels followed, presumably, by their disappearance in the complex plane. In the other models, such a phenomenon may exist and is usually interpreted as the so called spontaneous PT −symmetry breaking [22]. Here, Figure 6 with the fairly small distance a = 0.35 (and with the original very strong ω = 15 · 10 3 ) illustrates the situation where one gets very close to such a possibility and where several pairs of the real energies appear to be almost degenerate.
In our last Figure 7 where ω = 150, we finally demonstrate that the "chaotic" character of the eigenvalues in our schematic and exactly solvable PT −symmetric point-interaction double wells may be weakened and partially removed by a return to their weaker strengths. We see in the picture that the characteristic pairwise "irregular" quasi-degeneracies persist. This type of an irregularity has a slightly different source in an ultimate slowdown of the oscillations of the envelopes, the "wobbles" of which become comparable to the nodal distances.
We may summarize that the double-well-like structure of our complex, non-Hermitian example complies with some of our intuitive expectations. Thus, the "robust" reality of the spectrum or an emergence of the quasi-degeneracy have been re-confirmed. At the same time, a few other observations (say, of the multiple and/or super-imposed "beats" in the curve F (κ)) wait for a deeper understanding and/or a generic explanation in the future.          Figure 6, with a "wobbling-type" irregularity in the energy nodes for "shallow" wells with ω = 150.