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Matching method and exact solvability of discrete PT-symmetric square wells

Discrete PT-symmetric square wells are studied. Their wave functions are found proportional to classical Tshebyshev polynomials of complex argument. The compact secular equations for energies are derived giving the real spectra in certain intervals of non-Hermiticity strengths Z. It is amusing to notice that although the known square well re-emerges in the usual continuum limit, a twice as rich, upside-down symmetric spectrum is exhibited by all its present discretized predecessors.


Introduction
Undoubtedly, solvable models represent one of the most important and inspiring sources of insight in the physical properties of real quantum systems [1]. For illustration, one may recollect how the concept of the so called shape-invariant analytic potentials V (x) [2] opened the path towards a better understanding of supersymmetry [3]. Recently, a similar major progress in our understanding of some less standard structures admitted by the standard Quantum Mechanics [4] has been initiated by a few numerical as well as non-numerical analyses [5,6,7] of certain complex potentials exhibiting another interesting invariance property, It has been argued [7] that the latter invariance may prove highly relevant in physics as it makes all the Hamiltonian PT −symmetric, i.e., commuting with the product of parity P and complex conjugation T while, in its turn, the latter operator represents the time-reversal operation in the language of phenomenological considerations.
In the resulting slightly innovated formalism of quantization (called, for definiteness, PT −symmetric Quantum Mechanics, PTSQM [8,9]), you admit that Hamiltonians need not necessarily be Hermitian in the usual trivial metric I in Hilbert space, H = H † . As an illustrative example one may recollect the ordinary differential Schrödinger equation on a finite interval, where PTSQM admits potentials V (x) of eq. (1) which are not real but merely "parity plus time-reversal" -symmetric.
One of the most immediate consequences of the invariance (1) replacing Hermiticity is that the corresponding Hamiltonian H = H † admits the existence of complex eigenvalues E n = E * n . In the language of experimental physics this simply means a potential instability of the whole system, a phenomenon truly encountered, e.g., in relativistic Quantum Mechanics [10] or in certain magnetohydrodynamical systems [11]. In the former case the instability may either mean a complete collapse of the system or a mere controlled creation of the particle-antiparticle pairs [12]. An even less catastrophic scenario is encountered in experimental magnetohydrodynamics where the transitions to the unstable regime prove reversible as a rule [13].
The potential theoretical consequences of PT −symmetry seem equally appealing at present. One of the reasons is that in eq. (2) the coordinates x themselves may be complex and, hence, manifestly unobservable! This means that one gets quite close to the relativistic field theory where the corresponding analogue of x (viz., the field amplitude ϕ) is also an auxiliary (i.e., in principle, not necessarily observable) quantity [7,14].
In a way inspired by the latter comment the auxiliary role of the complex, unobservable coordinate x may equally well be played by any other element x k of any other unobservable set D = (−∞, ∞). Without getting too deeply involved in this theoretical question we decided to parallel the most popular strategy (where D is being specified as a suitable complex curve [6,7]) and to try to treat the (admissible) replacement (−∞, ∞) → D as a simplification, discretization of the real axis.
Our present paper summarizes a few of our most interesting results. Firstly, we set the scene in section 2 where we list some supplementary motivations and show how we intend to construct our illustrative examples. In the subsequent section 3 both the physical and mathematical background of our considerations is outlined in some necessary detail.
Up to the final brief summary of our results in section 7, the rest of our paper becomes more technical and it shows how our simplest discrete models can be solved and how their various generalizations can be treated and remain, in a current vague sense of the word, exactly solvable.

Non-analytic Schrödinger equations
The admissibility of a complexification of spectra is a puzzling possibility which inspired qualitative analyses of many concrete models. For many analytic potentials V (x) of an immediate physical interest the constructive demonstrations of the stability (i.e., of the reality of the spectrum) have been performed by perturbative [15], quasi-classical [16] as well as purely numerical [17] methods.
At an early stage of development the non-analytic solvable models have been considered conspicuous. Many of them -typically, potential V (x) ∼ |x| on the whole real line, solvable in terms of Airy functions by the matching method -happened to exhibit instabilities (i.e., complex energies) at all the non-vanishing values of the corresponding coupling constants [8]. This caused a certain delay of return to some of the most elementary exactly solvable piecewise-constant PT −symmetric potentials in eq. (2). Still, their later studies [18,19] clarified that their spectrum of bound states may remain real in a fairly large domain of their coupling constants (cf. also [20,21]). In ref. [22] it has been proved that our PT −symmetric Schrödinger eq. (2) possesses solely the discrete spectrum of real energies E = E n , n = 0, 1, . . . for all the complex potentials which are "not too strong", At the same time, the possible emergence of the instabilities at all the non-zero couplings has been confirmed for certain piecewise-constant PT −symmetric potentials acting on the complex, curved contours [23].

Runge-Kutta option
The indefiniteness of conclusions of the above observations inspired out present extension of the study of the piecewise-constant potentials towards the models which are defined over a mere discrete, equidistant lattice of points These models possessing PT −symmetry and real spectra will be characterized by the variability of the number N + 1 of grid points in (4). We believe that in this perspective our understanding of the generic PT −symmetric models may acquire a new dimension, returning us to the usual differential-equation framework in the N → ∞ limit.
One of the most natural possibilities of the discretization of the differential eq. (2) is given by the well known Runge-Kutta recipe [24] − Using the standard boundary conditions we shall employ here just the piecewise constant, purely imaginary antisymmetric potentials of refs. [18,21], , n = 1, 2, . . . , q + 1 .
Their discontinuities lie at the matching points ℓ 0 = 0 < ℓ 1 < . . . < ℓ q+1 = 1 so that at q = 0 we have to solve the discrete Schrödinger bound-state problem etc. It is worth noticing that the first nontrivial N = 4 version of eq. (7) coincides with the Weigert's matrix model studied in ref. [25], in more detail, as "the simplest nontrivial" Runge-Kutta discretization of the PT −symmetric square well of ref. [18].
3 Physics and mathematics behind the discrete Schrödinger equations

The definition of the physical metric
In the light of reviews [4,9], the PT −symmetric Hamiltonians are not compatible with the standard "Dirac's" metric I in Hilbert space (indeed, we have H = H † ).
A construction of an appropriate (i.e., of a Hamiltonian-dependent) generalization Θ = I of the metric is necessary for a correct probabilistic interpretation of the measurements. This means that our Hamiltonian only must be Hermitian (or, more precisely, self-adjoint) with respect to the generalized metric, All the physical contents of our models must be derived from the new metric of course. For this reason it makes sense to avoid confusion by calling eq. (9) with Θ = I a "quasi-Hermiticity" condition [4].
One can notice that almost exclusively, standard textbooks train our "quantum intuition" on the representations of the Hilbert space where the metric remains trivial. This means that all the nonstandard PTSQM considerations profit enormously from any construction of a sufficiently flexible class of solvable models. In such a setting our present finite-dimensional PT −symmetric models may play the role of a guide towards the interpretation, especially due to their facilitated connection with the continuous N → ∞ limit.
A parallel reason for the attention paid here to the discretized systems lies in the well known difficulty of the problem of the efficient construction of Θ = I for a given H = H † . Using the Dirac-inspired "brabraket" compact notation of ref. [26] and employing the biorthogonal basis composed of the left and right eigenstates of our non-Hermitian Hamiltonian H we may recollect (cf., e.g., [27]) that the knowledge of the spectral representation of the Hamiltonian, almost immediately leads to the parallel multiparametric formula which specifies the correct metric as a (non-unique, parameter-sequence-dependent) solution of the operator eq. (9). Thus, we can employ this formula and construct Θ, more or less comfortably, for the majority of the exactly solvable models of the form (2) [28]. This is in contrast with the situation where one must construct the states |n by some approximative method. This would make formula (10) practically useless. Complicated computations would be needed which, typically, search for Θ in the form of a product where the parity P is multiplied by a charge C (from the left, [9]) or by a quasi-parity Q (from the right, [29]).

Matching method
Let us recollect that the N = ∞ continuous-limit models (6) are exactly solvable at any q ≥ 0 and that their practical solution remains feasible and transparent at q = 0 [18], at q = 1 [30] and at q = 3 [31] at least. The solvability in closed form also characterizes the parallel modifications of these models characterized by the periodic boundary conditions [32].
It would be highly desirable to extend the available exact solution of the discrete Weigert's "minimal" finite-dimensional model (8) with q = 0 and N = 4 to some higher integers N and/or q. A key purpose of our present paper is to show that such extensions are feasible, indeed, once we adapt the matching method, so efficient in its standard differential-equation form, to the needs of difference equations.
In the next section our constructive considerations will start from one of the most of the Weigert's model represented by the N − 1 = 2n + 3 dimensional matrix transcription of our difference Schrödinger eq. (7) with the re-scaled energy eigenvalues F = E h 2 − 2 and with the re-scaled strength ξ = Z h 2 of the non-Hermiticity.
We shall see that such a matrix re-formulation of the matching recipe proves solvable in closed form at all n = 0, 1, . . .. In the Hermitian context of Quantum Chemistry we just arrive at the well known Hückel's solvable models in the limit 4 Finite-dimensional Schrödinger equation (11) 4

.1 Closed solvability
Recollecting the definition of the classical Tshebyshev polynomials of the second kind [34], (cf. a collection of their most relevant properties in Appendix A) and using the real elements γ = ψ(0), a k = Re ψ (x k+1 ) and b k = Im ψ (x k+1 ) we are now prepared to formulate our first result.

Theorem 1.
Whenever the PT −symmetry remains unbroken, closed solutions of eq. (11) are defined by the formulae and complemented by the selfconsistency constraint Proof.
In the first step of our analysis we get the relation β k = α * k due to the PT −symmetry of our eigenvectors. We then identify eq. (11) with the three-term recurrences satisfied by the Tshebyshev polynomials (cf. Appendix A) and verify formula (12) for eigenvectors by showing that it is compatible with the corresponding boundary conditions (at the smallest subscripts k). As long as all this reduces the full tridiagonal (N −1)×(N −1)−dimensional matrix eq. (11) to the mere three matching conditions, it remains for us to show that the first and the third line may be simplified to give eq. (13) while the middle line defines the product (14). QED

Remark.
Without an assumption of an unbroken PT −symmetry one would have to admit that the eigenvalues F are complex. A more explicit illustration of what happens has been described, in the continuum limit, in ref. [35].

Corollary.
A nontrivial solution always exists at F = 0.

Proof.
In a discussion of consequence of Theorem 1 one has to notice that formula (14) forces us to distinguish between two regimes where F = 0 and F = 0, respectively. Once we set, tentatively, F = 0, it is easy to deduce from eq. (13) that the parameter a must which acquires the following two alternative forms, These conditions are satisfied identically at any m = 0, 1, . . .. Thus, our tentative "guess of the energy" was correct and that F = 0 is always an eigenvalue. QED.

Remark.
In spite of the non-Hermiticity of the Hamiltonian, the F = 0 eigenvalue is "robust" [30] and remains real at all the real couplings Z ∈ (−∞, ∞). The spectrum remains symmetric with respect to this "middle point" [note that F = 0 corresponds to the energy E = E n+2 = 2/h 2 = N 2 /2 = 2(n + 2) 2 ]. The existence of such a "central" eigenvalue is not in contradiction with the differential-equation results of ref. [22].
Indeed, this level moves up with dimension N and disappears in infinity in the limit N → ∞. We may conclude that in this sense a "richer" structure is exhibited by the spectrum at the finite dimensions.

Properties of the "fragile" F = 0 solutions
Whenever F = 0 we may treat eq. (13) not only as the condition of vanishing of the imaginary part of γ, but also as an explicit definition of the non-vanishing left-hand-side quantity F γ in eq. (14). Its insertion simplifies the latter relation, where T k (z) denotes the k−th Tshebyshev polynomial of the first kind.
One of the latter two relations defines the normalization vector (a, b) = (a 0 , b 0 ) while their ratio gives This represents our final secular equation which defines, in an implicit manner, the energies F as functions of the couplings ξ.
An efficient numerical treatment of the latter eigenvalue problem may be based on the re-parametrization i.e., 1 2 F = − cos α cosh β, and, in the opposite direction,
Its roots must be determined, in general, numerically.

Proof.
The constant value of the coupling ξ > 0 is mapped upon a downwards-oriented halfoval in the α − β plane. Its top lies at α = π/4 while its two asymptotes α = 0 and α = π/2 are reached in the limit β → −∞. In such a representation the "robust" and ξ−independent energy level F = 0 lies on the top of the half-oval while its decreasing and increasing neighbors are found displaced to the left and right, respectively, along the half-oval downwards. QED.

Remark.
At the first few lowest dimensions N − 1 = 2n + 3 the roots of eq. (22) may be written in closed form, In comparison with ref. [25] a "one-step easier" discretization of eq. (2) emerges at q = 0 and at the odd N = 3, with the two energy roots. A non-equivalent discrete alternative to eq. (11) could then read, in the same notation, This leads to the following alternative result.

Theorem 2.
Whenever the PT −symmetry remains unbroken, closed solutions of eq. (23) are defined by formulae (12) accompanied by the alternative matching condition and by the odd−N counterpart of eq. (19), Proof.
Definition (12) of the eigenvectors remains unchanged but the matching condition generated by the subproblem is just one, α n+1 = α * n . The ratio between this equation and its Hermitian conjugate eliminates all the normalization ambiguities and leads to the secular equation (25).

Remark.
Secular equation (25) is an implicit definition of the N = 2n + 3 energy levels The respective elementary expressions for the critical constants above we notice that the numerical convergence towards n = ∞ is perceivably slower and from the opposite side at the odd N (cf. Figure 1 once more).

Models with more matching points
New features of the spectrum emerge when we choose q > 0 in eq. (6). In the light of the differential-equation results [30,31] one may expect the existence of the so called 'robust' levels at q ≥ 1 and N < ∞. By definition they remain real at the arbitrarily large coupling constants Z k . A priori, the position of these levels may be expected to be controlled by both the parameters Z k and ℓ k .
Obviously, the interpretation of the discretization becomes facilitated when we choose the discontinuities at the simple rational numbers ℓ k . It is interesting to note, marginally, that their choice simplifies even the trigonometric secular determinants in many differential-equation models. Nevertheless, in our present context it is much more important that the choice of the simplest ℓ k at q ≥ 1 enables us to continue working with the not too large matrix dimensions N − 1.

ℓ = 1/2 and N = 6
For the q = 1 interaction model (6) with ℓ 1 = 1/2, we may start from its discretization (5) with the most elementary choice of N = 6, This equation determines the following five eigenvalues, Besides the constant and ξ−independent level F 0 = 0 one can show that the two "external" levels remain robustly real for all the values of ξ and never complexify, At N = 10 the secular algebraic equation for the energy-level characteristics y = F 2 has the following four roots in the Hermitian limit with ξ = 0, Numerically one finds that the real quadruplet  [30] in the N → ∞ continuum limit.
6 The role of the shift-parameter ℓ 6.1 Suppressing the non-Hermiticity using ℓ = 5/8 In order to illustrate the emergence of a pair of the robust levels let us re-consider the q = 1 interaction model (6) and employ a larger shift ℓ 1 = 5/8. In the resulting potential the non-Hermiticity is weakened at a fixed coupling Z. We get the matrix analogue of problem (28) from the discretized eq. (5) with N = 8, An inspection of the explicit form of the secular determinant of this equation, confirms that all of its seven roots remain real in the Hermitian ξ → 0 limit, Moreover, in a way illustrated in Figure 2, only two roots F ±,− merge and complexify beyond ξ crit ≈ 1.15470. In the other words, as many as five of the roots remain real for Z ≫ 1, This observation parallels the similar results obtained in the continuous limit. All this also agrees with the a priori expectations, reflecting the weakened influence of the non-Hermitian part of the potential at the comparatively large shift ℓ = 5/8.

6.2
Strengthening the non-Hermiticity using ℓ = 3/8 The number of the robust levels may vary with the width ℓ of suppression of the imaginary interaction. Using a weaker suppression ℓ 1 = 3/8 we may modify eq. (29) accordingly and we get the precise N = 8 analogue of problem (30). It is easy to evaluate the related secular determinant which gives the same roots (32) in the weak-non-Hermiticity-limit ξ → 0. A change may be expected to emerge in the more non-Hermitian regime. Quantitatively this change is being well illustrated by Figure 3 where we see that merely two nonvanishing roots (viz., F ±,+ → ± √ 2) remain robust and real at large Z → ∞. We see that two complexifications take place at a certain finite coupling Z, i.e., at a unique critical point ξ crit ≈ 0.5875691807. In contrast to the previous case we witness two distinct mergers of the eigenvalues F ±,0 with F ±,− at the two different exceptional points F exc. ± ≈ ±1.140716421.

Summary
A tentative weakening of the standard Hermiticity H = H † to the mere P−pseudo-

Hermiticity of the Hamiltonian,
employs only too often the ordinary differential Hamiltonian H and the parity operator P [7]. Here we intended to emphasize that it is by far not the only possible scenario. For this reason we replaced eq. (2) by its discrete counterpart (5) and explained that its use may bring a few important advantages.
We feel satisfied by the fact that the purely formal application of the PTSQM recipe survives quite easily our present narrowing of one's attention to the very specific and solvable square-well models. We have shown that the use of various specific discrete versions of the Hamiltonians leads to very relevant formal simplifications emphasized throughout the text.
Within the dynamical region where our models possess the real spectrum, very close parallels were shown to emerge between the bound states for the standard continuous coordinate x [18,30,31] and for its present less standard discrete counterpart(s).
We have seen that in the discrete case it is particularly easy to complement the validity of the auxiliary, technical pseudo-Hermiticity condition (35)