Asymptotic non-Hermitian degeneracy phenomenon and its exactly solvable simulation

Asymptotic non-Hermitian deg e neracy phenomenon and its exactly solv able sim ulation Milosla v Zno jil 1 , 2 , 3 1 The Czec h Academ y of Sciences, Nuclear Phy sics Institute, ˇ Re ˇ z 292, 250 68 Husinec, Czec h Republic, e-mail: zno jil@ujf.cas.cz 2 Departmen t of Ph ysics, F acult y of Science, Univ ersit y o f Hradec Kr´ alo v´ e, Roki- tansk ´ eho 62, 50003 Hradec K r´ alov ´ e, Czec h Republic 3 Sc ho ol for Data Science and Computational Thinking, Stellen b osc h Univ ersity , 760 0 Stellen b osc h, South Africa Abstract A conceptually consisten t understanding is sough t for the in teractions sampled by the imag- inary cubic oscillator with p oten tia l V ( I C O ) ( x ) = i x 3 whic h is, b y it self, not a cceptable as a meaningful quan tum mo del due to a com binatio n of its non-Hermiticit y , un b oundedness and, first of all, due to the R iesz-basis non- diagonalizabilit y of the Hamiltonia n kno wn a s its in trinsic exce ptio na l p oint (IE P) feature. F or the purp oses of a p erturbation-theory-based sim ulation of the emergence of such a singular sy stem , a simplifie d (though no t to o strictly related) to y-mo del Hamiltonian is proposed. It com bines an N − p oin t discretization of the real line of coor dina t es with an ad ho c inte r a ction in a t w o-parametric N -by - N -matrix Hamiltonian H = H ( N ) ( A, B ). After suc h a simplification, one can still encounter a some- what w eak er form of non-diagona lizabilit y a t the con ve ntional Kato’s exceptional-p o int (EP) limit of parameters ( A, B ) → ( A ( E P ) , B ( E P ) ). The IEP-non-diagonalizabilit y phe- nomenon itself a pp ears mimic ked b y the less enigmatic EP degeneracy of the discrete to y-mo del, esp ecially at large N ≫ 1 . What we gain is that the regularization of the simplified to y- mo del in a vicinit y of the con v en tio nal EP b ecomes, in con trast to the IEP case, feasible. Keyw ords: non-Hermitian quan tum physic s of degeneracy phenomena; degeneracies realized as the Kato’s exceptional p oin ts (EPs); construction of EPs in a solv able N b y N matrix mo del; mec hanism of r emov al of the degeneracy via an ad ho c p erturbation; re-confirmation of the unph ysical in t r insic-EP status o f the imaginar y cubic oscillator 2 1 In tro duction The concept of asymptotic non-Hermitian degeneracy called intrinsic exceptional p oin t (IEP) w as introduced b y Siegl and Krej ˇ ci ˇ r ´ ık [1]. They detected the presence of this char- acteristics in the imag ina ry cubic oscillator (ICO) o p erator H ( I C O ) = − d 2 dx 2 + i x 3 . (1) They pro v ed tha t its eigenstates | ψ ( I C O ) n i do not form Riesz basis, so they concluded that suc h an o p erator cannot be considered diag onalizable and that “there is no quan tum- mec hanical Hamiltonian asso ciated with it” [1]. I n this contex t w e reconfirmed, in [2], that the applicabilit y of quan tum theory a t the non-Hermitian IEP dynamical extreme is questionable . W e found that man y sophisticated construction tec hniques whic h still do w ork for finite matrices cannot b e tr ansferred t o the IEP dynamical regime. A brief explanation is that near an N -b y- N -matrix singularity with N < ∞ , the “corri- dor of unitarit y” (cf. [3]) b ecomes, in the N → ∞ limit, unacceptably (i.e., exponentially) narro w. As a consequence, w e felt forced t o conclude that “the practical realizatio ns of the standa r d quantum-mec hanical ICO mo del remain . . . elusiv e” [2], and that “the cur- ren tly unresolv ed status . . . of the IEP-related instabilities do es not seem to ha ve an easy resolution” [2]. In our presen t pap er w e in tend to prop ose another (partial) clarification of the puzzle . The not quite exp ected existence of suc h a new explanation of the problem has t w o ro ots. First, w e imagined that in the ICO mo del of Eq. (1) as we ll as in its m ultiple IEP-singular analogues, a regularization of their IEP singularit y cannot pro ceed directly , via an imme- diate p erturbative regularization of the IEP-singular differen tial op erato rs themse lves . In a “preparatory-step” o f our prop osed treatment of t he problem w e decided to mimic the relev a n t features o d H ( I C O ) via certain limits of its discrete-co ordinate N by N mat r ix al- ternativ es. F o r this purp ose we a re going to consider certain N -by- N -matrix Hamiltonians of the form H ( N ) = T ( N ) + V ( N ) in whic h the kinetic energy is represen ted b y the discrete Laplacean, T ( N ) =             2 − 1 0 . . . 0 − 1 2 − 1 . . . . . . 0 . . . . . . . . . 0 . . . . . . − 1 2 − 1 0 . . . 0 − 1 2             . (2) Suc h an op erato r is, f or an y lo cal and real p otential V ( N ) = V ( N ) ( x ), Hermitian. Moreo v er, 3 its kinetic-energy comp onen t (2) can b e in terpreted a s an immediate discrete analo gue of its con tin uous-co ordinate partner T = − d 2 /dx 2 (cf., e.g., the recen t preprin t [4] for details and/or further references ; here w e use units suc h that ~ = 1 and mass m = 1 / 2) . The second ro ot and step of our presen t IEP-sim ulation prop osal w as inspired b y our other pap er [5]. In a w ay propo sed in this pap er, Hamiltonians H ( N ) = T ( N ) + V ( N ) can be also p erceiv ed as the discrete-coo r dinate analogues of certain more general non-Hermitian differen tial-op erator partners as sampled b y the ICO mo del o f Eq. ( 1): The corresp ondence ma y b e mediated, e.g., b y the ev a luations of a suitable giv en p ot ential at the g rid-p oint co ordinates x 1 , x 2 , . . . , x N (for more details see section 2 below ). This may b e expected to simplify the analysis. Thus , in [5] w e studied these approximan ts, with the basic message b eing, from our presen t p oint of view, just a mixture of goo d and bad news (see also a few further related commen ts b elow ). What remained encouraging w as an observ ation that the construction of the necessary N by N matrix approx imants migh t b e a feasible task, esp ecially when one admits an ample use of computer-assisted sym bolic manipulations. In parallel, a new discouragemen t emerged when w e noticed that with the gro wth o f N , the memory-and-time costs of the construction app eared to increase s o quic kly that we were only able to t est its p erformance up to comparativ ely small N = 6. Th us, w e had to conc lude, temp orarily , tha t the desirable “extrapo lation to an y h yp othetical con tin uous-co or dina t e limit N → ∞ do es not seem to b e feasible a t presen t” [5 ]. V ery recen tly , our ske pticism faded aw a y when we imagined that for a sim ulation of the IEP degeneracy it is in fact not necessary fo r us to insist on the en tirely general form of the family of the discrete p otentials V ( N ) ( x i ), esp ecially b ecause the IEP degeneracy itself is merely a n asymptotic, higher-excitation phenomenon. T he consequenc es of the g r owth of our in teger parameter N could equally w ell b e studied , therefore, using any suitable to y-mo del V ( N ) ( x i ). This idea immediately led to a n innov ated formulation o f our “ pa rtial-remedy” pro ject (cf. section 3). The essence of the inno v ation was that for the purpo ses o f the IEP- degeneracy simulation during the contin uous-co ordinate limit N → ∞ , t he structure of the degeneracy itself may and will be k ept elemen tary and fixed. This led to the results whic h will b e described in sections 4 a nd 5, with a compact summary added in section 6. 2 Difference -op erator Hamiltonians Although the IEP-singular ICO op erato r (1) cannot se rve as quantum Hamiltonian, its eigenstates form a complete set [1 ] so that one feels tempted to regularize t he model using a suitable p erturbation. Along suc h a pa th of conside r a tions (see , e.g., [2]) one 4 almost immediately disco ve r s that b esides the IEP-related asymptotic degeneracy (6 ), there emerges a lso ano ther and, p ossibly , equ a lly serious tec hnical obstacle whic h is related to the un b oundedness of t he op era t o r. This is a c hallenge which was already kno wn to Dieudonn´ e (cf. [6]). One o f its resolutions w as recommended a nd describ ed in review [7] (cf. also [8]). As w e already men tioned, a no t to o dissimilar tric k based on the discretization of co ordinates will b e also used in what follow s. 2.1 Discrete large − N ve rsion of square w ell 0 100 200 300 400 500 600 (arbitrary units) N=450 N=500 N=550 (N=infinite) E(n) n Figure 1: Curv es E ( n ) fitting the discrete-sq uare- w ell sp ectra (4) of Sc hr¨ odinger equation living on an equidistan t grid-p oin t lattice of a fixed length and v ariable mesh-size λ . 100 200 300 400 500 600 (arbitrary units) N=300 N=550 N=infinite E(n) n Figure 2: Same as Figure 1, with the small circles marking the cen tral lev el. Bey ond this “privileged” lev el, the N < ∞ approxim a t ion of the con tinuous-coordinat e deep-square- w ell sp ectrum ceases to b e meaningful. A t any sufficien tly large N < ∞ , the cen tral lev el can b e p erceiv ed as mimic king a t ypical “hig hly excited” state of t he N = ∞ sys tem with n ≫ 1. A t a fixed N < ∞ in Schr¨ odinger equation  T ( N ) + V ( N )  | ψ n i = E ( n ) | ψ n i n = 1 , 2 , . . . (3) w e may start our considerations by letting the p oten tia l, inside a finite in terv al of a fixed length L , v anish, V ( N ) = V ( N ) sq uar e w ell = 0. The n, the energy sp ectrum b ecomes strictly 5 p ositiv e and can b e written in closed form. In terested readers ma y find the details in the recen t preprin t [4] – for our presen t purp oses, it is sufficien t to recall just t he ultimate energy-sp ecifying formula Nr. 3 .1 5 o f lo c. cit. , viz., E ( sq uare we ll ) ( n ) =  sin[ π n λ/L ] λ  2 , n = 1 , 2 , . . . . (4) The real width of the infinitely deep square w ell is equal here to the pro duct L = N λ of N with the distance λ b et w een the grid p oin ts. This means that the decrease of λ → 0 is equiv alent to the gro wth of N → ∞ . Tw o aspects of the latter form ula are relev an t. First, the formula repro duces the con tinu o us-co ordinate square-w ell sp ectrum in the limit of N → ∞ (see Figure 1). Second, the w hole contin uous-co ordinate square-w ell sp ectrum can b e p erceiv ed as a limit of the mere lo w er part of the resp ectiv e N < ∞ sp ectrum (see Figure 2). 2.2 Inclusion of non-Hermitian p oten tials One of the characteris tic features of t he square-we ll sp ectrum with N = ∞ is the steady gro wth of the energy-lev el differenc es E ( sq w ) n +1 − E ( sq w ) n with n . The insp ection of the pictures rev eals t ha t at an y finite N < ∞ , suc h a feature is inadv erten tly lost b ey ond n = n max where n max = [ N / 2]. Th us, o nce one decides to tak e a non trivial V ( N ) 6 = 0 and study the sp ectra at the large N ≫ 1, one can still safely ignore the excited states with n > n max as irrelev an t. F or our presen t purp o ses it is import a n t to kno w that all of the latter observ ations remain applicable esp ecially after the inclusion of small p erturbations V ( N ) 6 = 0. One should only keep in mind that ev en then , the sp ectrum need not b e p ositiv e definite. In this sense the ab ov e-men tioned idea of the asymptotic ir r elev ance of the upp er half of the sp ectrum of the discrete-co ordinate toy mo dels with finite N < ∞ m ust be pr o p erly mo dified . The elusiv e IEP degeneracy can b e no w p erceiv ed as paralleled and mimic k ed b y its finite − N sim ulatio n mediated b y the Kato’s degeneracy-caus ing exceptional p oin ts (EP , [9]) or, more precisely , b y the Kato’s exceptional p oints of order M (EPM). In this direction w e already p o in ted out, in [2], that the ICO-related singularity finds its analogue in the b e- ha vior of eigenstates | ψ n i of a pa r ameter-dep enden t no n-Hermitian matrix H ( N ) = H ( N ) ( κ ) in the vicinit y of its exceptional-p oin t singularit y H ( N ) ( κ ( E P M ) ) at a su it a ble degree of de- generacy M ≥ 2. Some of these observ ations we re already dev elop ed in paper [5]. W e rev ealed there that a consequen t and systematic analysis o f the m ultiparametric and P T − symmetric ma t r ix mo dels H ( N ) = T ( N ) + V ( N ) of the form dep ending on a real [ N/ 2] − plet of parameters, 6 viz., H ( N ) = H ( N ) ( A, B , C, . . . ) =               − iA − 1 0 . . . . . . 0 − 1 − iB − 1 0 . . . . . . 0 0 − 1 − iC − 1 . . . . . . 0 0 . . . . . . . . . 0 0 . . . . . . − 1 iC − 1 0 0 . . . . . . 0 − 1 iB − 1 0 . . . . . . 0 − 1 iA               (5) ceases t o b e easy ev en in t he first no ntrivial three-par a metric case with N = 6 and M = 6. On these grounds we decided t o circum v ent the obstacles via a we akening of the strength of the assumptions. In place of the o ver-am bitious searc h for the EPMs “with a sufficien tly large M ”, w e will consider a reduced task in whic h the n umber of the free parameters (determining also the maximal order M o f the EPM degeneracies) will b e k ept fixed and restricted to t he first few smallest integers . 2.3 Purely imaginary d iscrete p oten tials In any ordinary differen tia l Schr¨ odinger eq uat io n H ψ n = E n ψ n and for an arbitrary un b ounded Hamiltonian H = − d 2 /dx 2 + V ( x ), as we already p ointed out, the real line of co ordinates x ∈ R (or, possibly , its finite segmen t) ma y b e replaced by a discrete and equidis ta n t ( a nd, sa y , finite) grid-p oin t lattice { x 1 , x 2 , . . . , x N } . In suc h a setting, a return to the con tin uous-co o rdinate limit can b e mediated by the growth of N → ∞ in com bination with a sim ultaneous decrease of the grid-p oin t distance λ ∼ 1 / N . F or the special, ICO-inspired class of mo dels o f our presen t in terest, the con tin uous- co ordinate p oten tials exhibiting the parit y-time symmetry H P T = P T H will b e assumed purely imaginary . Then, the b ounded-op erator a v atar of the initial Hamilto nia n with suc h a symm etry acquires the mat r ix form of Eq. (5) exhibiting the same antilinear symmetry . The kno wledge of the parameters might enable us to reconstruct (or, b etter, in terp o - late/appro ximate) the contin uous-co ordinate p oten tial V ( x ) in the limit of large N → ∞ . F or the sak e of definiteness, we will assume that all of the constan ts A , B , . . . are real. A t a fixed N our non-Hermitian but P T symmetric quan tum Hamiltonia n will then ha ve the N b y N matrix form (5). Its sp ectrum will b e real, dis crete and non-degenerate in an N − dep enden t domain D of parameters whic h ma y b e called “phy sical”. This do- main is not empty since it con tains a subset of the parameters whic h remain sufficien tly small. Mo del (5) can b e then p erceiv ed as a p erturbation of the con v en t io nal square w ell with Hamiltonian H ( N ) ( sq w ) = H ( N ) (0 , 0 , . . . ) represen ted b y a matrix whic h is Hermitian, H ( N ) ( sq w ) = h H ( N ) ( sq w ) i † . 7 2.4 Differen tial - d ifference-op erator c orr esp ondence Inciden tally , mo del (1) play ed, for y ears, the role of an imp ortant b enc hmark example in sev eral branc hes of phys ics [10, 11, 12, 13]. The dispro of of its probabilistic quan tum- mec hanical tractability w as, therefore, disturbing. Its unacceptabilit y has also b een re- confirmed by G ¨ un t her with Stefani [14] who pro vided a “clear complemen tary evidence” that the mo dels o f suc h a t yp e “are not equiv alen t to Hermitian models”, mainly due to the IEP prop erty . They called this prop ert y “non-Rieszian mo de b eha vior” and, in a w ay whic h inspired also our presen t constructive considerations, they concluded that “what is still lac king is a simple ph ysical explanation sc heme for the non-Rieszian b eha vior o f the eigenfunction sets” [14]. W e found it imp ortant that the latter IEP-degeneracy prop ert y of the eigenstates | ψ ( I C O ) n i o f H ( I C O ) can b e visualized, roughly sp eaking, as a steady weak ening of t heir m utual linear indep endence with the grow th of excitation, | ψ ( I C O ) n i ≈ | ψ ( I C O ) n +1 i , n ≫ 1 . (6) A deep er insigh t is obtained when o ne recalls the notion of exceptional p oin t (EP) as in tro duced in the classical Kato ’s monograph on p erturbat ion theory [9]. In spite of the fact that the Kato ’s atten tion has only b een restricted to the finite and para meter-dep endent N b y N matrices H ( N ) ( κ ), his concept of the EP degeneracy can really b e found analogous to its “intrinsic”, asymptotic v ersion of Eq. (6). The analogy is imp erfect of course. It o nly finds a partial supp ort in the fact that at a finite N < ∞ , the EP can b e a lso defined as an instant of parallelization of sev eral (i.e., in general, of M ) eigen vec to r s | ψ ( N ) n ( κ ) i of H ( N ) ( κ ) at κ = κ ( E P M ) with suitable M ≤ N . In comparison with IEP , t he other formal difference is that in the EP limit (i.e., in the EPM limit) κ → κ ( E P M ) , the Kato’s finite − N degeneracy inv olv es not only an M − plet of certain prop erly normalized eigen v ectors, lim κ → κ ( E P M ) | ψ ( N ) m j ( κ ) i = | ψ ( N ) ( E P M ) i , j = 1 , 2 , . . . , M , (7) but also the related eigenv alues, lim κ → κ ( E P M ) E ( N ) m j ( κ ) = E ( N ) ( E P M ) , j = 1 , 2 , . . . , M . (8) In [2] we found the “simplification” pro vided by the absence of the energy-degeneracy (8) to b e mo r e than comp ensated, in the IEP context, by the tec hnical complications arising from the “non-Rieszian mo de b eha vior” (6). Sie gl with Krejˇ ci ˇ r ´ ık emphasized, fo r this reason, that t he IEP singularit y is “m uch stronger than any EP asso ciated with finite Jordan blo c ks” [1]. Th us, our presen t ten tat iv e replacemen t of the IEP-singular systems b y their discrete EPM ana lo gues could help us t o understand at least some of the deep er mathematical ro ots of the rather subtle IEP-unacceptabilit y enigma . 8 3 The Kato’s exception al p oints at finit e N 3.1 The domains of unitarit y In our prese nt pap er w e in tend to study the regula r izat io n of the IEP mo dels based on a simu la tion of t heir prop erties using matrices (5). Our motiv ation is threefold . Firstly , w e feel impressed b y the empirical observ atio n that all o f these matrices share certain not quite exp ected “exact solv abilit y” features. Secondly , we find it imp ortan t that a ll of these mo dels seem t o admit a fall in to an EPM singularity via a smo oth, unitarit y-non- violating passage through a strictly ph ysical domain D . F orcing the system to ev olv e in to a loss-of- the-observ abilit y collapse, i.e., tow ards the mathematically rather sp ecific EPM-singularity extreme. Thirdly , b esides the most elemen tary discrete-co ordinate lo cal- p oten tial phy sics b ehind mo del (5), the study of t he same finite-matrix form of a realistic Ha milto nian could also find its motiv a t io n in a differen t phenomenological bac kground. Pars pr o toto , let us men- tion pap er [15] in whic h Jin and Song in tro duced suc h a Hamiltonian (5) (in the sp ecial case with v anishing B = C = . . . = 0) as a tigh t- binding c hain mo del with p ossible appli- cations in condensed matter ph ysics sa y , of the Blo ch’s electronic systems with impurities. These autho rs a lso men tioned that the same one-parametric matrices H ( N ) ( A, 0 , 0 , . . . ) find also another, en tirely different field of applicabilit y in quan tum information theory where they describ e the arr ays of qubits. It is also w o r th adding that another immediate one-parametric generalization of the mo del has b een pr o p osed b y Joglek ar et al [16]. –1 0 1 –1 0 1 N=4 N=5 A B Figure 3: The star-shap ed domains D of the realit y of sp ectra for the first t wo non trivial mo dels (5) with N = 4 and N = 5 In the virtually trivial one-parametric ve r sion of our mo del with A 6 = 0 and with v anishing B = C = . . . = 0 the lo calization of the central EP = EP2 singularit y is feasible, in closed form, at a n arbitr a ry finite matr ix dime nsion N (cf. [17]). In the further, non trivial versions of the t wo-parametric systems of o ur presen t in terest, the lo calization of the ph ysical do ma ins D = D ( N ) is, in contrast, no t a trivial task an ymore. This is illustrated 9 b y Figur e 3 where w e see a p erceiv able difference b et w een the shap es of b oundaries ∂ D ( N ) at the t wo smallest N . The picture sho ws that in comparison with N = 4, the domain of unitarit y is more protruded and rotated to t he left a t N = 5. In the light of the ab ov e-cited “no-go” observ ation that the c hoice of the triplet of v ariable parameters A 6 = 0, B 6 = 0 a nd C 6 = 0 leading to the extreme EPM with M = 6 already lies b ey ond t he area o f pra ctical feasibilit y of the constructions at the larger matrix dimensions N > 6 (see [5]) , w e are confronted with the r emaining op en question concerning the feasibilit y of the EPM constructions at arbitrary N in the t w o-parametric regime with A 6 = 0 and B 6 = 0 while C = D = . . . = 0. This is the question whic h is to b e addressed, and whic h will b e affirmativ ely answ ered in what follo ws. 3.2 Secular p olynomials In an o ve ra ll introduction to our forthcoming lo calization of the no n- Hermitian EPM de- generacies w e will consider, sep a rately , the tw o-parametric N b y N matrix Hamiltonia ns of Eq. ( 5 ) with the ev en and o dd N . The differences are caused b y the P T − symmetry of the matrices whic h enables us to ev aluate the resp ectiv e energy-dependen t secular p olynomials P ( N ) ( A,B ) ( E ) in the slightly differen t resp ectiv e fo rms. 3.2.1 Ev en N = 2 K Ev en a brief insp ection o f Figure 3 mak es the differences b et w een the ev en and o dd N s clearly visible. Th us, once w e restrict our attention, for the start, to the ev en-dimensional mo dels with N = 2 K , w e may immediately reduce the solution of our basic eigenv alue problem (3) to t he searc h of the ro ots o f the related secular p olynomial P (2 K ) ( A,B ) ( E ) = x K + c 1 ( A, B ) x K − 1 + c 2 ( A, B ) x K − 2 + . . . + c K − 1 ( A, B ) x + c K ( A, B ) . (9) By induction w e can then immeidately pro ve the following result. Lemma 1 A t even N = 2 K = 4 , 6 , 8 , . . . , the c o efficients in (9) c an b e evaluate d in close d form, yielding ( − 1) K c K ( A, B ) = (1 + AB ) 2 − A 2 (10) and ( − 1) K c K − 1 ( A, B ) = B 2 + K ( K − 1) A 2 / 2 − (2 K − 1) − ( K − 1)( K − 2)(1 + AB ) 2 / 2 (11) etc. 10 Pro of is simplified b y the K − indep endence of Eq. (10). This mak es the induction-step elemen tary b ecause it b ecomes restricted just to the single fo rm ula of Eq. (11).  Due to the P T − symmetry , the EPM mergers of the M − plets of the energies E n → E ( E P M ) of o ur in terest o ccur at the v ery cen ter of the sp ectrum, E ( E P M ) = 0. This leads to our follo wing, easily prov ed k ey conclusion. Corollary 2 The sufficient c ondition of the quadruple sp e ctr al de gener acy at even N (with M = 4 ) has the form of c ouple d p air c K − 1 ( A, B ) = 0 , c K ( A, B ) = 0 (12) of algebr aic p olynomial e quations for A = A ( E P M ) and B = B ( E P M ) in which the r esp e ctive p olynomials as sp e cifie d in L emma 1. The latter Corollary is slightly formal b ecause at least some of the solutions A = A ( E P M ) and B = B ( E P M ) of Eq. (12) ma y happ en to b e complex and, henc e, not of our presen t in terest. Secondly , in principle at least, some of the M − tuple spectral degeneracies ma y reflect just the presence of an exceptional p oin t of the order smaller than M [9 ]. An explicit classification of these degeneracies as w ell as the analysis of the prop erties of the energies and/or w a ve functions near these boundar ies of a cceptability ha v e to b e p erfo rmed case b y case. The pro cess will b e sampled in what follo ws. 3.2.2 Odd N = 2 K + 1 A t the o dd N = 2 K + 1, obv iously , it mak es sense to set P (2 K +1) ( A,B ) ( E ) = E φ ( E 2 ) . where w e can still expand φ ( x ) = x K + c 1 ( A, B ) x K − 1 + c 2 ( A, B ) x K − 2 + . . . + c K − 1 ( A, B ) x + c K ( A, B ) . (13) Then, the follo wing result can b e deduce d. Lemma 3 A t o dd N = 2 K + 1 = 5 , 7 , 9 , . . . , the c o efficients in (13) c an b e eval uate d in close d form, yielding ( − 1) K c K = ( K − 1)( 1 + AB ) 2 + 2 − K A 2 (14) and ( − 1) K c K − 1 = ( K − 1) B 2 + ( K + 1) K ( K − 1) A 2 / 6 − K 2 − K ( K − 1)( K − 2)(1 + AB ) 2 / 6 (15) etc. 11 Pro of , by induction a gain, b ecomes more complicated due to the explicit K − dependence of b oth of t he formulae. Still, no real complications emerge b ecause w e may immediately use the tridiagonality of the matrix and the insp ection of the underlying secular determi- nan t.  Corollary 4 The sufficie n t c ondition of the quintuple sp e ctr al de gener acy at o dd N (with M = 5 ) has the form of c ouple d p air c K − 1 ( A, B ) = 0 , c K ( A, B ) = 0 (16) of algebr aic p olynomial e quations for A = A ( E P M ) and B = B ( E P M ) in which the r esp e ctive p olynomials as sp e cifie d in L emma 3. No w we are finally prepared to mo v e from the IEP-singular, manifestly unph ysical mo dels with, sc hematically , N = ∞ to their EPM-singular analogues with a suitable (and not to o small) N < ∞ , a nd with j ust a fixed and not to o large M ≤ 4 or M ≤ 5. 4 Tw o-parametric mo dels with arbit rary ev en N = 2 K One of the beneficial consequences of the not to o complicated structure of our Hamiltonian matrices (5) is t ha t w e alw ays hav e to searc h, irresp ectiv ely of the parit y of N , fo r the squared-energy ro o t s x = E 2 of a p olynomial of degree K . A t the same time, a deeper insp ection of the problem rev eals a not quite exp ected structural difference b etw een the systems with the ev en N = 2 K and with the o dd N = 2 K + 1. In and only in the latter case, for example, t here exists an o dd cen tral constan t energy lev el E 0 = 0 . F or this reason, w e will study the resp ectiv e tw o subsets of the Hamiltonians separately . 4.1 The next-to-elemen tary Hamiltonian with N = 6 Once w e decided to k eep our Hamiltonians just t w o-par ametric, the most complicated three-parametric N = 6 mo del of Ref. [5 ] b ecomes simplified, H (6) ( A, B ) = T (6) + V (6) ( A, B ) =                − iA − 1 0 0 0 0 − 1 − iB − 1 0 0 0 0 − 1 0 − 1 0 0 0 0 − 1 0 − 1 0 0 0 0 − 1 iB − 1 0 0 0 0 − 1 iA                . 12 It is written here in partitio ned form whic h emphasizes not only the absence of the thir d parameter, C = 0, but a lso a certain enhancemen t of the symmetry of the matrix which encourages us to recall the N = 4 analysis as a metho dical guide. A t N = 6, our first task is t he ev aluation of the sec ula r p olynomial P (6) ( A,B ) ( E ) = E 6 +  − 5 + A 2 + B 2  E 4 +  6 − B 2 + 2 B A − 3 A 2 + B 2 A 2  E 2 − − 1 − 2 B A + A 2 − B 2 A 2 . Its real and non- degenerate ro ot s E ± m ≥ 0 with m = 1 , 2 , 3 ( i.e., the sextuplet of b ound- state energies ) c o uld b e written in closed form. The fo r mulae (easily generated using computer-assisted sym b olic manipulations) b ecome to o long fo r a prin ted displa y . Still, whenev er needed, their graphical presen tation remains instructiv e and straightforw ard (cf., e.g., [18]). In particular, it is imp ortan t to notice and emphasize that the purely n umerically determined star-shaped domain D of the realit y of t he energy sp ectrum at N = 6 has b een found to b e similar and v ery close to its N = 4 predecessor of Fig ure 3. Th us, although w e are not going to pro vide a rigorous formal pro of (whic h could be based on the increase- of-precision metho d of pap er [19] and w ould b e, t herefore, feasible), w e b eliev e that all of the similar n umerical tests seem to confirm a h yp othesis that the spik es of the b oundaries of D really represen t the non-degenerate EPM singularities of order four (a t the ev en N s) or fiv e (at the o dd N s). 4.2 Arbitrary N = 2 K and the EPMs with M = 4 In what follo ws, the ev en matrix dimensions N = 2 K will b e a llo w ed to be arbitry , ren- dering our understanding of the b eha vior of the la rge-matrix mo dels p ossible. –6 –4 –2 0 2 4 6 8 –2 2 N=4 N=4 N=12 N=12 Z(x) x Figure 4: The graphs of p o lynomials Z ( − N ) ( x ) o f Eq. (1 8) with x = x − , N = 2 K and K = 2 , 3 , 4 , 5 , 6. 13 Theorem 5 The de gener acy of the fo ur c entr al eigenvalues (i.e., of the four smal lest eigenvalues) to E ( E P 4) ( A ( E P 4) , B ( E P 4) ) = 0 is enc ounter e d, at any even N = 2 K ≥ 8 , when one c onsiders the p air of p olynomials Z (2 K ) ( x + ) = ( K − 1) x 4 + − 2 ( K − 1 ) x 2 + + 2 x + + 1 , (17) and Z ( − 2 K ) ( x − ) = ( K − 1) x 4 − − 2( K − 1) x 2 − − 2 x − + 1 , (18) and, mak i n g use of their ex act solvability, w h en one determine s the r esp e ctive quadruplets of their r o ots x + = x ( j ) + and x − = x ( j ) − with j = 1 , 2 , 3 , 4 . Whenever one fin d s that these r o ots ar e r e al (which c an b e shown to b e true for K ≥ 4 ), the s e ts of the eligible EP4- supp orting p ar ameters in H (2 K ) b e c ome sp e cifie d b y the formulae A ( E P 4) = A ( j ) ± = x ± , B ( E P 4) = B ( j ) ± = ( K − 1) x 3 ± − 2 ( K − 1 ) x ± ± 1 , j = 1 , 2 , 3 , 4 . Pro of . At an y N = 2 K , the coupled pair of Eqs. (16) can immediately b e solv ed using the elimination of the unkno wn quan tit y B from the second item of Eq. (16). T his can b e done b y t a king the t w o alternativ e (viz., p ositive or negativ e) square ro ots of the latter constrain t which may b e c har acterized b y the subscripted sign to yield t w o options, viz., B = B ± = ± 1 − 1 / A . After the elimination and insertion of B ± = B ± ( A ) into the first item of Eq. ( 1 6) we get the resp ectiv e t w o p olynomials (17), (18), and the tw o r espective relations Z ( ± 2 K ) ( x ± ) = 0. The quadruple degeneracy of the eigenv alues at E ( E P 4) = 0 is guarante ed.  A detailed classific at io n based on the explicit proo fs of the reality of the ro ots x ± of the resp ectiv e p olynomials Z ( ± 2 K ) ( x ± ) remains K -dep enden t (i.e., N -dep enden t). Ev en t hough the p olynomial is of the mere fourth order in x (i.e., exactly solv able), b y far the most straigh tfor ward insight in the K - or N -dep endence of the r o ots is pro vided graphically . In Figure 4 we displa y ed the six shap es of p o lynomial f unctions Z ( x ) = Z ( − 2 K ) ( x ) with K = 2 , 3 , 4 , 5 and 6. As long as b oth of the lo cal minima of these curv es decrease with the gro wth of K , the picture clearly sho ws the con v ergence of the ro ots x = A ( E P 4) − in the limit of K → ∞ . Numerically , this con v ergence is also confirmed b y T able 1. The v alues o f B = B ( E P 4) − = − 1 − 1 / A ( E P 4) − are unique. F or completeness, the T able could ha v e been also extended to contain the ot her sets of ro ots with opp osite sign and subscript, viz., with A ( E P 4) + = − A ( E P 4) − as w ell as with the related B ( E P 4) + = 1 + 1 / A ( E P 4) + . 14 T able 1: Eligible EP4- supp o r t ing real r o ots A = A ( E P 4) − of p olynomials Z ( − 2 K ) ( x ) and their K → ∞ con v ergence ( incomplete illustrative list, see also Eq. (18) and Figure 4 ). F or a completion o f the list, the opp o site-sign v alues A ( E P 4) + = − A ( E P 4) − w ould o nly hav e to be added. K 2 -1.68377 1565 -0.37150697 4 0 - - 3 -1.56060 2400 -0.31330985 5 9 - - 4 -1.51486 8938 -0.27764827 5 5 0.7925172140 1.000000000 5 -1.49093 7129 -0.25270910 8 6 0.5611034016 1.182542836 6 -1.47620 7086 -0.23391142 7 3 0.4652866201 1.244831894 7 -1.46622 4803 -0.21903848 2 7 0.4055099224 1.279753364 ∞ - √ 2 0 0 √ 2 4.3 Numerical detour T able 1 together with Figure 4 can b e r ecalled as a sample of the use of a solv able mo del admitting an internally consisten t extrap ola t io n of an EPM - supp orting quan tum mo del to the la rge N ≫ 1 or eve n tow ards t he limit of N = ∞ . This transition can b e read as a ten tative in terp olation, appro ximation or ev en just a mere sim ulation of a singular differ- en tial o p erator, with the idea of its p ossible (though not ye t sufficien tly w ell understo o d) regularization realized via a return to the simplified, non-EPM (i.e., diagonalizable and, as quan tum Hamiltonians, acceptable) discrete p erturb ed-EPM N < ∞ mo dels. In the light of R efs. [3 ] or [20], for instance, a t least a few highly-excited eigenv ectors of the N ≫ 1 mo del tend to o v erlap at the EPM singularit y . Mimic king, in this man- ner, the b eha vior o f the deeply singular and manifestly unacceptable IEP-singular mo del and of its eigenv ectors, with their a symptotically-high-excitation degeneracy as charac- terized b y Eq. (6) ab ov e . Suc h an argumen t, still just as merely in tuitiv e as it is at presen t, migh t explain the asymptotic (i.e., high-energy) parallelization of eigenv ectors via its EPM- mimic k ed rein terpretation. Sim ultaneously , w e also circum v en t, due to the finite-dimensional form of the EPM mo dels , the v ery essence of the quan tum-theoretical unacceptabilit y o f the IEP-singular op erators as sampled b y H ( I C O ) : According to the dedicated p erturbatio n theory as outlined in [3 ], the reason lies in the existence of a small-p erturbation-induced and quasi-unitarit y-compatible [7] regularization of the EPM singularit y in any N < ∞ model. F or example, in the ligh t o f Figure 4 one can exp ect that at large N = 2 K ≫ 1 the quic k est asymptotic K → ∞ con ve rg ence will b e encountere d in the case of the leftmost EP4 ro o t A = A ( E P 4) − ≈ − √ 2 while the slo we st 15 asymptotic K → ∞ conv ergence can b e exp ected to o ccur for the t w o cental ro ots. F or illustrativ e purp oses we will pic k up, therefore, the v alue “in b et w een” i.e., the righ tmost ro ot A ( E P 4) + ≈ + √ 2 of Z ( − N ) ( x ) alias , due to the left- r ig h t symmetry , the leftmost ro ot A ( E P 4) − ≈ − √ 2 of the o t her, plus-subsc ripted p olynomial Z = x 4 − 2 x 2 + (2 x + 1) g where w e c ha ng ed the normalization factor, and where w e in tro duced a new, asymptotically small parameter g = 1 / ( K − 1). W e will alw ays ha v e g ≤ 1 so that for a purely n umerical lo calization of an y one of the four EP4 ro ots we may try to use an asymptotic-series ansatz x = x ( K ) = − √ 2 + c 1 g + c 2 g 2 + c 3 g 3 + . . . . (19) As long as the v ery c hoice of our ansatz has b een based on our kno wledge o f the solution of equation Z = 0 in the limit of small g → 0 (i.e., of large K → ∞ , cf. the last line of T able 1), the zero-order O ( g 0 ) form of equation Z = 0 is an iden tity . The next, first-order O ( g ) comp onent can b e read as an explicit definition of co efficien t c 1 = −  4 − √ 2  / 8 ≈ − 0 . 323223304 5 . Similarly , the second-order O ( g 2 ) constraint leads t o co efficien t c 2 =  29 √ 2 − 32  / 128 ≈ 0 . 070407760 3 0 while on the third-o rder lev el of precision O ( g 3 ) we get c 3 = − 7  64 − 43 √ 2  / 1024 ≈ − 0 . 02179855231 etc. In this con text we hav e to emphasiz e that although equation Z = 0 is solv able in closed fo rm, it still mak es sense to construct and w ork with our “ redundan t” asymptotic expansion (19) b ecause (a) all of its coefficien ts can b e giv en a close d, exact form, and (b) ev en after its drastic truncation, the expansion yields a fairly reasonable n umerical precision ev en when N = 4 and g = 1 is not to o small (see T able 2). 5 Tw o-parametric mo dels w ith o dd N = 2 K + 1 5.1 Algebraic constructions at small K In pap er [5] w e described, thoroughly , the o dd − N mo dels with K = 1 (i.e., the one- parametric case, se e section IV of lo c. cit. ) and with K = 2 (whic h is the simplest t w o - parametric case yielding EPM with M = 5, see subsection V. C in lo c. cit. ). This allo ws us to start directly fr om the o dd − N system with K = 3. 16 T able 2: The sample of con v ergence of the asymptotic-expansion approximan ts of the leftmost ro ot of p olynomial Z (+ N ) ( x ) at the smallest N = 2 K (due to symmetry , w e could just cop y the last column from T able 1). N first order second order third order exact 14 -1.466128 -1.46808 -1.46622 92 -1.4662 24803 12 -1.47604 -1.4788 -1.47621 6 -1.47620 7086 10 -1.49062 -1.4950 -1.49095 9 -1.49093 7129 8 -1.51413 - 1.5219 -1.51494 -1.51486 8938 6 -1.5582 -1.5758 -1.56095 -1.5606024 0 0 4 -1.6670 -1.737 -1.6888 -1.6 83771565 5.1.1 K = 3 and N = 7 . Hamiltonian H (7) ( A, B ) =                   − iA − 1 0 0 0 0 0 − 1 − iB − 1 0 0 0 0 0 − 1 0 − 1 0 0 0 0 0 − 1 0 − 1 0 0 0 0 0 − 1 0 − 1 0 0 0 0 0 − 1 iB − 1 0 0 0 0 0 − 1 iA                   is, for us, an eligible candidate for an energy-represen ting observ able in quasi-Hermitian quan tum theory [7] only after we manage to sp ecify the phys ical domain D of its real parameters A a nd B fo r which the sp ectrum remains real and non-degenerate. The task requires the ev aluation of secular p o lynomial P (7) ( A,B ) ( E ) = E 7 −  6 − A 2 − B 2  E 5 −  − 10 + 2 B 2 − 2 B A + 4 A 2 − B 2 A 2  E 3 − −  4 + 4 B A − 3 A 2 + 2 B 2 A 2  E . Th us, one ob vious a nd constan t cen tral-energy ro ot E 0 = 0 is accompanied b y the sextuplet E ± m with m = 1 , 2 , 3. In closed form, t hese ene rg ies can be expressed using the well kno wn Cardano form ulae. Although these form ulae are already to o long for a displa y in prin t , they ma y b e stored in the computer so that an y form of the graphical or num erical represen tation of the energies still remains to b e an en tirely ro utine task. In the same sense, it is also en tirely straig h tforward to find t he shape of the ph ysical para metric domain D , most easily 17 b y the use of requiremen t of the reality and non-negativity o f the squares of the eigen v alues E 2 ± m at m = 1 , 2 and 3. What one obtains is j ust a sligh tly mo dified analogue of Figure 3. 5.1.2 K = 4 and N = 9 . The same t ec hniques apply to the N = 9 mo del yielding the secular p olynomial P (9) ( A,B ) ( E ) = E 9 −  8 − A 2 − B 2  E 7 −  − 21 + 4 B 2 − 2 B A + 6 A 2 − B 2 A 2  E 5 − −  20 − 3 B 2 + 8 B A − 10 A 2 + 4 B 2 A 2  E 3 −  − 5 + 4 A 2 − 6 B A − 3 B 2 A 2  E . Again, the use of the (still existing) closed form ulae would b e impractical. F or the same reason, also the determination of the b oundaries of the phy sical parametric domain D w ould b e a purely n umerical task. F or t he purp oses of the description of the energy-lev el mergers, fortunately , one only has to kno w the b oundaries of D in a small vicinity of the EPM singularity . This mak es the use of approximate metho ds sufficien t and efficien t. 5.2 Graphical and n umerical constructions W e ha v e noticed that a n enormous growth of the complexit y of the form ulae already emerges at N = 2 K + 1 with K as small as three. In con trast to the ab ov e-desc rib ed elemen tary elimination o f B f r o m Eq. (10) at an a rbitrary ev en N = 2 K , the o dd − N v ersion of constraint c K ( A, B ) = 0 ceases to offer a sufficien tly elemen tary elimination of one of the parameters. A full-fledged computer-assisted elimination tec hnique m ust b e used to solve t he system of the t w o coupled p olynomial algebraic equations (1 6 ). –1.5 –1.0 –0.5 0.0 0.5 1.0 1 2 N=5 N=7 N=9 B Z Figure 5: Graphical determination of the p ositive pairs of ro ots B = B ( E P 5) of the renormalized p olynomials Z = Z (5) ( B 2 ) / 25, Z (7) ( B 2 ) / 49 and Z (9) ( B 2 ) / 900 as defined by Eqs. (20), (21) a nd (22 ), resp ectiv ely . Still, w e can rely up on the computer-generated formulae. What is encouraging is that if o ne t r ies to formulate an o dd- N analogue of Theorem 5, one still arrive s at certain s tr ict 18 analogues of the resp ectiv e auxiliary p olynomials (17) and (18) whic h hav e to v anish at the EPM singularit y . After an abbreviation B 2 = y w e get the rule Z (2 K +1) ( y ( E P M ) ) = 0 with Z (5) ( y ) = y 4 − 1 2 y 3 + 50 y 2 − 76 y + 2 5 , (20) Z (7) ( y ) = 4 y 4 − 4 4 y 3 + 169 y 2 − 2 34 y + 4 9 , (21) Z (9) ( y ) = 8 1 y 4 − 936 y 3 + 3748 y 2 − 5360 y + 900 , (22) etc. The shap es of the latter three curv es are displa y ed in Figure 5, with the resp ectiv e ro ots listed in T able 3. T able 3: Numerical zeros of functions Z = Z ( N ) ( B 2 ) o f F igure 5. N B = B ( E P 5) 5 0 .668317806 2 1.60720 8567 7 0 .502479400 9 1.68667 9121 9 0 .438896023 2 1.81868 7904 In con trast to the left-right asymmetry of the curv es at N = 2 K , their o dd − N ana- logues are left- righ t symmetric (that ’s wh y just their right-semi-axis halv es a re displa y ed in F ig ure 5). This implies tha t the ro ots of the new auxiliary p o lynomials Z (2 K +1) ( y ) no w define the eligible EPM parameters as the pairs of square ro ots B = B ( E P 5) ± = ± √ y . The go o d news is that the EPM construction is reduced to the searc h for ro ots of a p olynomial of the fourth order in y . This means that these ro ots (cf. T able 3) ) as we ll as the further EP5 par a meters are exact. –3 –2 –1 0 1 0.2 0.4 1.0 1.2 1.4 N=11 N=13 A Z Figure 6: Graphical determination of the p ositiv e pairs of ro ots A = A ( E P 5) of certain computer-generated p o lynomials Z ( A ) whic h are, at N = 11 and 13, not suitable for a prin ted display anymore (see also T able 4) . A few less pleasan t complications emerge at the larger K s. In con tra st to the ev en − N constructions, the higher- K p olynomials Z (2 K +1) ( y ) app ear to b e extrap olation-unfriendly , 19 and they do not lo ok sufficien tly elemen tary an ymore. Moreov er, their optimal forms m ust b e computer-generated at ev ery separate v alue of K . The task becomes more and more difficult with the growth of K . The computer- generated p olynomials Z ( N ) ( y ) with N = 11 and N = 13 cease to lo ok nice (so w e do not display them here in prin t). Ev en the rather r o utine computer-assisted n umerical lo calization of their EP5 ro ots b ecomes more costly in b oth o f its graphical forms as sampled in Figure 6, and in its purely n umerical forms as sampled in T able 4 . T able 4: Numerical zeros of the t w o functions of Figure 6. N A = A ( E P 5) 11 0.8 2 44776746 1.60 5 982629 13 0.9 0 57668676 1.57 4 311228 F ortunately , all of these calculations are still routine. One can ev en decide to mo ve b ey ond the scop e of the presen t pap er, tr ying to p erform the computer-assisted calculations at the larger N and M . Our r eaders ma y find a few tec hnical commen ts on suc h a possible future pro ject in App endix A. 6 Summary Detailed analysis o f a family of comparativ ely elemen tary b ound-state mo dels enabled us to r eve a l and describ e an in timat e correspondence b etw een the asymptotic b ehavior of certain non-Hermitian p otentials V ( x ) a nd a no n-Hermiticit y-related EP degeneracy o f the spectra. As one of b ypro ducts, this led to a deep er understanding of the so called in trinsic exceptional p oin t ( IEP) feature of certain maximally non-Hermitian but still P T − symmetry-exhibiting op erators. The IEP singular b eha vior ha s recen tly b een noticed to c haracterize a fairly broad family of ill- conceiv ed candidates for quantum Hamiltonians. Ev en though suc h a prop erty (i.e., basically , the Riesz-basis loss of the diagonalizability) mak es eve ry such an op erator unacceptable as a quantum observ able, w e proposed a partial remedy whic h lies in a p erturbation-theory based w eak ening if not remo v al of its singular nature. The regularization pro cess has three parts. In the first one w e hav e to follow the theory and w e ha v e to circum v en t the unbounded-op erator status of H ( I E P ) [7]. In our presen t pap er the goal has b een ac hiev ed b y the discretization of Sc hr¨ odinger equation, due to whic h w e could rein terpret op erator H ( I E P ) as an N → ∞ limit of a sequence o f certain N b y N matrices H ( N ) . 20 In the second step w e made use of the elemen tary matrix nature of ev ery H ( N ) , and w e rein terpreted and generalized it as parameter-dep enden t, H ( N ) → H ( N ) ( g ) , admitting that the sym b ol g ma y represen t an arbitra r y m ultiplet of auxiliary v ariables, g = { A, B , . . . } . Ev en though our new matrices H ( N ) ( g ) fo rming a broader family are, by construction, non-Hermitian, their sp ectra should remain real and non- degenerate: This restricts the v ariabilit y of g to a certain non- empt y ph ysical domain D . In the third step w e recalled the fact that the b oundary of the latter phy sical parametric domain is formed b y the EP v alues a t whic h one encoun ters the mergers o f energy leve ls. A lo calization of these degeneracies app eared to b e the most difficult part of o ur regularization recip e. Still, in our mo dels w e managed to lo calize the most relev an t part of the EP b oundary ∂ D at whic h the n um b er M of the merging energy leve ls w as maximal. The latter maxima app eared to b e reac hed at the isolated EPs of order M (EPMs), and for the energies forming, at a g iv en matrix-dimension N < ∞ , a precise cente r of the sp ectrum. In other w ords, the k ey p oint was t ha t at eve ry separate N and in spite of the singular na t ur e of ev ery auxiliary op erator H ( N ) ( g ) in the N − and M − dep enden t EP limit of g → g ( E P M ( N )) , the phys ical unitary-ev olution compatibility o f eve ry H ( N ) ( g ) b ecomes restored, for g ∈ D , in an a rbitrarily small vicinit y of g ( E P M ) ∈ ∂ D . On these grounds it w as possible to conclude that in spite of the strictly unphys ical nature of our preselected o p erator H ( I E P ) , its phenomenologically acceptable vicinit y could b e understo o d a s op erators o btained as an N → ∞ limit o f their phenomenologically acceptable N < ∞ predecess o r s H ( N ) ( g ) , prov ided o nly that one would b e able to construct suc h a limit. In our presen t pap er w e managed to p erform suc h a constructiv e regularization, us- ing a subset of matrices H ( N ) ( g ) whic h v aried with tw o-parametric g = { A, B } . As a consequenc e, t he finite − N EPM-related cen tra l-sp ectral parallelization of the middle-of- sp ectrum eigenstates of H ( N ) ( g ( E P M ) ) app eared success fully tractable as mimic king, in the con tinu o us-co ordinate limit of N → ∞ , the IEP-related asymptotic pa r allelization, i.e., as mimic king o ne of the most relev an t characteristics of the IEP degeneracy singularity . F unding This researc h receiv ed no external funding. Data Av ailabilit y Statemen t Data are con ta ined within the article. Conflict of In ter ests The autho r declare no conflict of in terest. References [1] Siegl, P .; Krej ˇ ci ˇ r ´ ık, D. On the metric op erator for the imaginary cubic oscillator. Phys. R ev. D 2012 , 86 , 121702(R). 21 [2] Zno jil, M. The in trinsic exceptional p oin t – a c hallenge in quan tum theory . F ound a - tions 2025 , 5 , 8. [3] Zno jil, M. Unitarity corridors to ex ceptional p oin ts. Phys. R ev. A 2019 , 100 , 032124. [4] P op ov, D. F ree particle trapp ed in an infinite quan tum w ell examined through the discrete calculus mo del. [5] Zno jil, M. Construction of maximally non-Hermitian p otentials under un broken PT- symmetry constrain t. J. Math. Phys. 2025 , 66 , 082102. [6] Dieudonne, J. Quasi-Hermitian Op erators. In Pro c. Int. Symp. Lin. Spaces (P ergamon, Oxford, 1961), pp. 11 5 - 122 . [7] Scholtz, F . G .; Gey er, H. B.; Hahne, F. J. W. 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(5) w e rev ealed tha t eve n after a n ample use o f computers, the practical constructiv e determination of the exceptional-p oin t- represen ting maxima A ( E C M ) , B ( E C M ) , . . . of the parameters is complicated, requiring a lot of computer’s memory and t ime. With the gro wth of N and M , moreov er, the results b ecome also difficult to displa y in prin t. In lo c. cit. , this w a s the reason wh y w e only described, in full detail, the simplest t w o - parametric small-matrix mo dels with N ≤ 5 and M ≤ 5 . Only a sk etch y note has b een added on the N = 6 Hamiltonian with three non-v anishing parameters A , B and C yielding M = 6. The sudden emergence of difficulties persuaded us that one only ha s to rely upo n the computer-stor ed results at N ≥ 6. W e only managed to ov ercome suc h a rather strong ske pticism when w e dec ided to test, in our presen t pap er, the feasibilit y of the constructions in the only remaining unexplored subset of the mo dels with the mere pa ir o f pa rameters A and B and with M ≤ 5 and arbitrary N . W e succeeded, so that our ske pticism w as softened. A t presen t w e started b elieving t ha t a t least some of the tec hnical obstacles will b e circum v en ted using some amended vers io ns of the Gr¨ obner-elimination techniq ues. A practical constructiv e v erification of suc h a b elief is just a c hallenge and op en problem at presen t, forming a bac kground for a deeper study o f the solv abilit y o f the coupled p olynomial a lgebraic equations fo r the ECM maxima A ( E C M ) , B ( E C M ) , . . . in the nearest future. In this setting it mak es sens e to list the first few samples of secu la r p olynomials whic h are asso ciated with mat r ices (5) at N = 7 and N = 8. A.1. Three-parametric mo del with N = 7 In the case of models with A 6 = 0, B 6 = 0 and C 6 = 0 and with arbit r a ry N , our presen t tw o Lemmas and Corolla ry 2 w ould hav e t o b e prop erly g eneralized. Using the same f o rmal 23 represen tation o f the corresp onding sec ular p o lynomial as ab o v e (cf. Eq. (9)), one w ould ha v e to replace our presen t fundamen tal alg ebraic set (16) b y the triplet c K − 2 ( A, B , C ) = 0 , c K − 1 ( A, B , C ) = 0 , c K ( A, B , C ) = 0 . (23) W e exp ect that suc h a set o f coupled p olynomial equations for A = A ( E P M ) , B = B ( E P M ) and C = C ( E P M ) migh t still b e solv able b y a suitable and comm ercially accessible Gr¨ obner elimination techniq ue. A t the same time, the t est as p erformed at N = 6 in [5] p ersuaded us that the standard G r¨ o bnerian reduction of the set (23) to a single p olynomial w ould lead to suc h a high degree of this p olynomial that one could contemplate a direct nume rical solution of the coupled set (23) itself. F or the latter purp ose the c hoice of N = 7 (i.e., of K = 3) w ould immediately lead to the explicit form of relatio ns (23) where one only has to insert c 3 = − 4 + C 2 − 4 B A + 2 C A + 2 C 2 B A + 3 A 2 + 2 C B A 2 − 2 B 2 A 2 + B 2 C 2 A 2 , c 2 = 10 − 2 B 2 + 2 B A − 2 C 2 − 4 A 2 + B 2 A 2 + C 2 A 2 + 2 C B + C 2 B 2 , c 1 = − 6 + A 2 + C 2 + B 2 . A v erification of the h yp othesis that suc h an a pproac h would yield b etter results than the Gr¨ obner basis approach already lies b ey ond the scop e of the presen t pap er. A.2. Three-parametric mo del with N = 8 In the same spirit as a b o ve, the c hoice of the next v alue of K = 4 and of the even N = 8 yields the secular p olynomial with co efficien ts c 4 = 1 + 2 B A − C 2 − 2 C A − 2 C 2 B A − A 2 − 2 C B A 2 + B 2 A 2 − B 2 C 2 A 2 , c 3 = − 10 − 6 B A +3 C 2 − 2 C B +2 C A +2 C 2 B A + B 2 − C 2 B 2 +6 A 2 +2 C B A 2 − 3 B 2 A 2 − C 2 A 2 + + B 2 C 2 A 2 , c 2 = 15 − 3 B 2 + 2 B A − 3 C 2 − 5 A 2 + B 2 A 2 + C 2 A 2 + 2 C B + C 2 B 2 , c 1 = − 7 + A 2 + B 2 + C 2 . One ma y exp ect the existence of the EPM degeneracy with M = 6. The guarante e and lo calization of suc h a degene ra cy will again require the (computer-pro vided) solution o f the triplet of Eqs. (23) so that, for t he purp ose, o ur knowle dge of c 1 remains redundan t. 24 A.3. F our-parametric mo del with N = 8 In the four-parametric models the con trol of the existence o f the EPM extreme o f non- Hermiticit y with M = 8 is prov ided b y the quadruplet of the coupled p olynomial equations c K − 3 ( A, B , C, D ) = 0 , c K − 2 ( A, B , C, D ) = 0 , c K − 1 ( A, B , C, D ) = 0 , c K ( A, B , C, D ) = 0 . (24) A t N = 8 and { A, B , C , D } ∈ D the exact b ound-state-energy ro ots of secular p olynomial P (8) ( A,B ,C,D ) ( E ) ma y still b e defined algebraically and in closed form, in principle at least. In con trast, w e hav e no es tima t e concerning the computer time needed for the solution of the EPM-determining set (24). A t the presen t c hoice of N = 2 K = 8, the separate items of this set o f equations already b ecome to o long to b e prin ted without a bbreviations. Th us, w e will decomp ose c j = k 0 + k 1 D + k 2 D 2 and obtain c 4 = 1 + 2 D 2 C A + 4 D C B A + 2 D 2 C 2 B A + 2 B A − C 2 − 2 C A + 2 D A + +2 D C − 2 C 2 B A − A 2 − 2 C B A 2 + B 2 A 2 − B 2 C 2 A 2 + D 2 C 2 +2 D B A 2 +2 B 2 D C A 2 + B 2 D 2 C 2 A 2 + +2 D 2 C B A 2 + D 2 A 2 i.e., k 0 = 1 + 2 B A − C 2 − 2 C A − 2 C 2 B A − A 2 − 2 C B A 2 + B 2 A 2 − B 2 C 2 A 2 , k 1 = 4 C B A + 2 A + 2 C + 2 B A 2 + 2 B 2 C A 2 and k 2 = 2 C A + 2 C 2 B A + C 2 + B 2 C 2 A 2 + 2 C B A 2 + A 2 . Similarly , in c 3 = − 10 + 2 D 2 B A + 2 D 2 C B + 4 D 2 − 6 B A + 3 C 2 − 2 C B + 2 C A + 2 D B − 4 D C + +2 C 2 B A + B 2 − C 2 B 2 + +6 A 2 + 2 C B A 2 + C 2 D 2 B 2 + 2 D C B 2 − 3 B 2 A 2 − C 2 A 2 + B 2 C 2 A 2 + B 2 D 2 A 2 + + C 2 D 2 A 2 + 2 D C A 2 − 2 D 2 C 2 − 2 D 2 A 2 w e mak e use of abbreviations k 0 = − 10 − 6 B A +3 C 2 − 2 C B +2 C A +2 C 2 B A + B 2 − C 2 B 2 +6 A 2 +2 C B A 2 − 3 B 2 A 2 − C 2 A 2 + + B 2 C 2 A 2 , 25 k 1 = 2 B − 4 C + 2 C B 2 + 2 C A 2 and k 2 = 2 B A + 2 C B + 4 + C 2 B 2 + B 2 A 2 + C 2 A 2 − 2 C 2 − 2 A 2 . Next, for c 2 = 1 5 − 3 B 2 + 2 B A − 3 C 2 − 4 D 2 − 5 A 2 + B 2 A 2 + C 2 A 2 + D 2 A 2 + +2 C B + C 2 B 2 + D 2 B 2 + 2 D C + D 2 C 2 w e may set k 0 = 15 − 3 B 2 + 2 B A − 3 C 2 − 5 A 2 + B 2 A 2 + C 2 A 2 + 2 C B + C 2 B 2 and k 1 = 2 C , k 2 = − 4 + A 2 + B 2 + C 2 while, finally , our last form ula fo r c 1 = − 7 + A 2 + B 2 + C 2 + D 2 do es not require any auxiliary a bbreviations. 26