A generalized family of discrete PT-symmetric square wells
aa r X i v : . [ qu a n t - ph ] F e b A generalized family of discrete
P T − symmetric square wells
Miloslav Znojil
Nuclear Physics Institute ASCR,250 68 ˇReˇz, Czech Republice-mail: [email protected]
Junde Wu
Department of Mathematics, College of Science, Zhejiang University,Hangzhou 310027, Zhejiang, P. R. Chinae-mail: [email protected] bstract N − site-lattice Hamiltonians H ( N ) are introduced and perceived as a set ofsystematic discrete approximants of a certain PT − symmetric square-well-potential model with the real spectrum and with a non-Hermiticity whichis localized near the boundaries of the interval. Its strength is controlledby one, two or three parameters. The problem of the explicit constructionof a nontrivial metric which makes the theory unitary is then addressed. Itis proposed and demonstrated that due to the not too complicated (viz.,tridiagonal matrix) form of our input Hamiltonians, the computation of themetric is straightforward and that its matrix elements prove obtainable, non-numerically, in elementary polynomial forms.
Keywords quantum mechanics; discrete lattices; non-Hermitian Hamiltonians; Hilbert-space metrics; solvable models; 2
Introduction
A priori it is clear that the traditional and most common physical Hilbertspaces of the admissible quantum states need not necessarily prove optimalfor computations. Once these “obvious” spaces H ( P ) become distinguishedby the superscript ( P ) which may be read as an abbreviation for “primaryspace”, one may find an explicit verification of this expectation in nuclearphysics cca twenty years ago [1]. The amended Schr¨odinger-representationHilbert space H ( S ) (where the superscript stands for “secondary”) has beenconstructed there via a fermion-boson-space correspondence P ↔ S .A perceivable simplification of the practical numerical evaluation and/orat least of the variational prediction of the bound-state energy levels E n hasbeen achieved for a number of heavy nuclei. In the notation as introduced inRef. [2] one can identify the underlying key mathematical idea as lying in aDyson-inspired ansatz connecting the P − superscripted and S − superscriptedket-vectors, | ψ ( P ) i = Ω | ψ ( S ) i ∈ H ( P ) , | ψ ( S ) i ∈ H ( S ) . (1)The manipulations with the original ket vectors | ψ ( P ) i became, by such aconstruction, facilitated.In particular, what appeared simplified was the evaluation of the innerproducts h φ ( P ) | ψ ( P ) i and of the P − space matrix elements, say, of the Hamil-tonian operator h acting in H ( P ) . After the unitary-equivalence transitionto H ( S ) the same quantities were represented by the new inner products h φ ( S ) | ψ ( S ) i and by the matrix elements h φ ( S ) | H | ψ ( S ) i , respectively.It is well known [3, 4, 5] that during the transition P ↔ S betweenHilbert spaces one must also guarantee the isospectrality between the re-spective Hamiltonians h and H . In other words,, we must define the new3amiltonian H acting in H ( S ) by formula H = Ω − h Ω. Then, it appearsnatural when the whole change of the representation P → S is followedby another, second-step simplification. Such a step is usually motivated bythe survival of certain cumbersome character of the work in the secondaryHilbert space H ( S ) . In the notation of Ref. [2], for example, it makes sense toreplace the latter space by its “friendlier”, auxiliary, manifestly unphysicalalternative H ( F ) .Due to a certain freedom in the construction, the latter, third Hilbertspace may be allowed to coincide with H ( S ) as a topological vector space (i.e.,as the space of kets, | ψ ( F ) i| := ψ ( S ) i ). What leads to the ultimate simplicityis then the replacement of the fairly complicated, S − superscripted operation T ( S ) of the Hermitian conjugation in H ( S ) by the standard and trivial (i.e.,transposition plus complex conjugation) F − superscripted operation T ( F ) ofthe Hermitian conjugation in the final friendly space H ( F ) .The net purpose of the second simplification step S → F is that thequantum system in question finds its optimal Schr¨odinger representation in H ( F ) . In this auxiliary and maximally friendly Hilbert space one merelydefines h φ ( S ) | ψ ( S ) i ≡ h φ ( F ) | Θ | ψ ( F ) i , Θ = Ω † Ω (2)This convention keeps trace of the S − superscripted definition of the physics-representing inner products in H ( S ) and it offers a guarantee of validity ofthe initial requirement of the unitary equivalence between H ( P ) and H ( S ) . Ina compact review [2] of the formalism we emphasized that a given quantumbound-state system is in fact characterized by a triplet of Hilbert spaces4ccording to the following diagram: primary , difficult space Pand Hamiltonian h = h † Dyson map Ω ր ցտ unitary equivalence friendly but false space Fand non − Hermitian H := Ω − h Ω = H † hermitization −→ secondary , ultimate space Sis correct and physical ,H = H ‡ := Θ − H † Θ (3)During the above-mentioned application of such a pattern to the variationalanalysis of heavy nuclei it has been emphasized that, firstly, the model it-self is introduced in the P-superscripted Hilbert space but it appeared thereprohibitively complicated [1]. Secondly, the successful choices of the suit-able simplification mappings Ω have been found dictated or inspired by theunderlying dynamics (i.e., in nuclei, by the tendency of fermions to form,effectively, certain boson-resembling clusters). Thirdly, in a way reaching farbeyond the particular nuclear physics context, the product Ω † Ω = Θ = I hasbeen noticed to play the role of the metric in the ultimate, S-superscriptedHilbert-space.Cca ten years ago, the metric-operator interpretation of nontrivial Θ = I became believed to apply to a very broad family of models including,typically, the imaginary-cubic oscillator H = − d dx + i x (4)as well as many other Hamiltonians H introduced as acting in H ( F ) := L ( R )and/or in H ( S ) = L ( R ) and relevant, typically, in the relativistic quantumfield theory (cf., e.g., [4] or [5] for extensive details).The basic ideas behind the pattern of Eq. (3) were broadly accepted and5he whole mathematical formalism (which we call, conveniently, the three-Hilbert-space (THS) representation of quantum states) started to be treatedas an old and well understood one. In the year 2012, this opinion has ratherdrastically been challenged by the results of Refs. [6] where it has been proved, rigorously , that for the most popular “benchmark” THS model (4) the class ofthe eligible Hilbert-space metric operators Θ is in fact empty . In other wordswe were all suddenly exposed to the necessity of reanalyzing the mathematicsbehind the differential-operator models as sampled by Eq. (4).This observation belongs to one of the key motivations of our presentstudy. The emergence of incompatibility of the overall methodical THS pat-tern (3) with the concrete unbounded-operator example (4) implies that theattention of mathematical physicists must immediately be redirected andreturned to the alternative, mathematically correct benchmark models like,e.g., the bounded-operator Hamiltonians of Ref. [1] and/or even to the mostschematic, exactly solvable finite-dimensional models as sampled, say, by thenon-numerical discrete square well of our preceding Paper 1 [7].The latter family of models was characterized by the sequence of the mostelementary finite-dimensional Hamiltonians H (3) ( λ ) = − − λ − λ − λ − − λ ,H (4) ( λ ) = − − λ − λ − − − λ − − λ (5) ( λ ) = − − λ − λ − − − − − λ − − λ i.e., by the matrix H ( N ) ( λ ) = − − λ . . . − λ − . . . − − − − λ . . . − − λ (5)considered at an arbitrary preselected Hilbert-space dimension N . As re-quired, this matrix appears non-Hermitian in the N − dimensional and mani-festly unphysical, auxiliary (and, in our case, real) Hilbert space H ( F )( N ) ≡ R N where the inner product remains trivial, h φ ( F ) | ψ ( F ) i = N X n =1 φ ( F ) n ψ ( F ) n . In Paper 1 we emphasized that one may try to deduce the physical context,contents and meaning of models (5) in their N → ∞ limiting coincidencewith certain usual single-parametric differential Schr¨odinger operators on theline [8].In the additional, methodical role of non-contradictory and exactly solv-able, non-numerical benchmark models, the most serious weakness of Hamil-tonians (5) may be seen in their trivial kinetic-operator nature inside thewhole interior of the interval of the spatial coordinate x (see also Paper 1 fora more explicit explanation and further references). This means that their7ontrivial dynamical content (i.e., their point-like-interaction component) ismerely one-parametric and restricted to the points of the spatial boundary.In our present paper we intend to extend this perspective in a systematicmanner by showing, first of all, that the latter weakness of the models ofPaper 1 is curable. We shall introduce and employ a few less elementarytoy-model interactions on the same N − site quantum lattice. In Section 2we select just a less trivial version of the one-parametric interaction whilein subsequent Sections 3 and 4, two and three parameters controlling theinteraction are introduced, respectively. Our overall message is finally sum-marized in Sections 5 (discussion and outlook) and 6 (summary). H ( N ) and metrics Θ ( N ) Let us consider a non-Hermitian and real N by N Hamiltonian matrix H inwhich the interaction connects the triplets of the next-to-the-boundary sites, H ( N ) ( λ ) = − − λ . . . . . . − λ − λ . . . ...0 − − λ − − − − − − λ . . . − λ − λ . . . . . . − − λ . (6)8ecalling the experience gained in Paper 1 we may expect that the bound-state-energy eigenvalues obtained from this Hamiltonian will be all real atthe sufficiently small values of the couplings λ ∈ ( − a, a ) with, presumably, a = 1.A rigorous proof of the above conjecture would be feasible albeit lengthy.Although we are not going to present it here due to the lack of space, Figure1 samples the whole spectrum at N = 11 and offers a persuasive numericalsupport of such an expectation. Moreover, a comparison of this picture withits predecessors of Paper 1 indicates that the use of a less trivial Hamilto-nian seems truly rewarding. In the past, the phenomenologically rich andpromising nontrivial structure of the parameter-dependence of the spectrumnear λ ≈ a motivated quite strongly the continuation of our study of similar,more complicated toy models. λΕ Figure 1: The λ − dependence of the eigenvalues of Hamiltonian (6). Obvi-ously, this spectrum stays real in the interval of λ ∈ ( − , N − parametric sets of metricsΘ ( N ) = N X k =1 µ k P ( k ) (7)to the mere evaluation of a characteristic sample of its individual Hermitian-matrix components P ( N ) k . These components may be interpreted as metric-resembling (i.e., not necessarily positive definite) matrices. Their main ped-agogical merit is that they remain sufficiently transparent matrices with,hopefully, sparse structure of the universal form which has been found anddescribed in Paper 1.With this purpose in mind we shall require that the individual com-ponents of the sum Eq. (7) satisfy the Dieudonn´e equation alias quasi-Hermiticity condition N X m =1 h(cid:0) H † (cid:1) jm P mn − P jm H mn i = 0 , j, n = 1 , , . . . , N . (8)In the light of the analysis of Paper 1 we shall, furthermore, save time and skipthe exhaustive discussion of the (more or less trivial) general N − dependenceof the model. In order to gain an overall insight into the structure of theTHS representability of our model, we found it sufficient to restrict attentionto a fixed value of dimension N which is neither too small (we have to avoidthe structural degeneracies at small N ) nor too large (we intend to displaysome matrices in print). 10 .2 Matrix P (6) at N = 11 Following the recipe described in Paper 1 we shall start from the ansatz P (6) = r s s v t v v w w v s w w s r t t r s w w s
00 0 v w w v v t v s s r (9)and, in the light of Eq. (8), we shall compare the matrix product P (6) H withthe matrix product H † P (6) . Element by element, their (row-wise running)comparison yields the nontrivial constraints s = sλ + r in the fifth and seventhstep, v = − vλ + s in the fifteenth step, etc. After the tedious though entirelystraightforward manipulations we obtain the final solution/formulae r = 1 − λ λ , s = 1 + λ λ , v = 11 + 3 λ t = 1 + λ λ , w = 1 + 2 λ λ (10)which indicate that the transition to the more-site interactions in the Hamil-tonian may still be expected to lead to the polynomial or rational-functiondependence of the matrix elements of the metric on the value of the couplingconstant. The second, methodically equally encouraging consequence of theconstruction of the sample pseudometric P (6) is that after a not too drastic11oss of the simplicity of the input matrix Hamiltonians the construction of theclass of admissible metric remains feasible by non-numerical means. Thirdly,via a deeper analysis of Dieudonn´e’s Eq. (8) it is easy to deduce that the ex-tension of the N = 11 results to any dimension N >
11 parallels the patternfound in Paper 1 and degenerates to a virtually trivial extrapolation of theinterior parts of individual items P ( k ) in the matrix sequences determiningthe general metric (7). Once we recall preceding section and disentangle the values of the respec-tive couplings between the two next-to-boundary and two next-to-next-to-boundary sites we obtain the following two-parametric N = 11 Hamiltonianmatrix − − λ − λ − µ − − µ − − − − − − − − − − − − − − µ
00 0 0 0 0 0 0 0 − µ − λ − − λ (11)12ts full display still almost fits in the printed page but what is certainlymore important is that the presence of the new variable coupling µ extendsthe capability of the model of being more useful in some phenomenologi-cally oriented considerations. This seems well illustrated by Fig. 2 wherewe restricted attention to a line in the plane of parameters defined by theconstraint µ → µ ( λ ) = λ + a constant. λΕ Figure 2: The λ − dependence of the eigenvalues of Hamiltonian (11) in whichwe selected the constantly shifted value of µ = µ ( λ ) := λ + 0 . λ ∈ ( − , b ) where b ≈ .
75 for our particularillustrative choice of the constant shift ∆ = µ − λ . The further inspection ofthe picture reveals many further and qualitatively interesting features of the“phase transition” during which the pairs of individual energy levels cross ormerge and, subsequently, complexify. Temporarily, some of the complexifiedpairs may even return to the reality later – notice, in the picture, that thereare as many as nine real level at λ s which lie slightly below the critical λ = 1.13 .2 Pseudometrics In a way paralleling the preceding section we shall now restrict attentionto the intervals of λ ∈ ( a ( µ ) , b ( µ )) and µ ∈ ( c ( λ ) , d ( λ )), i.e., to the two-dimensional physical domain D of “acceptable” parameters in the Hamilto-nian. Inside this domain the whole spectrum remains, by definition, com-pletely real and non-degenerate, i.e., potentially, physical, observable andcompatible with the unitarity of the time evolution.In this setting the obligatory construction of the suitable matrices of themetric may proceed along the same lines as above. In full parallel, we shalltherefore return to the independent variability of the two couplings in theHamiltonian and reopen the problem of the construction of the metric viaEq. (7). In the language of Ref. [1], the N − parametric ambiguity containedin the latter formula makes the related picture of physics flexible and adapt-able to our potential choice of further relevant operators (i.e., in our case, ofsome N by N matrices) of observables.Under our present restricted project, we shall again pay attention merelyto the explicit construction of the “most interesting” N by N pseudometric P ( J ) at J = 6 and N = 2 J + 1 = 13. The method of construction willremain the same. During its application we displayed, first of all, the non-vanishing matrix elements of the sparse difference matrix H † P (6) − P (6) H and made them equal to zero via the solution of the corresonding algebraicequations. At the end of this procedure which completely paralleled ourpreceding use of ansatz (9) as well the format of result (10) we obtained thematrix elements of our sample pseudometric P (6) in the following, equallycompact and comparably transparent form r = (1 + µ ) (1 − λ )1 + λ + 2 µ , s = 1 + µ λ + 2 µ , v = 11 + λ + 2 µ = 1 + λ λ + 2 µ , w = 1 + λ + µ λ + 2 µ . One should add here that due to the multiple symmetries of our Hamiltonianmatrix as well as of the metric, the inversion of the metric (or pseudometric)may be obtained by the simple change of the sign of the pair of our coupling-constant quantities λ and µ . The inspection of the latter formulae also revealsthat the numerators remain the same so that they might be all omitted orignored as an inessential overall multiplication factor. For a proper, non-degenerate tractability of the next family of some three-parametric Hamiltonians we need to deal with the dimensions N ≥
13 atleast. The full matrices will not fit in the printed page anymore. Fortu-nately, their numerous symmetries will still allow us to display the relevantinformation about their matrix elements. In particular, it proves sufficientto display just the upper part of the Hamiltonian matrix in full detail, H ( N ) = − − λ . . . . . . − λ − µ . . . ...0 − − µ − − ν . . . ... 0 − ν − − − λ . . . . . . − − λ . Similarly, the symmetries of the most interesting N = 13 pseudometric com-ponent P (7) of the N = 13 metric (9) enables us to search for its matrixelements via the thirteen-dimensional matrix ansatz15 (7) = . . . . . . r . . . . . . . . . s s . . . ... . . . p t p . . . . · . v q q v . . . . · . w m w . · . . · . u u . · . . · . . . . . . . r . . . . . . (12)It is worth adding that wherever we decide to choose N >
13, the tripledots may be read here as indicating, for all of the sharply larger dimensions,simply the repetition of the same (i.e., of the last) element until the symmetryof the matrix allows.Strictly the same procedure as above leads again to the final and stillamazingly compact solution r = (1 − ν ) (1 + µ ) (1 − λ )1 + λ + 2 µ + 3 ν + ν λ , s = (1 − ν ) (1 + µ )1 + λ + 2 µ + 3 ν + ν λ p = 1 − ν λ + 2 µ + 3 ν + ν λ , v = 11 + λ + 2 µ + 3 ν + ν λ t = (1 − ν ) (1 + λ )1 + λ + 2 µ + 3 ν + ν λ , q = 1 + µ + λ λ + 2 µ + 3 ν + ν λ w = 1 + µ + ν + λ λ + 2 µ + 3 ν + ν λ , m = 1 − ν λ + 2 µ + 3 ν + ν λ u = 1 − ν λ + 2 µ + 3 ν + ν λ From this set of formulae we may extract the similar messages as above.16
Discussion
In the sense of commentaries scattered over the preceding sections we nowintend to complement the preceding Summary section by an outline of a fewpossible future mathematical and methodical as well as purely phenomeno-logically motivated extensions of the model.In the corresponding list of the possible directions of a generalizationof the present model, the one which looks most worth pursuing lies in thesystematic search for the further exactly solvable finite-dimensional modelswhich would admit not only the closed-form representation of the real spec-trum of the energies but also the explicit construction of the metric operator.Even if one would be able to construct just some (i.e., not all) metrics (whichis, after all, most common in the literature), the scarcity of the exactly solv-able models in this field would certainly provide a ground for the publicationof this type of the results.By our recommendation one might particularly concentrate attention tothe preservation of the localized support of the interactions near the cornersof the tridiagonal Hamiltonian matrix. This idea was originally inspired bythe discovery of the tractability of the differential-equation N → ∞ modelswith point interactions at the boundaries [8]. At the finite dimensions, thesame features of the dynamics have now been found to survive even in themodels constructed at the not too large dimensions N ≪ ∞ . We believe,therefore, that the latter choice of the specific dynamics will gain furtherpopularity as a ground of an optimal solvable-model-building strategy in thenearest future.Certainly, there exist further interesting aspects of a systematic, model-based quantum mechanics of the elementary models which look non-Hermitianwhen solely considered in the most user-friendly, F-superscripted Hilbert17pace H ( F ) . One of the most obvious apparent paradoxes may be seen in themathematical non-uniqueness of the assignment of the metric Θ to a givenHamiltonian H . Fortunately, the answer has already been provided twentyyears ago when the authors of Ref. [1] gave the complete answer. Brieflystated: the ambiguity Θ = Θ( H ) merely reflects the open possibility of in-corporation of additional phenomenological information via an introductionof more observable quantities.The best known illustrative example of such an added observable is theBender’s “charge” [4]. Now, whenever one chooses this charge or anotherobservable as a phenomenological input, the possibility and feasibility of theconstruction of the complete family of the eligible metrics Θ = Θ( H ) ina closed, non-numerical form will always represent a significant advantageof the mathematical model. Plus, needless to add, the use of any analyticthough still flexible form of the metric which appears in the mean values, i.e.,in principle, which enters all of the measurable predictions would certainlyenhance the appeal of the theory in applications.Another apparent paradox concerns the “kinematical” multi-index pa-rameter α which reflects the above-mentioned ambiguity and which numbersthe alternative eligible metrics Θ( H ) = Θ α ( H ). It is obvious that for somevalues α critical of these parameters the metric itself may become singular andunacceptable. An interesting potential reward of the further study of a par-ticular quantum model characterized by an operator (or, in our case, matrix)doublet ( H, Θ α ) might be seen in the possible quantitative specification ofthe connections between the critical values of α critical as functions, say, of the(possibly, multi-index) dynamics-determining couplings λ in H = H ( λ ).In some sense, the closely related and/or complementary questions willalso emerge in connection with any toy-model H = H ( λ ) in which the com-plexification of the eigenenergies occurs at the so called Kato’s exceptional18oints λ cricical (at which the energies merge and subsequently complexify- for illustration see, e.g., the presence of the pair of exceptional points λ cricical = ± λ = λ cricical and, secondly, the “kinematical” loss of the existence of the pre-selected S-superscripted Hilbert space at α = α cricical in connection, thirdly, with thenecessary loss of the observability of some other dynamical observable at thesame α = α cricical (the readers should consult, first of all, Ref. [1] in thiscontext).Last but not least, another natural future continuation of research whichmay be expected exceptionally promising might concentrate upon the sce-nario in which the eigenvalues of H remain real while the metric re-regularizes“insufficiently”, becoming merely indefinite after the parameter α itself crosses,in an appropriate manner, the critical value of α cricical . In such a context,one might merely re-classify the resulting “wrong” or “indefinite” metric Θ α as the Bender’s “parity” P and search for his “charge” C in the “new metric”Θ changed = PC (cf. [4] for the complete recipe). On the background of comparison with the older results of Paper 1, one of themost surprising features of their present generalization may certainly be seenin the friendly nature of the more-parametric formulae. A completion andfurther extension of such constructions along the lines indicated in precedingsection seems to be a project with good chances for a success in the future,indeed.Our present first results in this direction may be briefly summarized as19ollows. Firstly, we revealed an emergent pattern of having, up to an overallfactor, the purely polynomial matrix elements of the “pseudometric” com-ponents P of the metrics. Our sample calculations found such a hypothesisreconfirmed.Secondly, we may feel impressed by the emergence of the pattern of mostnatural and obvious further generalizations of the Hamiltonians in whichone introduces new and new parameters at an increasing distance from theboundaries of the lattice. It is certainly encouraging that such a recipe leavesthe construction non-numerical and that it seems to offer unexpectedly com-pact and transparent benchmark-type results. Acknowledgments
Participation of MZ was supported by the GA ˇCR grant Nr. P203/11/1433.Participation of JW was supported by the Natural Science Foundations ofChina (11171301) and by the Doctoral Programs Foundation of Ministry ofEducation of China (J20130061). 20 eferenceseferences