Finite-gap systems, tri-supersymmetry and self-isospectrality

Finite-gap systems, tri-sup ersymmetry and self-isosp ectralit y F rancisco Co rrea , V ´ ıt Jakubsk ´ y and Mikhai l S. Plyushcha y Dep a rtamento de F ´ ısic a, Universidad de Santiago de Chile, Casil la 307 , Santiago 2, Chile E-mails: fco.correa.s@gmail .com, v.jakubsky@gmail.com, mplyush c@lauca.usac h.cl Abstract W e sho w that an n -gap p erio dic quan tum system with parit y-ev en smo oth p otent ial admits 2 n − 1 isosp ectral sup er-extensions. Eac h is describ ed b y a tri-sup ersymmetry that originates from a higher-order diﬀerent ial op erator of the Lax pair and t w o-te rm nonsingular decomp ositions of it; its lo cal part corresp ond s to a sp ontaneously par- tially broke n cen trally extended nonlinear N = 4 sup ers ymmetry . W e conjecture that an y ﬁnite-gap system ha ving an tip erio dic sin glet states admits a self-isosp ectral tri- sup ersy m metric extension with the p artner p oten tial to b e the original one trans lated for a half-p erio d. Applying the theory to a broad class of ﬁnite-gap elliptic systems de- scrib ed b y a t wo -parametric asso ciated Lam´ e equation, our co njecture is supp orted b y the explicit construction of the self-isosp ectral tri-sup ersymmetric pairs. W e ﬁnd that the sp on taneously b rok en tr i-sup ersymmetry of the self-isosp ectral p erio dic system is reco v ered in the inﬁnite p erio d limit. 1 In tro d uction Finite-gap p erio dic quan tum systems play an imp orta n t ro le in phys ics. They underly the theory of p erio dic solutions in nonlinear integrable systems, including the Kortew eg-de V ries, the nonlinear Sc hr¨ odinger, the Kadom tsev-P etviash vili, and the sine-Go rdon equations [1, 2, 4, 3, 5]. Being analytically solv able systems, they ﬁnd v a rious applications in div erse areas. The list of their applications is extensiv e, and among others includes the mo delling of crystals [6, 7, 8], the theory of monop oles [9], instan t o ns and sphalerons [10, 11], classical Ginzburg-Landau ﬁeld theory [12], Josephson junctions theory [13], magnetostatic problems [14], inh omogeneous cosmologies [15], Kaluza- Klein theories [16], c haos [17], preheating after inﬂation mo dern theories [18], string theory [19], matrix mo dels [20], sup ersymmetric Y ang-Mills t heory [21, 22 ] and AdS/CFT dualit y [23]. Some time ago it w as sho w ed b y Braden and Macfarlane [2 4], and in a more broad context b y Dunne a nd F einberg [25], that a usual N = 2 sup ersymmetric extension of a p erio dic quan t um system ma y pro duce a completely isosp ectral pair with a z e r o ener gy doublet of the ground states. Suc h a picture is completely diﬀeren t fro m that taking place in non- p erio dic systems described b y the same linear N = 2 sup eralgebraic structure { Q a , Q b } = 2 δ ab H , [ Q a , H ] = 0. There, the complete isosp ectrality of t he sup er-par t ners happ ens only 1 in the case of a sp on taneously brok en sup ersymmetry , characterize d b y a p ositiv e energy of t he low est sup ersymmetric doublet [2 6]. F urthermore, it was sho w ed that there exist p eculiar isosp ectral sup ersymmetric p erio dic systems, in whic h t he partner p oten tials are iden tical in shap e but mutually translated for a half-p erio d, or reﬂected, or translated and reﬂected. A pair of sup er-partner p oten tials with suc h a prop ert y w a s na med by Dunne and F ein b erg as self-isos p e ctr al . The phenomenon of self-isosp ectralit y with a half-p erio d shift w a s illustrated by some examples o f exactly soluble mo dels b elonging to a class of ﬁnite-gap p erio dic systems. Later on isosp ectral and self-isospectral sup ersymmetric ﬁnite- gap p erio dic systems we re studied in v arious asp ects [27, 28, 29], and it w as found in [30] that a prop ert y of the self-isosp ectralit y may also appear in some p erio dic ﬁnite-gap systems based on a nonlinear supersymmetry of the second order { Q a , Q b } = 2 δ ab P 2 ( H ), [ Q a , H ] = 0 [31, 32, 33], where P 2 ( H ) is a quadratic p o lynomial. The nature and origin of isospectrality and self-isosp ectrality in ﬁnite-gap systems ha v e remained, how ev er, to b e obscure. Recen tly w e show ed [34 ] that self-isosp ectrality may b e realized b y a non-relativistic electron in the p erio dic magnetic and electric ﬁelds of a sp ecial form, and indicated on a p eculiar non- linear sup ersymmetric structure a sso ciat ed with it. In the presen t pap er, we study sup erextension of quantum p erio dic systems with a parit y- ev en smo o th ﬁnite-ga p p ot ential of g eneral form, and show that it is c haracterized by an un usual tri-sup ersymmetric structure. This p eculiar sup ersymmetric structure originates from t he higher order diﬀeren tial op erat o r of the Lax pair, and its decomp osability in pairs of nonsingular op erators. The sup eralgebra, generated b y three indicated in tegrals of motion together with t rivial in tegrals associated with parity symmetry and matr ix extension, has a nonlinear nature, that reﬂects a nonlinearity of a sp ectral p olynomial of the orig inal ﬁnite-ga p system. The higher order op erator of the Lax pair of a non trivial n - gap ( n > 0) system admits 2 n − 1 nonsingular t w o-term decomp ositions. By means of the Crum-Darb o ux transformation, with eac h nonsingular decomp osition w e ass o ciate a particular tri-sup ersymmetric extension, and as a result get a fa mily of 2 n completely isosp ectral systems. W e sho w that a lo cal part of the t r i-sup ersymmetry is a sp on taneously partia lly brok en cen trally extended nonlinear N = 4 sup ersymmetry , tha t explains the nature and orig in of the complete isosp ectralit y . When the original ﬁnite-gap system has in its sp ectrum a nonzero n um b er of a n ti-p erio dic singlet states corresp onding to the edges of p ermitted bands, among all the non-singular decomp ositions o f the higher order op erato r of the La x pair t here is a sp ecial o ne which corresp onds to a separation of all the singlets in to orthogonal subspaces of perio dic and anti- p erio dic states. W e conjecture that it is this separation that pro duces a self-isosp ectral tri- sup ersymmetric system. This means particularly that all the set of 2 n completely isospectral systems we get, including the o riginal n -g a p system with the sp eciﬁed sp ecial prop ert y , is divided in t o 2 n − 1 self-isosp ectral tri-sup ersymmetric pairs. Then we apply a general theory to a broad class of ﬁnite-g ap elliptic (double perio dic) quantum systems described by a t wo- parametric family of asso ciated Lam ´ e equations. An y suc h a system has in its sp ectrum a non-empt y subspace of a nti-p erio dic singlet stat es, and we supp ort our conjecture by the explicit construction of the self-isosp ectral tri-sup ersymmetric pairs. W e also in v estigate a rather intricate picture of the inﬁnite-p erio d limit of the tri-sup ersymmetry . The pap er is org anized as follo ws. In the next section w e ﬁrst discuss general prop erties of the ﬁnite-gap p erio dic systems with a smo oth p otential, and sho w that an y parit y- even n - gap sys tem is c haracterized by a hidden b o sonized N = 2 nonlinear sup ersymmetry of order 2 2 n + 1. This sup ersymmetry reﬂects the p eculiarities of t he band structure. In Section 3 w e sho w ho w the tri-sup ersymmetric extensions of the syste m are constructed b y means of the Crum-Darb oux transformation. There w e also inv estigate a g eneral structure and prop erties of the tri- sup ersymmetry . In Section 4 w e apply a general theory to the case of ﬁnite-gap elliptic systems describ ed b y the asso ciated Lam ´ e equation. In section 5, the inﬁnite-p erio d limit of the tri- supersymmetry is studied. Section 6 is dev oted to concluding r emarks and outlo ok. 2 Hidden s up ersymmetry in ﬁ nite-gap syst ems T o ha v e a self-contained presen tation, in this section we ﬁrst summarize brieﬂy the prop erties of the quan tum p erio dic systems of a general f o rm. Then w e restrict the consideration to the case of the smo o th parit y-ev en ﬁnite-gap systems to rev eal in them a hidden b osonized nonlinear sup ersymmetry whose order is deﬁned b y the num b er of energy gaps. 2.1 General p rop erties of qu an tum p erio dic systems Consider a quan tum system giv en b y a Hamiltonian op erator H = − D 2 + u ( x ), D = d dx , with a real smo oth p erio dic p o ten tia l u ( x ), u ( x ) = u ( x + 2 L ). F or the corresp onding statio nary Sc hr¨ odinger equation, H Ψ( x ) = E Ψ ( x ) , (2.1) kno wn in the literature as Hill’s equation, w e c ho ose some real basis of solutio ns, ψ 1 ( x ; E ) , ψ 2 ( x ; E ). The o p erator of translation f o r the p erio d 2 L , or the mo no drom y o p erator, T Ψ( x ) = Ψ( x + 2 L ) , (2.2) comm utes with the Hamiltonian H , [ T , H ] = 0. It preserv es a t w o -dimensional linear v ector space of solutions of (2.1), and can b e represen t ed there by the second order mono drom y matrix M ( E ), T ψ a ( x ; E ) = ψ a ( x + 2 L ; E ) = M ab ( E ) ψ b ( x ; E ) . (2.3) The c ha ng e of the basis, ψ a ( x ; E ) → ˜ ψ a ( x ; E ) = A ab ψ b ( x ; E ), det A 6 = 0, generates a con- jugation of the mono drom y matrix, M ( E ) → ˜ M ( E ) = AM ( E ) A − 1 , but do not c hange its determinan t, det M ( E ) = det ˜ M ( E ), trace, T r M ( E ) = T r ˜ M ≡ D ( E ), and eigen v alues, giv en b y solutions of the c haracteristic equation det( M ( E ) − µ I ) = 0 . (2.4) Let us c ho ose a particular basis of solutions ﬁxed by conditions ψ 1 (0; E ) = 1 , ψ ′ 1 (0; E ) = 0 , ψ 2 (0; E ) = 0 , ψ ′ 2 (0; E ) = 1 , (2.5) where prime denotes the x -deriv ativ e. Diﬀeren tiating relation (2 .3) in x and putting then x = 0 in (2.3) and in the deriv ed relation, w e ﬁnd the form of the mono dromy matrix in basis (2 .5), M ( E ) =  ψ 1 (2 L ; E ) ψ ′ 1 (2 L ; E ) ψ 2 (2 L ; E ) ψ ′ 2 (2 L ; E )  . (2.6) 3 W ronskian W ( ψ 1 , ψ 2 ) = ψ 1 ψ ′ 2 − ψ ′ 1 ψ 2 of an y tw o linearly indep enden t solutions of equation (2.1) takes a nonzero x -indep enden t v alue, whic h for basis ( 2 .5) is equal to 1. The n the explicit f orm of the real mono dromy matrix (2.6 ) sho ws that a basis-indep enden t v a lue of its determinan t do es not depend o n energy either, det M ( E ) = 1, and so, M ( E ) ∈ sl (2 , R ) . Note t hat the change x = 0 → x 0 ∈ R in relations (2.5) gives a one-parametric family of the bases, ψ a ( x ; x 0 , E ) = A ( x 0 ) ab ψ b ( x ; E ), A ( x 0 ) ∈ sl (2 , R ) , pla ying a n imp ortant role in the theory of p erio dic quan tum systems [1]-[4]. In suc h a basis, the mono dromy matrix will include an additional dep endence on the mark ed p oin t x 0 , M ( E , x 0 ) ∈ sl (2 , R ). With taking into accoun t that det M = 1, the c har acteristic equation (2 .4) is reduced to 1 − D ( E ) µ + µ 2 = 0, and the basis-indep enden t eigen v a lues of the mono dromy matrix are giv en in terms of its trace 1 , µ 1 , 2 ( E ) = 1 2 D ( E ) ± q D ( E ) 2 / 4 − 1 . (2.7) In corresp ondence with det M ( E ) = 1, µ 1 µ 2 = 1. Common eigenstates of H and T are described by the Blo c h- Flo quet functions ψ ± ( x ; E ), whic h satisfy a relation T ψ ± ( x ; E ) = exp( ± iκ ( E )) ψ ± ( x ; E ) , (2.8 ) where µ 1 , 2 ( E ) = exp( ± iκ ( E )), and the quasi-momen tum κ ( E ) is give n b y 2 cos κ ( E ) = D ( E ) . (2.9) The v alues of the discriminant D ( E ) deﬁne the sp ectral prop erties of the p erio dic Sc hr¨ odinger equation. F or some energies E ∈ ( E 2 i − 1 , E 2 i ), i = 0 , . . . , E i < E i +1 , E − 1 = −∞ , the quasi-momen tum κ ( E ) takes complex v alues, and | D ( E ) | > 2. Solutions corresp onding to suc h E ’s a re not phy sically acceptable as they div erge in x = −∞ o r + ∞ . F or these v alues of E w e hav e a forbidden band, or an energy gap, see Fig . 1. In a generic case, a p erio dic quan tum system has an inﬁnite n um b er of gaps. The width o f the gaps decreases rapidly when energy increases, while the rate of decrease depends on the smo othness of the p oten tial. In the case o f ana lytic p oten tials, the gaps decrease exp onen tially . Energies E f o r whic h |D ( E ) | ≤ 2 , deﬁne p ermitted bands, or p ermitted zones. Here, the quasi-momen tum κ ( E ) tak es real v alues, and complex n um b ers exp( ± iκ ( E )) ha v e mo dulus equals to 1. All the energy lev els with |D ( E ) | < 2 are doubly degenerate, but for |D ( E ) | = 2 w e hav e tw o essen tially diﬀeren t cases. F or those E , whic h separate p ermitted and pro hibited ba nds, corresp onding eigenv alue of the mono dro m y matrix is non-de gener ate , t he matrix M has a form of Jordan matrix, a nd a phy sical singlet band-edge state is p erio dic, exp( iκ ( E )) = +1 , if D ( E ) = 2 , while for D ( E ) = − 2 a singlet state is an tip erio dic, exp( iκ ( E )) = − 1. When |D ( E ) | = 2 but the corresp onding eigen v alue of the mono dro my matrix is do ubly degenerate, M is diagonalizable on the tw o linearly indep enden t Blo ch-Flo quet states, which b oth ar e p erio dic if D ( E ) = 2, o r are antiperio dic when D ( E ) = − 2. This second situation, that corresp onds to p oints E 3 = E 4 and E 9 = E 10 on Fig. 1 , tak es place when a prohibited band disapp ears. Summarizing, the in terv al ( −∞ , E 0 ) constitutes the lo w est forbidden band. The p ermit- ted bands with |D ( E ) | ≤ 2 a re separated by prohibited bands, or energy gaps. All the energy 1 The trace of the monodro m y matrix is ca lled in the literature the Lyapuno v function, Hill deter minant, or discriminant of the Sc hr¨ o dinger equation. 4 +2 − 2 E D ( E ) E 0 0 E 1 1 E 2 1 E 3 = E 4 2 , 2 E 5 3 E 6 3 E 7 4 E 8 4 E 9 = E 10 5 , 5 Figure 1: The discriminan t D ( E ) in a generic situation of a p erio dic p otential. lev els in the in terior of p ermitted bands ha v e a double dege neration, while the states a t their edges ar e singlets. According to the oscillation theorem [35], the common eingenstates of H and mon- o drom y o p erator T with energies E 0 < E 1 ≤ E 2 < E 3 ≤ E 4 < E 5 ≤ E 6 < . . . such that |D ( E k ) | = 2 , are describ ed by the w a v e functions whic h a re characterized b y the p e- rio ds 2 L, 4 L, 4 L, 2 L, 2 L, 4 L, 4 L... , and by the no de nu m b ers in the p erio d 2 L equal to 0 , 1 , 1 , 2 , 2 , 3 , 3 , . . . , see Fig. 1. The o dd n um b er of no des corresponds to antiperio dic states, whereas the p erio dic states hav e an ev en num b er of no des in the p erio d 2 L . The singlet states a t the edges of the same prohibited band ha v e the same n um b er o f no des and the same p erio dicity , and their no des are alternating. 2.2 Finite-gap s y stems and hidden b osonized sup ersymmetry In some p erio dic p oten tials inﬁnite num b er of bands merge together so that only ﬁnite n um b er o f gaps remains in the sp ectrum. Suc h p otentials are called ﬁnite-ga p. The simplest case of a zero-g ap system corresp o nds here to a free particle with u ( x ) = const 2 . F or the Sc hr¨ odinger equation with a ﬁnite-ga p p o t ential the sp ectrum and eigenfunctions can b e presen ted in a n analytical form 3 . Hav ing also in mind that for analytical p oten tials the size o f the gaps decreases exp onen tially when energy increases, any p erio dic p oten tial can b e approximated by a ﬁnite-gap p oten tial if narro w gaps are disregarded. F rom no w on w e supp ose that a p erio dic p otential u ( x ) is ﬁnite-gap. A dditional ly , w e assume that it is an even function, u ( x ) = u ( − x ). Then a reﬂection (parity ) op erator R , Rψ ( x ) = ψ ( − x ), is a nonlo cal inte gral of motion, [ R, H ] = 0. P erio dicit y and pa rit y symmetry together imply that the p oten tial p o ssess es also a middle-p oint reﬂection symmetry u ( L + x ) = u ( L − x ). 2 Here and in wha t follows w e do not co un t the prohibited band ( −∞ , E 0 ) that alwa ys presents in any per io dic system with a smo oth p otent ial. 3 In this sense, a nd in a con trast with, for exa mple, the Kr o nig-Penney mo del, ﬁnite-gap potentials play the same role in s olid-state ph y sics as the K epler problem in a tomic theory . 5 The sp ectrum σ ( H ) of a nontrivial n -gap ( n > 0) system is c haracterized by the band structure, σ ( H ) = [ E 0 , E 1 ] ∪ . . . ∪ [ E 2 n − 2 , E 2 n − 1 ] ∪ [ E 2 n , ∞ ), where E 0 < E 1 < . . . < E 2 n are the non-degenerate energies cor r esponding to the 2 n + 1 sin glet band-edge states Ψ i ( x ), H Ψ i = E i Ψ i , i = 0 , 1 , . . . , 2 n . Since parity op erator R is an integral, each singlet state Ψ i ( x ) has a deﬁnite parit y , +1 or − 1. The energy levels in the in terior of p ermitted ba nds, E ∈ ( E 2 i , E 2 i +1 ), i = 0 , . . . , n , are doubly degenerate, and certain linear com binations of corresp onding Bloch-Flo quet doublet states a re the eigenstates of R with eigen v a lues +1 and − 1. These properties indicate on the presenc e of a hidden b osonize d N = 2 sup ersymm etry in an y ﬁnite-gap system, for whic h op erator R has to play the role of the grading opera t or. The presence of 2 n + 1 ≥ 3 singlet states indicates, ho w ev er, on its nonlinear nature [31, 32, 33]. The sup erc harges a nd the form of the corresp onding nonlinear sup eralg ebra can easily b e iden tiﬁed. An y ﬁnite-gap system is characterize d by the presenc e of a nontrivial in tegral of motio n in the form of an anti-Hermitian diﬀeren t ia l op erator of order 2 n + 1, A 2 n +1 = D 2 n +1 + c A 2 ( x ) D 2 n − 1 + c A 3 ( x ) D 2 n − 2 + . . . + c A 2 n ( x ) , (2.10) where the co eﬃcien t functions c A i ( x ) are real. The absence of the term pro p ortional to D 2 n in its structure, i.e. an equalit y c A 1 ( x ) = 0, is dictated b y the condition [ A 2 n +1 , H ] = 0. Other co eﬃcien ts c A j ( x ) a re ﬁxed in the form of p olynomials in the p oten tial u ( x ) and its deriv ativ es [5]. Thu s, for p erio dic p oten tial, A 2 n +1 is a perio dic op era t o r, i.e. [ A 2 n +1 , T ] = 0. ( A 2 n +1 , H ) is kno wn as the Lax pair of t he n -t h o rder Kortew eg-de V ries (KdV) equation. A p ossible for m of the n -gap p oten tial is ﬁxed b y a nonlinear equation, whic h has a sense of the n -th equation o f the stationary KdV hierar ch y [2, 5]. This equation can b e represen t ed alternativ ely as ˜ L ( J ˜ L ) n 1 = 0 , ˜ L = D 3 + 2( uD + D u ) , (2.11) where J is the op erato r o f indeﬁnite integration J = D − 1 [4]. The form of a o n e-gap p oten tial is ﬁxed by this equation in a unique manner, u ( x ) = 2 P ( x + ω 2 + c ), where P ( x ) is the doubly p erio dic (elliptic) W eierstrass function [36], a nd c is a constan t. T o hav e a real- v alued p otential, one of the p erio ds of P ( x ) is c hosen to b e real, 2 ω 1 = 2 L , while ano t her p erio d 2 ω 2 is assumed to b e pure imaginary , and c ∈ R . In the case n > 1 the form of the p oten tial u ( x ) is no t ﬁxed uniquely ev en if it is restricted to a class of elliptic functions. The mutually comm uting op erators A 2 n +1 and H satisfy the relation − A 2 2 n +1 = P 2 n +1 ( H ) , P 2 n +1 ( H ) = 2 n Y j = 0 ( H − E j ) , (2.12) where P 2 n +1 ( H ) is a sp ectral p olynomial give n in t erms of singlet energies . It is in accordance with Burc hnall-Chaundy theorem [37, 38], whic h say s that if tw o diﬀeren t ial in x op erators A and B of m utually prime orders l and k , resp ectiv ely , comm ute, [ A, B ] = 0, they satisfy a relation P ( A, B ) = 0, where P is a p olynomial of order k in A , and of order l in B . Equation (2.12) corresp o nds to a non-degenerate ( E i 6 = E j for i 6 = j ) sp ectral elliptic curv e of gen us n asso ciated with an n -g a p p erio dic system [1]-[4] 4 . 4 Because of the describ ed properties , u ( x ) is called algebr o-geometric ﬁnite-g a p potential. 6 As a consequence of (2.1 2), the op erator A 2 n +1 annihilates all the 2 n + 1 singlet band-edge states. Indeed, fro m [ A 2 n +1 , H ] = 0 we hav e A 2 n +1 Ψ j = α Ψ j + β Φ j , where Ψ j is a ph ysical ( T Ψ j = γ Ψ j , γ ∈ {− 1 , 1 } ) and Φ j is a non-ph ysical solution corresp o nding to a band-edge energy E j . Acting fro m the left by T , w e get γ A 2 n +1 Ψ j = γ α Ψ j + β T Φ j , and, therefore, β ( γ T − 1)Φ j = 0 . As Φ j is neither p erio dic nor an tip erio dic, the last equation can b e satisﬁe d if and only if β = 0. Then, equation (2.12) dictates that α = 0. Consider the W ronskian of the singlet states, W A ≡ W (Ψ 0 , ..., Ψ 2 n ). In a generic case the W ronskian of s linearly independen t functions that fo rm a k ernel of an arbitra ry linear diﬀeren tia l op erator of order s , L = D s + c 1 ( x ) D s − 1 + . . . , satisﬁes the Ab el iden tity W ′ ( x ) = − c 1 ( x ) W [38]. F or op erator (2.10) a corr esp onding co eﬃcien t function is c A 1 ( x ) = 0, and b ecause of the linear indep endence of the singlet band-edge states w e ﬁnd that W A ( x ) = C 6 = 0 , (2.13) where C is a constant. When s linearly indep enden t zero mo des ϕ j , j = 1 , . . . , s , o f op erator L a re kno wn, the form of this op erator can b e reconstructed in their terms. The co eﬃcien ts c k ( x ) are deﬁned b y relations c k ( x ) = − W k W , k = 1 , . . . , s, where the functions W k ( x ) are obtained fr o m W ronskian W = W ( ϕ 1 , . . . , ϕ s ) b y replacing in it ϕ ( s − k ) j ≡ D s − k ϕ j b y ϕ ( s ) j [40] , see Appendix A. In our case, eac h singlet band-edge state Ψ i ( x ), b eing a zero mo de o f A 2 n +1 , p ossesses a deﬁnite parit y . As a result, with taking into accoun t (2 .13), w e ﬁnd that the co eﬃcien ts c A 2 r ( x ) are o dd , while the coeﬃcien ts c A 2 r +1 ( x ) are even non-singular functions. Hence the in t egral A 2 n +1 is parity o dd, { R, A 2 n +1 } = 0 . (2.14) In tro ducing t wo Hermitian op erators Z = Z 1 = iA 2 n +1 , Z 2 = iRZ, (2.15) and identifyin g them as o dd sup erc harges, w e conclude that any ﬁnite-gap p erio dic sys- tem with ev en smo oth p otential is c haracterized by a hidden b osonized nonlinear N = 2 sup ersymmetry of order 2 n + 1 [32, 33, 39], { Z a , Z b } = 2 δ ab P 2 n +1 ( H ) , a, b = 1 , 2 . (2.16) 3 T ri-sup ersymmetric exten sions of ﬁnit e-gap systems In this section w e show that the application of a non-singular Crum-D arb oux transformation to a ﬁnite-gap system pro duces a partner system with iden tical sp ectrum, a nd study a p eculiar sup ersymmetry app earing in the obtained isosp ectral pair. 3.1 Darb oux transformations and su p ersymmetry A usual mo del of sup ersymmetric quan tum mec hanics is based on a Darb oux transformation, b y whic h an ( almost) isosp ectral system can b e a sso ciat ed with a giv en quantum system. Consider a Hamiltonia n H = − d 2 dx 2 + u ( x ), and a n eigenstate ψ ⋆ corresp onding to a ﬁxed eigen v a lue E ⋆ , H ψ ⋆ = E ⋆ ψ ⋆ . Here w e do not assume an y regularit y conditions fo r ψ ⋆ . It can 7 b e a ph ysical eigenstate, or a second, non-ph ysical solution of the second o rder diﬀeren tial equation, corresp onding to a ph ysical energy lev el E ⋆ , or can be a solution corresp onding to a nonph ysical v alue E ⋆ . The Darb oux transformatio n is generated b y a ﬁrst order diﬀeren tial op erator A 1 = d dx − ( ln ψ ⋆ ) ′ , whic h annihilates ψ ⋆ , A 1 ψ ⋆ = 0, a nd relates H with another Hamiltonian ˜ H = − d 2 dx 2 + ˜ u ( x ) , ˜ u ( x ) = u ( x ) − 2 d 2 dx 2 ln ψ ⋆ , (3.1) b y means of an in tert wining relation A 1 H = ˜ H A 1 . (3.2) Then for eigenstates of t w o Hamiltonians corresp onding t o the same arbitrary v alue of energy E , w e ha v e H ψ E = E ψ E , ˜ H ˜ ψ E = E ˜ ψ E , (3.3) ˜ ψ E = 1 √ E − E ⋆ A 1 ψ E , ψ E = 1 √ E − E ⋆ A † 1 ˜ ψ E . ( 3 .4) Relations (3.3), (3.4) ha v e a symmetry H ↔ ˜ H , ψ E ↔ ˜ ψ E , A ↔ A † . This reﬂects a prop ert y that the transformation corresponding to the adjoint in tertwining relation A † 1 ˜ H = H A † 1 (3.5) is generated b y the o p erator A † 1 , whic h annihilates a state ˜ ψ ⋆ = 1 /ψ ⋆ , A † 1 ˜ ψ ⋆ = 0 , and acts in an opp o site direction by relating ˜ H with H . It is easy to see that the b o th Hamiltonians can b e represen ted in terms of op erators A 1 and A † 1 , H = A † 1 A 1 + E ⋆ , ˜ H = A 1 A † 1 + E ⋆ . Usually , the Darb oux transformatio n is c hosen to annihilate a no deless ph ysical gr o und state ψ 0 with energy E 0 . In suc h a case, the p otentials u ( x ) and ˜ u ( x ) ar e b oth smo oth and regular, or b o th hav e the same singularities 5 . In a non- p erio dic case, the ph ysical no deless ground state ψ 0 v anishes at the ends o f a (p o ssibly inﬁnite) in terv al. As a consequence, there is no phy sical part ner state with the same energy in the sp ectrum of ˜ H . Indeed, the state ˜ ψ 0 = 1 /ψ 0 annihilated by A † 1 is div ergen t at inﬁnit y and is not phys ical. In this case b oth systems are almost isosp ectral, their sp ectrum is the same except the energy lev el E 0 to b e absen t fro m the sp ectrum of ˜ H . Note tha t from the viewpoint o f the adjoint in t ertwining relation (3.5), t he transformatio n from ˜ H to H is generated by the op erator A † 1 asso ciated with a no nph ysical state ˜ ψ 0 = 1 /ψ 0 , whic h cor r esp onds to a nonphys ical for ˜ H eigenv alue E 0 . On the other hand, in corresp ondence with (3 .4), fo r E = E 0 w e still hav e relations ψ 0 = A † 1 ˜ η 0 and ˜ ψ 0 = 1 /ψ 0 = A 1 η 0 , but ˜ η 0 = − 1 ψ 0 R x ψ 2 0 ( x ) dx, η 0 = ψ 0 R x ψ − 2 0 dx a re the non-ph ysical, non-normalizable solutions of t he equations H η 0 = E 0 η 0 and ˜ H ˜ η 0 = E 0 ˜ η 0 . In the p erio dic case with ψ 0 corresp onding to the singlet band-edge state of the low est energy , η 0 and ˜ η 0 are the non-phy sical, non-p erio dic div ergen t solutions. F ro m this discus sion it is also clear that if the Dar b oux transformation is realized with a no deless state ψ ⋆ suc h that b oth states ψ ⋆ and 1 /ψ ⋆ are not phys ical (non-no rmalizable), the energy lev el E ⋆ is absen t from the sp ectra of b oth partner systems , and ph ysical energy lev els satisfy a relatio n E > E ⋆ . 5 Singular Darb oux tr a nsformations genera ted by the s ta tes with no des a lso ﬁnd some a pplications, see [41]. 8 The relation b et w een the Darb oux transformation and the usual sup ersymmetric quan- tum mec hanics is direct. The Hamiltonians H and ˜ H shifted for the constan t E ⋆ are kno wn as sup erpartner Hamiltonians, and form a sup erextended system described by the matrix Hamiltonian H =  H + 0 0 H −  , (3.6) where H − ≡ H − E ⋆ = − d 2 dx 2 + W 2 − W ′ , H + ≡ ˜ H − E ⋆ = − d 2 dx 2 + W 2 + W ′ , (3.7) and W ( x ) is a superp o ten tia l, W = − d dx ln ψ ⋆ . With the Darb oux tra nsformation, tw o Hermitian linear diﬀeren tia l matrix op erato rs Q 1 =  0 A 1 A † 1 0  , Q 2 = iσ 3 Q 1 (3.8) are asso ciated, in terms of whic h in tert wining relations (3.2) and (3 .5 ) take a form of con- serv ation la ws f o r supercharges Q a , [ Q a , H ] = 0, a = 1 , 2. T ogether with Ha milto nian (3.6) they generate the linear N = 2 superalgebra [ Q a , H ] = 0 , { Q a , Q b } = 2 δ ab H . (3.9) A diagonal Pauli matrix σ 3 pla ys here the ro le of the g rading o p erator, [ σ 3 , H ] = 0, { σ 3 , Q a } = 0. In a non-p erio dic case, if ψ ⋆ or 1 /ψ ⋆ is normalizable, there exists a tw o-comp o nen t ph ys- ical state annihilated by b oth matrix sup erc harges, whic h is a ground state of zero energy of one of the sup er-partner subsystems. It is in v ariant under corresp onding sup ersymme try transformations generated b y Q a , and w e hav e the case of exact, unbrok en supersymmetry . The sup ersymmetric doublets of states corresp onding to p o sitiv e energies are m utually tra ns- formed b y sup erc harges Q a in correspo ndence with (3.4). In the case if b oth ψ ⋆ and 1 /ψ ⋆ are not ph ysical, all the states o f sup ersymmetric system (3.6) are organized in sup ersymmetric doublets, including the states of the low est energy , that takes here a nonzero, p ositiv e v alue. This picture corresp onds to the broke n supersymmetry , whic h describes a pair of completely isosp ectral super-pa rtner systems. In the case of a p erio dic quantum system with a smo oth p otential, the sup ersymmetric system (3.6) constructed o n the base of the Darb oux transformation with a no deless singlet band-edge state Ψ 0 will b e characterize d by zero energy doublet of the states g iv en by the columns (0 , Ψ 0 ) t and (1 / Ψ 0 , 0) t . Both these states are a nnihilat ed b y t he sup erc harges Q a , and corresp onding N = 2 sup ersymmetry is unbrok en. Here the sup er-partner systems are completely isosp ectral as in the no n-p erio dic case with broken sup ersymmetry . 3.2 Higher-order Crum-Darb oux transformations (Almost) isosp ectral systems can a lso b e r elated by diﬀeren tial o p erators of higher order, that corresponds to the situation w ell describ ed b y the generalization of the Darb o ux trans- formation due to Crum. 9 Let a diﬀeren tial op erator A k of order k , A k = D k + P k j = 1 c A j D k − j , annihilates a space V spanned by k eigenstates of t he Hamiltonian H , V = span { ψ 1 , . . . , ψ k } , which a re not obligatory to b e phy sical. Then, there holds the relation A k H = ˜ H A k , ˜ H = H + 2( c A 1 ) ′ = H − 2(ln W ( ψ 1 , . . . , ψ k )) ′′ . (3.10) In the case of the Darb oux tr a nsformation ( k = 1), the W ronskian of a single function is the function itself, and (3.10) reduces to the in tert wining relation of the standard sup ersymmetry . The op erat ors A k and A † k pro duce the relations of t he form ( 3 .4) for energies E 6 = E i , i = 1 , . . . , k . F or k > 1, Eq. (3.10) underlies a higher-order (nonlinear) generalization of sup ersymmetric quantum mec ha nics, see [31, 32, 33]. In a generic case the sp ectra of H and ˜ H are almost identical, their sp ectra can be diﬀeren t in k or less n umber of ph ysical eigen v alues. F or a quan tum system describ ed b y H , one can obtain v arious partner Hamiltonians ˜ H , b y c ho osing diﬀeren t sets of eigenstates ψ 1 , . . . , ψ k . How ev er, if w e wan t to g et the asso ciated partner Hamiltonian ˜ H with the same r egularit y prop erties as the initial Hamilto nia n H , these states ha v e to b e c hosen in a special w ay . The higher-order Crum-Darb oux transformations can b e factorized in to the consecutiv e c ha in of the ﬁrst o r der Darb oux, or the second-order Crum transformations, see [42]. W e are in terested in the conditions whic h not only ensure the regularity of the transformations for a p erio dic ev en ﬁnite-gap sys tem, but also pro duce an isosp ectral even partner p oten tial. In a p erio dic sys tem, a r egula r transformation of the second order can b e obtained if the k ernel of the op erato r A 2 consists of the states corr esp onding to the edges of t he same prohibited band [43]. These tw o states, due to the oscillation theorem mentioned in the previous section, ha ve the same p erio d and the same n umber of alternating no des. The W ronskian of the functions selected in this w a y is a function of a deﬁnite sign, not taking the zero v alue. Consider a Crum-Darb oux tra nsformation that a nnihilates the indicated pairs of the edge-stat es. In the case if the order of the transformation is o dd, it has also to annihilate the no deless ground stat e Ψ 0 . This g uaran tees a smo o th and singularity-free p o ten tia l of the part ner Hamiltonian. Concluding, we can construct a whole family of the partner ﬁnite-g a p p erio dic systems b y means of the Crum-Darb o ux transformatio ns, just b y c ho osing appropriately the singlet states of the o riginal system, resp ecting the rules describ ed ab ov e. F or instance, the gen- erator of a hidden b osonized sup ersymmetry Z pro duces a Crum-Darb oux transformation asso ciated with the trivial selection: it annihilates all t he singlets. Since the W ronskian of all the singlet states is a non trivial constan t, see Eq. (2 .13), w e ﬁnd that the pa rtner Hamil- tonian coincides with the original one, and the in tert wining relation reduces to the relation of comm utation of Z and H . Belo w, by means o f non trivial Crum-Darb oux transformatio ns, we shall construct a family of 2 n − 1 diﬀerent completely isos p ectral partner systems for a giv en arbitrary n -gap p erio dic system, and rev eal a sp ecial nonlinear sup ersymmetry app earing in any pair of the total family of 2 n systems . The k ey role in the construction will b elong to the already describ ed hidden b osonized sup ersymmetry . It is worth to notice here that a regular Crum-Darb oux tra nsforma t ion for ﬁnite-gap p erio dic systems can a lso b e pro duced by making use of certain Blo ch functions [2 8, 29, 44, 45, 46]. A partner for par it y-ev en p oten tial obtained in suc h a w a y in a generic case, ho w ev er, will not b e an ev en function. W e shall return to this p oin t in the last section. 10 3.3 T ri-sup ersymmetric extensions and cen trally extended nonlin- ear N = 4 sup ersymmetry Consider an n -ga p p erio dic system, a nd mark r ≤ n prohibited bands in its sp ectrum. 2 r singlet ph ysical states at the edges of these prohibited ba nds span a 2 r -dimensional linear v ector space whic h w e denote by V + . The W ronskian of the corr espo nding 2 r singlet band- edge states is a no deless 2 L -p erio dic ev en f unction. Let Q + b e a linear diﬀeren tial op erato r of order 2 r that a nnihilates the space V + , Q + = D 2 r + 2 r X j = 1 c + j ( x ) D 2 r − j , Q + V + = 0 . (3.11) Singlet band- edge states are p erio dic o r an ti-p erio dic, and can b e presen ted b y real w a v e functions. So , the co eﬃcien t functions in (3 .1 1) are real. T aking also in t o a ccoun t that an y band-edge state has a deﬁnite par ity , one can show that (3.11) is a 2 L -p erio dic ev en op erator, [ T , Q + ] = [ R, Q + ] = 0, see App endix A. The k ernel o f the in tegral Z has a form Ker Z = V + ⊕ V − , where V − is a supplemen ta ry 2( n − r ) + 1-dimensional linear vec tor space spanned b y the rest of singlet band- edge states. Then Z can b e decomp osed as Z = S † Q + , where S † is a diﬀeren tial op erator of order 2( n − r ) + 1 with the prop ert y S † Q + V − = 0. Hermiticity of Z and Eqs. (2.10), (2.15), (3 .1 1) mean that Z = S † Q + = Q † + S, (3.12) and − iS = D 2( n − r )+1 + 2( n − r )+1 X j = 1 c S j ( x ) D 2( n − r )+1 − j , (3.13) where t he co eﬃcien t functions are real and c S 1 ( x ) = c + 1 ( x ). F rom the prop erties of Z and Q + , we also ﬁnd that S is a 2 L -p erio dic parit y-o dd op erator. No w w e show that Ker S = V − . T o t his end w e note t hat in accordance with Eq. (3.10) and the Ab el identit y W ′ = − c 1 ( x ) W , the equalit y c S 1 ( x ) = c + 1 ( x ) obtained directly from (3.12) means that the application of the Crum-Da rb oux transformations with the op erators Q + and S pro duces the same non-singular sup er-partner Hamilto nia n ˜ H = H + 2( c + 1 ) ′ satisfying the in tert wining relations Q + H = ˜ H Q + , S H = ˜ H S, (3.14) Q † + ˜ H = H Q † + , S † ˜ H = H S † . (3.15) The Hamiltonian ˜ H describ es a p erio dic system with an ev en p oten tial of the p erio d 2 L with n ga ps in the sp ectrum. Th us, there exists an o dd Hermitian diﬀeren tial op erator ˜ Z of the fo r m (2.10) comm uting with ˜ H . The intert wining relations (3.14 ), (3.15) pro vide an alternativ e, tw o-term decomp osed form for ˜ Z . Indeed, we get [ ˜ H , S Q † + ] = [ ˜ H , Q + S † ] = 0. Both op erato r s S Q † + and Q + S † are of order 2 n + 1, and should coincide with ˜ Z up to some p olynomial in ˜ H . Ho w ev er, they anticomm ute with reﬂection op erator R , and this implies that ˜ Z = S Q † + = Q + S † . (3.16) 11 T ak e an y of 2( n − r ) + 1 singlet states Ψ − ∈ V − not annihilated b y Q + . Multiplying the equation Z Ψ − = 0 b y Q + from the left and using (3.12) a nd (3.16), w e get S Q † + Q + Ψ − = 0. The op erator Q † + Q + is 2 L -p erio dic op erator , and as it fo llows from (3.14), (3.15), comm utes with the Hamiltonia n H . It changes neither energy nor the p erio d of a singlet state Ψ − , and then Q † + Q + Ψ − = α Ψ − , where α is a non-zero num b er. Hence, S Q † + Q + Ψ − = αS Ψ − = 0 , and we conclude that Ker S = V − . Changing the notation, Q − ≡ S , we ha v e Z = Q † + Q − = Q † − Q + , ˜ Z = Q + Q † − = Q − Q † + , (3.17) and Ker Q + ⊕ Ker Q − = Ker Z , Ker Q † + ⊕ Ker Q † − = Ker ˜ Z . (3.18) This result means a complete isosp ectralit y of the ﬁnite-gap p erio dic systems described by the Ha miltonians H and ˜ H . Indeed, in accordance with the prop erties of the Crum-Darb oux transformations, the action of b o th op erators Q + and Q − on an y doublet of eigenstates of H from the in terior of p ermitted bands transforms it in to a doublet of eigenstates of ˜ H with the same energy v a lue. The adjoin t o p erators Q † + and Q † − act in the same w a y in the opp osite direction. The singlet states of H annihilated by Q + (or Q − ) are tra nsformed b y Q − (or Q + ) into zero mo des of Q † + (or Q † − ) b eing the singlet states of ˜ H o f the same energy . The same picture is v alid for the singlet states of ˜ H annihilated b y Q † + (or Q † − ) and transformed b y Q − (or Q + ) in to the corresp onding singlet states of H . The in tertw ining relations (3.14), (3.15) a s w ell as factorization of the sup erc ha r g es of b osonized supersymmetry can b e rewritten in a compact fo rm once w e use the matrix for- malism a nd deﬁne a n extended Hamiltonain H and op erators Q ± and Z , H =  ˜ H 0 0 H  , Q ± =  0 Q ± Q † ± 0  , Z =  ˜ Z 0 0 Z  . ( 3 .19) Here relations (3.14), (3.15 ) and (3.17) can b e presen ted a s [ H , Z ] = 0 , [ H , Q ± ] = 0 , (3.20) Z = Q − Q + = Q + Q − . (3.21) The tr iplet Q + , Q − and Z is a set of c ommuting integrals for the sup erextended system described b y the matrix Hamiltonian H . There exists a common basis, in whic h Q ± , Z and H are diagona l, and since all these op erato rs are self-adjoin t, their eigenv alues are real. W e ha v e a c hain of equalities Z 2 = Q + Q − Q + Q − = Q 2 + Q 2 − = P Z ( H ) = 2 n +1 Y j = 1 ( H − E j ) , (3.22) where P Z is a p ositiv e- semideﬁnite sp ectral p olynomial, and E j are energies of the singlet states of subsyste ms. In corresp ondence with (3.22), the band-edge states of the extended system are organized in sup ersymmetric doublets, on whic h Z take s zero v alues. The states of t he in terior of p ermitted bands are organized in energy quadruplets, on certain pairs of whic h Z take s nonzero v alues ± p P Z ( E ). The diagonal comp onents of Q 2 ± consist of diﬀeren tia l op erators of orders 4 r and 4( n − r ) + 2. One of these tw o n um b ers is less than 12 2 n + 1. Supp o se that it is t he case of the op erator Q 2 + . Its low er diag onal comp o nen t Q † + Q + satisﬁes a relation [ H , Q † + Q + ] = 0. According to t he general theory of ﬁnite-gap systems , the only o p erators comm uting with H of order lo w er then 2 n + 1 a re p o lynomials in t he Hamiltonian, and w e conclude that Q † + Q + is suc h a p olynomial, whic h has to tak e zero v alues on 2 r singlets b elonging to Ker Q + . The same argumen ts hold for the upp er comp onen t of the op erator Q 2 + , and w e ﬁnd t ha t Q 2 + = P + ( H ) = 2 r Y j = 1 ( H − E + j ) . (3.23) With making use of (3.22) and (3.23), w e get also Q 2 − = P − ( H ) = 2( n − r )+1 Y j = 1 ( H − E − j ) , (3.24) where P ± ( H ) are p ositiv e-semideﬁnite op erators, and E ± are the energies of the corresp ond- ing band- edge states annihilated b y Q ± . The eigen v alues of Q ± are ± p P ± ( E ), where the signs of square ro ots are correlated with the square ro ot sign of corresp onding eigen v alue of Z in accordance with Eq. (3.21). In addition to t he non-trivial in tegrals of motion Z and Q ± , the Hamiltonian H possesses another tr iplet of trivial, m utually comm uting, in tegrals Γ 1 = σ 3 , Γ 2 = R, Γ 3 = Rσ 3 , (3.25) whic h satisfy the relations Γ 2 i = 1, i = 1 , 2 , 3, Γ 1 Γ 2 Γ 3 = 1. An y of Γ i can b e c hosen a s a Z 2 -grading op erato r Γ ∗ . An y of the non-trivial in tegrals of motion either comm utes o r an ti-comm utes with any of the tr ivial in tegrals. Fixing t he grading op erator, w e classify an y non trivial inte gral as a b osonic or fermionic, while the Hamiltonian and the t rivial in tegrals (3.25) are alw ays iden tiﬁed as b osonic o p erators. In correspondence with this identiﬁcation, a certain sup eralgebra is generated. F or all the three p ossible c hoices of the grading op erato r , one of the no ntrivial in tegrals pla ys the role of a b osonic, Z 2 -ev en op erator, while the other t w o in tegrals are classiﬁed as fermionic, Z 2 -o dd op erat o rs, see the T able b elo w. W e name this structure the tri-sup ersymm etry 6 . Grading op erator σ 3 R σ 3 R Bosonic integral Z Q + Q − F ermionic integrals Q + , Q − Z , Q − Z , Q + The complete structure of the tr i- sup ersymmetry will b e describ ed in the next subsection. Here, let us c ho ose Γ ∗ = σ 3 as the grading op erator, and discuss the corresp onding nonlinear 6 Such a structure was o bserved for the ﬁrst time in the N = 2 sup erextended Dirac de lta p otential problem [47], where the basic triplet of non trivial integrals has a completely diﬀerent nature. In particular, there b oth s up er charges of the hidden b osonized N = 2 linear supers ymmetry of the form (3.9) are nonlocal op erators , cf. (2.16) and (2.15) 13 sup ersymmetric subalgebra generated by the lo c al in t egrals of motio n, fo rgetting for the momen t the nonlo cal integral R . Intro duce the notatio n Q (1) ± = Q ± , Q (2) ± = iσ 3 Q ± . (3.26) These fermionic sup erc harges together with b osonic o p erators Z and H g enerate the sup er- algebra {Q ( a ) + , Q ( b ) + } = 2 δ ab P + ( H ) , {Q ( a ) − , Q ( b ) − } = 2 δ ab P − ( H ) , (3.27) {Q ( a ) + , Q ( b ) − } = 2 δ ab Z , (3.28) [ H , Q ( a ) ± ] = [ H , Z ] = [ Z , Q ( a ) ± ] = 0 . (3.29) Sup eralgebra (3.27), (3.28 ), (3.29) is iden tiﬁed as a centrally extended non line ar N = 4 sup ersymmetry , in whic h Z pla ys a role of the b osonic cen tral c harge 7 . The sup ercharges Q ( a ) + annihilate a part of the band-edge stat es organized in sup ersym- metric do ublets, while another part of sup ersymmetric doublets is annihilated by t he sup er- c ha rges Q ( a ) − . The band-edge states whic h do not b elong to the k ernel of the sup erc harges Q ( a ) + (or Q ( a ) − ) ar e t r ansformed (rotated) by these sup erc harges within the corresp onding su- p ersymmetric doublet. The b osonic central c harge Z annihilates all the band-edge states. So, we hav e here the picture reminiscen t somehow of the partia l sup ersymmetry breaking app earing in sup ersymmetric ﬁeld theories with BPS-monop o les [49]. 3.4 T ri-sup ersymmetry and su (2 | 2 ) Let us study the complete alg ebraic structure of the tri-sup ersymmetry . T o do this, consider the set of integrals of motion created by the multiplicativ e com binations of the trivial in tegrals with nontrivial ones, H , Γ i , Γ α Z , Γ α Q + , Γ α Q − , (3.30) where α = 0 , 1 , 2 , 3 , and b y Γ 0 w e denote a unit t w o-dimensional matrix. Eac h of these in tegrals either comm utes or anticomm utes with a n y of Γ i deﬁned in (3.25). Identifying one of Γ i as the Z 2 -grading op erator Γ ∗ , w e separate the set (3.30) into eight Z 2 -ev en (b osonic) op erators commuting with Γ ∗ , and eigh t Z 2 -o dd (f ermionic) op erators, whic h anticomm ute with Γ ∗ . Though this separation dep ends on the c hoice of Γ ∗ , the sup eralgebra in all three cases has, in fact, the same structure. T o reveal this common sup eralgebraic structure for a ll the three p ossible c hoices of the grading op erator, we denote the fermionic op erators as ( F 1 , . . . , F 8 ). The set of b o sonic op erators w e write as ( H , Γ ∗ , Σ 1 , Σ 2 , B 1 , . . . , B 4 ), where Σ 1 , 2 are t w o trivial in tegrals from the set (3.25) to b e diﬀeren t from Γ ∗ . D enote b y P B ( H ) a univ ersal p olynomial pro duced b y the square of an y o f the f o ur in tegrals B a , B 2 a = P B ( H ), a = 1 , . . . , 4. Given Γ ∗ , we can separate the set of fermionic op erators in tw o subsets dep ending on the commutation 7 The basic structure o f the a lg ebra o f ﬁnite order diﬀerential o per ators of a form mo re genera l than (3.27)–(3.29) was discussed by Andrianov and Sokolo v [4 8], but outside the c o nt ext of ﬁnite-gap perio dic systems and parity-ev en p otentials. In compariso n with (3.27)–(3.29), the sup era lgebraic structure o f Ref. [48] includes some additional independent p olyno mial of the Ha milto nia n. As a consequence, instead of the relation Z 2 = P + ( H ) P − ( H ), which follows from Eqs. (3.22)–(3.24) and reﬂects the nature a nd peculiarities of the band s tructure [34], its analog in [48] has a diﬀeren t form, see Eqs. (40 ), (4 1), (43) and (46 ) there. 14 relations with the in tegrals Σ 1 and Σ 2 . The subset which comm utes with Σ 2 , we lab el as F µ , µ = 1 , . . . , 4, [Σ 2 , F µ ] = 0, and denote a univ ersal p olynomial corresp onding to a square of an y o f these fermionic op erators by P 22 , F 2 µ = P 22 . F o r the subset of fermionic op erators whic h comm ute with Σ 1 , we put the index F λ , λ = 5 , . . . , 8, [Σ 1 , F λ ] = 0, a nd denote t he analogous univ ersal p olynomial b y P 11 , F 2 λ = P 11 . Iden tifying in t he describ ed w a y the in tegrals F µ , F λ , B a , Σ 1 and Σ 2 , and computing directly all the (an ti)comm utators, w e ﬁnd the sup eralgebra of tri-sup ersymmetry , whic h may b e presen ted in the same for m mo dulo some sp ecial p o lynomials in dep endence on the c hoice of Γ ∗ . These additional p olynomials w e denote b y P 12 , P 1 B and P 2 B , where the subindex with t w o en tries indicates the origin of the (an ti)comm utation relation. P olynomial P 12 comes from the an ticommutators o f F λ with F µ , p olynomial P 2 B comes from the commutators b et w een the integrals F µ , ([Σ 2 , F µ ] = 0) and B a , p olynomial P 1 B has analogous sense. Since the p olynomials P 12 , P 1 B and P 2 B dep end on the grading, their explicit form together with explicit form of bo sonic and fermionic op erato rs for all thr ee choices o f the gr ading operat o r are pr esen ted in App endix B, see T ables 1, 2 and 3. With the describ ed notations, the a nti-comm utatio n relations for f ermionic op erator s, and comm utation relations b etw een b osonic and fermionic op erators are presen ted in T ables 4 and 5. The iden tiﬁcation of the complete sup eralgebra of t he tri-sup ersymmetry can b e ac hieve d no w if we analyze the still missing comm uta tion relations b et w een Z 2 -ev en generators. In- tro duce the follow ing linear comb inations of t hem, G ( ± ) 1 = 1 4 ( B 1 ± B 3 ) = 1 2 B 1 Π ± , G ( ± ) 2 = − 1 4 ( B 2 ± B 4 ) = − 1 2 B 2 Π ± , (3.31) J ( ± ) 3 = 1 4 (Σ 1 ± Σ 2 ) = 1 2 Σ 1 Π ± , (3.32) where Π ± = 1 2 (1 ± Γ ∗ ) ar e the pro jectors. These op erators satisfy the following algebra h G ( ± ) 1 , G ( ± ) 2 i = iJ ( ± ) 3 P B ( H ) , (3.33) h J ( ± ) 3 , G ( ± ) a i = iǫ ab G ( ± ) b , a, b = 1 , 2 , (3.34) [ G (+) a , G ( − ) b ] = [ J (+) 3 , G ( − ) a ] = [ J ( − ) 3 , G (+) a ] = [ J (+) 3 , J ( − ) 3 ] = 0 , (3.35) The commutation relations (3.33)–(3.35) corresp ond to the direct sum of tw o deformed su (2) algebras in whic h H pla ys a ro le of a m ultiplicativ e cen tral c harg e. This b osonic subalgebra is reminiscen t of the nonlinear a lg ebra satisﬁed by the La place-Runge-Lenz and angular momen tum ve ctors in the quan tum Kepler problem [50, 51]. It is kno wn that in the case of the quantum Kepler problem, its nonlinear symmetry algebra is reduced on the subspaces of ﬁxed energy E < 0, E = 0 and E > 0 to the Lie algebras so (4), so (3 , 1) and e (3), resp ectiv ely , where e (3) is the 3D Euc lidean algebra. Let us see what happ ens with our tri-sup ersymmetry under similar reduction. First, consider an y 4-fold degenerate energy lev el E 6 = E i corresp onding to the inte rior part o f any p ermitted band. Rescaling the o p erators, G ± a → J ( ± ) a = G ( ± ) a /P B ( E ), w e ﬁnd that together with J ( ± ) 3 they generate the Lie algebra su (2) ⊕ su (2). These op erators satisfy the relations J (+) i J (+) i = 3 4 Π + , J ( − ) i J ( − ) i = 3 4 Π − , where the summation in i = 1 , 2 , 3 is assumed. The t w o 15 common eigenstates of the Hamiltonian, with energy E 6 = E i , and of the grading op erator Γ ∗ , with eigenv alue +1 or − 1, carry the 1 / 2 ⊕ 0, or 0 ⊕ 1 / 2 represe n tations of su (2 ) ⊕ su (2), where t he ﬁrst (second) term corr esp onds to the generators J (+) i ( J ( − ) i ). The f ermionic generators mutually transform the states from the t wo eigenspaces of the grading o p erator. In accordance with the total n umber of indep enden t fermionic generators, the energy subspace with E 6 = E i carries an irreducible represen tation of the su (2 | 2) sup erunitary symmetry , whic h is a sup ersymmetric extension of the b o sonic symmetry u (1) ⊕ su (2) ⊕ su (2 ) , where the u (1) subalgebra is g enerated by the grading op erator, see Ref. [52]. Having in mind that the Hamiltonian app ears in a g eneric form of the sup eralgebra as a m ultiplicativ e central c ha rge, w e conclude tha t the system p ossesses a nonlinear su (2 | 2) sup erunitary symmetry in the sense of R efs. [3 1 , 32, 51]. If w e reduce our extende d system to the subspace corresp onding to any doubly degener- ate energy leve l E i corresp onding to a doublet of ba nd- edge states, the b osonic part o f the sup eralgebra is reduced to the alg ebra u (1) ⊕ e (2) ⊕ e (2), where the ﬁrst t erm corresp onds to the integral Γ ∗ , while other t w o corresp ond t o the tw o copies of the 2D Euclidean alge- bras g enerated in the eigensubspaces of Γ ∗ b y the rotat ion op erato r s J ( ± ) 3 and comm uting translation generators G ( ± ) a . Note that the sup ersymmetry of the form similar to the presen t one reduced t o a lev el E i w a s a nalyzed in [53] in the context of sp on taneous sup ersymmetry breaking in 3+1 dimensions. 3.5 Self-isosp ectralit y conjecture In the realm of sup ersymmetric quan tum mec ha nics asso ciated with linear sup eralg ebraic structure, the complete isosp ectrality in the non-p erio dic systems is related to the sup er- symmetry breaking, that means that the doublet of the ground states is not annihilated b y sup erc harg es. In [25], Dunne and F einberg considered supersymmetric extensions o f perio dic p oten tials. They argued that in contrary to the usual situation, the complete isosp ectrality of sup er-partner Hamiltonians could app ear without violation of the sup ersymmetry . As one of the examples of the situatio n they presen ted a one-gap Lam ´ e Hamiltonian, where the sup er- symmetric extension w as pro vided b y the ﬁrst-order Darb oux transformation corresp onding to the ﬁrst o rder sup erc harg es Q ( a ) − deﬁned b y (3.26). The super- partner Hamiltonian sho w ed to b e the or ig inal one but displaced for a half o f the p erio d. As w e men tioned at the b e- ginning, suc h a phenomenon of a half-p erio d displacemen t of sup er-partners w as named in [25] the self-isosp ectralit y . W e explained ab ov e how the complete isosp ectrality emerges due to the tri- sup ersymmetry , namely , its lo cal part (3.27)–(3.29). In this framew ork, the symmetries of the N = 2 sup erextended one-gap Lam ´ e system hav e to b e completed by adding tw o o ther sup erc har ges Q ( a ) + of order 2 and a b osonic in tegral Z of order 3. The second order sup erc harges Q ( a ) − do not annihilate the doublet of the ground states, and t he tri-sup ersymmetry is sp ontaneously partially bro k en. As w e sho w ed, this turns out to b e a general feature of the tri-sup ersymmetric systems constucted by extension of a ﬁnite-gap p erio dic syste m b y means of a regular Crum-Darb oux tra nsfor ma t io n. W e could ask, motiv at ed b y [25], for the indications on the self-isosp ectralit y in the tri-sup ersymmetric extensions of the ﬁnite-ga p systems. In our curren t setting, the self- isosp ectralit y ar ises if the translation in the half-p erio d L prov ok es in v ersion of the W ron- 16 skian, W ± ( x + L ) = C ± 1 W ± ( x ) , (3.36) where C ± are some nonzero constan ts. Indeed, suc h a displacemen t pro duces changes in the sign for co eﬃcien t functions c ± 1 ( x ) = − (ln W ± ) ′ of t he op erators Q ± , a nd t herefore transforms the latters into their conjug ates, c ± 1 ( x + L ) = − c ± 1 ( x ) , Q ± ( x + L ) = Q † ± ( x ) . (3.37) Making the tra nslation for L in the intert wining relations a nd comparing the result with their conjugat es, w e rev eal that ˜ H ( x ) = H ( x + L ) , (3.38) and therefore the self-isosp ectrality do es app ear. The construction considered in this section provides a receipt ho w to get isosp ectral tri- sup er-symmetric partners for a given n - g ap Hamiltonia n. W e hav e seen how the partner Hamiltonian ˜ H is determined uniquely once we make a separation of the singlet states in to t w o disjoint families. There exist P n k =0 ( n k ) = 2 n distinct separations whic h resp ect the rules explained in the subse ction on the D arb oux-Crum transformatio ns. Since one of them is trivial (includes all the singlet states and corresp onding in tegral Z comm utes with the Hamiltonian H ) w e end up with 2 n − 1 tri-sup ersymmetric isosp ectral extensions of the giv en n -gap system. All the isosp ectral extensions can b e obtained b y successiv e ﬁrst order Darb oux a nd second-order Crum transformations. If antiperio dic singlet states are presen t in the sp ectrum, among the p ossible separations of the singlet states there exists an exceptional o ne, give n by sorting out the singlets into m utually ortho g onal families of p erio dic a nd an tip erio dic states. Despite the lack of the pro of, w e conjecture that this “natur a l” separation leads t o the s elf-isosp e c tr a l sup e rs ymm etry c ha racterized b y the partner Ha miltonian ˜ H to b e the original one but displaced in the half of the p erio d. Supp ose an n -g ap system H with n > 1 has antiperio dic singlet states in its sp ectrum, and ˜ H is a shifted for the half-p erio d Hamiltonian obtainable b y the Darb o ux-Crum trans- formation asso ciated with the sp eciﬁed natural separation of the singlets. Let Q ± b e the generators of the Crum-Darb oux tr ansformation asso ciated with a separation of singlets dif- feren t from t he natura l one. Then shifted for the half-p erio d o p erators ˜ Q ± will generate a corresp onding Crum-Darb oux transformation for the system ˜ H . In suc h a w a y w e obtain a new pair of self-isosp ectral systems H Q and H ˜ Q : ˜ H H H ˜ Q H Q Q ± ˜ Q ± x → x + L x → x + L ❄ ❄ ✲ ✲ Including H and ˜ H , we can get 2 n − 1 distinct pa ir s of self-isospectral Ha miltonians. Starting with an y n - gap system system that has nonzero num b er of an tip erio dic singlet states, w e w o uld b e able to construct 2 n − 1 extended self-isosp ectral Hamitonians H . 17 In the next section, our self-isosp ectralit y conjecture will b e supp orted b y the study of the tri- sup ersymmetry of the asso ciated Lam ´ e equation. 4 Asso ciated Lam ´ e equatio n and i t s isosp ectral exten- siones W e apply here a g eneral theory dev elop ed in the previous sections to a broad class of ﬁnite-ga p systems describ ed b y the asso ciated Lam ´ e equation. In particular w e study the isosp ectral extension ba sed on the natural se paration of the singlet states in to p erio dic and an ti-p erio dic ones, a nd sho w that it leads to the self-isospectral t r i-sup ersymmetric systems . W e pro vide an explicit form of b oth the diagonal and non-diagona l in tegra ls of motion of the extended system. The examples of isosp ectral extens ions not p ossessing a prop ert y of self-isospectrality are presen ted as we ll. Asso ciated Lam ´ e equation is a t wo-parametric sec ond order diﬀeren tial equation of F uch- sian type with four singularities a nd doubly-p erio dic co eﬃcien ts, − ψ ′′ −  C m dn 2 x + C l k ′ 2 dn 2 x + E  ψ = 0 , (4.1) where C m = m ( m + 1) , C l = l ( l + 1) are r eal n um b ers and dn x ≡ dn ( x, k ) is Jacobi elliptic f unction with mo dular parameter k ∈ (0 , 1); k ′ ∈ (0 , 1) is a complemen tary mo dular parameter, k ′ 2 = 1 − k 2 . Lam ´ e equation ( l = 0 case), obta ined orig inally b y separation of the Laplace equation in elliptical co ordinates, has b een a sub ject of extensiv e studies with use of b oth a na lytical [5 4, 55 ] and a lgebraical [8] metho ds. Due to app ealing prop erties of its solutions, t he equation (4.1) found the applications in divers e areas of phy sics. In solid-state phys ics [7], it represen ts a stationary Schr¨ odinger equation of a mo del of one- dimensional crystal with a more realistic p oten tial than Kronig-Pe nney o r Scarf p oten t ials. This equation, esp ecially its l = 0 case, pla ys an imp ortan t role in man y other ﬁelds of ph ysics as w ell. F or instance, it app eared in some expansions of scattering amplitudes [56], in the study of bifurcatio ns in c haotic hamiltonian system s [17], it go verns distance red-shift for partially ﬁlled-b eam optics in pressure-free FLR W cosmology [57 ], it was used in the study of static S U (2) BPS monop oles [9] and kink solutions [58] in the ﬁeld theory . 4.1 Construction of self-isosp ectral extension The sp ectrum of the one-dimensional p erio dic system gov erned b y the Hamiltonian op erato r corresp onding to (4.1) H − m,l = − D 2 − C m dn 2 x − C l k ′ 2 dn 2 x (4.2) consists of the v alence bands and the prohibited zones ( g aps) whic h alternate m utually un t il energy reac hes a semi-inﬁnite band of conductance. Conﬁguration of the sp ectral bands dep ends sensitiv ely on the constan t parameters. As long as m and l a cquire in teger v alues, whic h w e supp ose to b e the case from no w on, the sp ectral bands are arrang ed suc h that only ﬁnite num b er of ga ps app ear. The p erio d 2 L of the p oten tial with C m 6 = C l in ( 4 .2) is equal to 2 K , where K = R π / 2 0 (1 − k 2 sin 2 φ ) − 1 2 dφ is the complete elliptic in tegral of the 18 ﬁrst kind. The case C m = C l corresp onds to t he L a m ´ e system with the same v alue of C m but C l = 0 and the p erio d 2 L = K ; it is discussed separately in the App endix C. The indep enden t c hange of parameters m → − m − 1, l → − l − 1 leav es the Hamiltonian (4.2) in v arian t so that we can consider the case m > l ≥ 0 without the loss of generalit y . In this case the sys tem is m -gap. T o start on the construction of the tri-sup ersymmetric extension, w e fo cus to t he sep- arations of the band- edge states. As w e announced, the separation in to the p erio dic and an ti-p erio dic singlets will b e used here. Construction of the op erators Q + and Q − asso ciated with an y separatio n w ould require an explicit kno wledge of the band-edge states, i.e. an explicit solution of the stationary Sc hr¨ odinger equation. F ortunately , in the case o f natural separation this rather comp elling w ork can b e passed with the use of p eculiar prop erties of the mo del. The presen t one-dimensional system is closely related to the ﬁnite-dimensional represen- tations of Lie algebra sl (2 , R ). The Hamiltonian (4.2) can b e written as a second order p olynomial in g enerato r s of a ﬁnite-dimensional ir r educible represen tation of sl (2 , R ). This imp ortant feature underlies quasi-exact solv ability of the mo del, implying that a ﬁnite num- b er of eigenstates corr esponding to band edges can b e found b y purely algebraic means [59, 60]. F or integer v alues of m a nd l , m > l ≥ 0, the space of 2 m + 1 singlet states of the asso ciated La m´ e system can b e treated as a direct sum of t w o irreducible non- unitary represen tations of sl (2 , R ) algebra of dimensions m − l (spin j − = 1 2 ( m − l − 1)) and m + l + 1 (spin j + = 1 2 ( m + l )) [61, 62]. This fact is deeply related to the structure of the band-edge wa v e functions of the system, m + l + 1 of whic h can b e factorized fo r ma lly as Ψ µ = µ F µ ( ξ ) , (4.3) whereas the remaining m − l singlets acquire the fo rm Ψ ν = ν F ν ( ξ ) . (4.4) Here we intro duced the functions µ = cn m + l x dn l x , ν = cn m − l − 1 x dn l +1 x and a new v aria ble ξ ( x ) = sn x cn x , that v aries smo othly fro m −∞ to + ∞ in the p erio d in terv al ( − K , K ). The functions F µ ( ξ ) and F ν ( ξ ) a r e, in general, p olynomials of order m + l and m − l − 1 in ξ and, as w e will see, lie in ve ctor spaces of the irreducible represen tations of sl (2 , R ) of the dimensions m + l + 1 and m − l . As ξ = ξ ( x ) is p erio dic, the factors µ a nd ν dictate p erio dicit y or a ntiperio dicit y of the eigenfunctions; fo r eve n m + l the wa v efunctions (4.3) are p erio dic while the functions (4.4) are anti-perio dic. F or m > l ≥ 0, function µ has one no de in t he in terv al ( − K , K ), while ν can hav e a t most one no de there. Being the p olynomial in ξ , the function F µ ( F ν ) can acquire at most m + l ( m − l − 1) zeros in this interv al. Com bine these facts with the general prop erties of p erio dic and anti-perio dic states of Hill’s equation discussed in Section 2. The resulting picture show s that starting with p erio dic ground state and anti-perio dic states at the edges o f the ﬁr st gap, the gaps with p erio dic and an ti- p erio dic stat es at their edges alter with energy increasing un til the ( m − l ) t h ga p is reached. F or higher gaps, all the remaining singlets at the edges are of the same nature as edge states of the ( m − l )th gap, p erio dic or an ti-p erio dic with ev en or o dd n um b er of no des, see a lso ref. [34]. 19 T o reveal the algebraic form of ( 4 .2), w e shall reco v er ho w the Hamiltonian acts on the “dynamical” part F µ ( F ν ) of the w av efunctions. T ransforming out the function µ and writing the result in the v ariable ξ , w e obta in h µ = ( µ ) − 1 H m,l µ = −  k ′ 2 ( T + ) 2 + (1 + k ′ 2 )( T 0 ) 2 + ( T − ) 2 + k 2 ( l − m ) T 0  = −  1 + k ′ 2 ξ 2   1 + ξ 2  d 2 dξ 2 + ξ (2 k ′ 2 ( m + l − 1) ξ 2 + 2( m + l k ′ 2 − 1) + k 2 ) d dξ − k ′ 2 ( m + l )( m + l − 1) ξ 2 + const. (4.5) Here the op erators T + = ξ 2 ∂ ξ − ( m + l ) ξ , T 0 = ξ ∂ ξ − m + l 2 , T − = ∂ ξ , (4.6)  T + , T −  = − 2 T 0 ,  T 0 , T ±  = ± T ± , (4.7) are the generators of irreducible represen tation of sl (2 , R ) acting on the v ector space spanned b y monoms { 1 , ξ , .., ξ m + l } . Represen tation (4.6) is sp eciﬁed b y the eigenv alue j + ( j + + 1) of the Casimir C = − ( T 0 ) 2 + 1 2 ( T + T − + T − T + ) corresp onding to sl (2 , R ) spin j + = 1 2 ( m + l ). The ot her algebraic form of (4.2), h ν , that acts o n the “dynamical” part F ν of the w av e functions ( 4.4), can b e obt a ined b y p erforming the gauge transformation with the other common factor ν . Alternative ly , we can use the apparent symmetry µ | l →− l − 1 = ν and write do wn h ν immediately just b y substituting l → − l − 1 in (4.5) and (4.7), h ν = h µ | l →− l − 1 = ( ν ) − 1 H − m,l ν = −  k ′ 2 ( ˜ T + ) 2 + (1 + k ′ 2 )( ˜ T 0 ) 2 + ( ˜ T − ) 2 + k 2 ( − l − 1 − m ) ˜ T 0  . (4.8) W e denoted by ˜ T ρ = T ρ | l →− l − 1 , ρ = 0 , + , − , the sl (2 , R ) generators of ( m − l )- dimensional represen tation, where T ρ are the generators (4.6) of spin- j + represen tation. Note that the “eﬀectiv e” Hamiltonian h µ is Hermitian with resp ect to a scalar pro duct deﬁned with a non trivial w eigh t, ( f , g ) = R ∞ −∞ f ∗ ( ξ ) g ( ξ )(1 + k ′ 2 ξ 2 ) − l + 1 2 (1 + ξ 2 ) − m + 1 2 dξ ; the same is true fo r h ν with the c hange l → − l − 1. No w the background of the natural separation o f the singlet states is clear. The p erio dic and an tip erio dic singlet states carry tw o diﬀerent irreducible represen tations of sl (2 , R ) of dimensions m + l + 1 a nd m − l . The n umber of the p erio dic and an ti-p erio dic singlet states dep ends on the v alues of m and l , while their tot a l n um b er 2 m + 1 is ﬁxed b y the n um b er of ga ps m . F or instance, fo r m = 3 , l = 0 there are m − l = 3 p erio dic band-edge states while for m = 3 , l = 1 w e hav e m + l + 1 = 5 p erio dic singlet states. T o a v oid p ossible confusions during the construction of the sup erc harges, let us change the notation sligh tly . W e will denote b y X − m,l an op erator whic h a nnihilates all the functions (4.4), and the op erato r annihilating all the states (4.3) will b e Y − m,l . First, let us consider eigens tates (4.3) co ve red by the m + l + 1 dimensional repres en tation of sl (2 , R ). An op erator of the order m + l + 1 which annihilat es the represen tation space spanned by the monoms { 1 , ξ , . . . , ξ m + l } has the follow ing general for m y − m,l = α m,l ∂ m + l +1 ξ . (4.9) The function α m,l is ﬁxed uniquely as w e require t he co eﬃcien t a t D m + l +1 of the op erator Y − m,l = µ y − m,l 1 µ | ξ = ξ ( x ) to b e equal t o one. It reads explicitly α m,l =  dn x cn 2 x  m + l +1 . W e presen t 20 b elo w tw o equiv alen t forms of the op erator Y − m,l . The second, factorized expression, will b e particularly helpful in study of the limit case k → 1. An explication ho w to get it from (4.9) can b e found in [63], Y − m,l = D m + l +1 + m + l +1 X j = 1 c Y j D m + l +1 − j = dn m +1 x cn m + l +2 x  cn 2 x dn x D  m + l +1 dn l x cn m + l x = ( m + l ) / 2 Y j = − ( m + l ) / 2  D +  k 2 ( m − l )cn 2 x 2 − j ( k ′ 2 + dn 2 x )  sn x cn x dn x  . (4.10) The upp er index of the ordered pro duct corresponds t o the ﬁrst term on the left side while the lo w er index denotes the last term on the right side of the pro duct. W e can construct the op erator X − m,l in the same w ay or just b y making the substitution l → − l − 1 in (4 .10) whic h inte rc hanges considered algebraic sc hemes. Explicitly , we get X − m,l = D m − l + m − l X j = 1 c X j D m − l − j = dn m +1 x cn m − l +1 x  cn 2 x dn x D  m − l dn − l − 1 x cn m − l − 1 x = ( m − l − 1) / 2 Y j = − ( m − l − 1) / 2  D +  k 2 ( m + l + 1)cn 2 x 2 − j ( k ′ 2 + dn 2 x )  sn x cn x dn x  . (4.11) As w e explained in the preceding section, the co eﬃcien ts of the second highest deriv ativ e of X − m,l and Y − m,l coincide and en ter the explicit construction of the sup erpartner Hamiltonian, c 1 ≡ c X 1 = c Y 1 = − W ′ m,l W m,l = k 2 ( C m − C l ) 2 sn x cn x dn x = 1 2 ( m − l )( m + l + 1) k 2 cn x sn x dn x . (4.12) The equalit y c X 1 = c Y 1 reﬂects the coincidence of the W ronskians of the k ernels of X − m,l and Y − m,l up to inessen tial n umerical fa ctor related to the ar bit r a riness in normalization of their zero mo des. The essen tial part of these W ronskians is giv en by the no deless function W m,l ( x ) = (dn x ) 1 2 ( m − l )( m + l +1) , ( 4 .13) whose in v ariance with resp ect to the c ha ng e l → − l − 1 just reﬂects the indicated equalit y of the co eﬃcien ts. The Jacobi function prop ert y dn ( x + K ) = k ′ / dn x sho ws that the W ronskian W m,l ( x ) sat- isﬁes the relation (3.36), and supports our conjecture that the natura l separation of the singlet states in to perio dic and anti-perio dic states results in the self-isosp ectral tri-sup ersymmetric system with the partner Ha miltonian op erator ˜ H ≡ H + m,l to b e the or ig inal Hamiltonia n translated fo r the half- p erio d, H + m,l ( x ) = H − m,l ( x + K ) . Its explicit form is H + m,l = H − m,l + 2 c ′ 1 = − D 2 − C l dn 2 x − C m k ′ 2 dn 2 x . (4.14) Recalling (3.17), the generato r o f the hidden b osonized sup ersymmetry of the asso ciated Lam ´ e system (4 .2) acquires the following factorized form 21 Z − m,l = dn − l x cn m − l +1 x  cn 2 x dn x d dx  m − l  dn x cn x  2 m +1  cn 2 x dn x d dx  m + l +1 dn l x cn m + l x (4.15) = dn − l x sn m − l +1 x  sn 2 x dn x d dx  m − l  dn x sn x  2 m +1  sn 2 x dn x d dx  m + l +1 dn l x sn m + l x , (4.16) where we used alternativ e expressions X − m,l ( x ) = dn m +1 x sn m − l +1 x  sn 2 x dn x D  m − l dn − l − 1 x sn m − l − 1 x and Y − m,l ( x ) = dn m +1 x sn m + l +2 x  sn 2 x dn x D  m + l +1 dn l x sn m + l x , obtained with the use of a sp eciﬁc iden tity 1 sn j + 1 x  sn 2 x dn x D  j 1 sn j − 1 x = 1 cn j + 1 x  cn 2 x dn x D  j 1 cn j − 1 x . (4.17) With making use of the same iden tity w e can prov e that Y − m,l ( x + K ) = ( − 1) m + l +1 ( Y − m,l ( x )) † and X − m,l ( x + K ) = ( − 1) m − l ( X − m,l ( x )) † . Fina lly , w e write do wn t he obtained extend ed Hamil- tonian H as w ell as the Hermitian diag onal and an tidiagonal integrals H m,l =  H + m,l 0 0 H − m,l  =  H − m,l ( x + K ) 0 0 H − m,l ( x )  , ( 4 .18) Z m,l = i 2 m +1  Z + m,l 0 0 Z − m,l  = i 2 m +1  Z − m,l ( x + K ) 0 0 Z − m,l ( x )  , (4.19) X m,l = i m − l  0 X − m,l X + m,l 0  = i m − l  0 X − m,l ( x ) X − m,l ( x + K ) 0  , (4.20) Y m,l = i m + l +1  0 Y − m,l Y + m,l 0  = i m + l +1  0 Y − m,l ( x ) Y − m,l ( x + K ) 0  , (4.21) whic h represen t an explicit realization of (3.1 9 ). In accordance with the analysis of Section 3, the in tegrals Q ± are identiﬁed with X m,l and Y m,l in the following wa y: Q − = X m,l and Q + = Y m,l when m − l is o dd, and their roles are in terchanged when m − l is ev en. 4.2 Self-isosp ectral pairs and “sup erp oten tial” Let us indicate on an in teresting represen t a tion of the self-isosp ectral pairs of the Hamiltoni- ans that generalizes a represen tation H ± = − D 2 + W 2 ± W ′ of the sup er-partner Hamilto nians in terms of the sup erp o ten tia l in the case of the usual ( linear) N = 2 supersymmetry . The self-isosp ectral pair ( 4 .2), (4.14) can b e presen ted in t he equiv alen t form H ± m,l = − D 2 + 2 C m + C l ( C m − C l ) 2 (ln W m,l ) ′ 2 ± (ln W m,l ) ′′ + (1 + k ′ 2 ) 1 2 ( C m + C l ) , (4.22) where W m,l is the W ronskian (4.13) corresp onding to the kerne ls of op erators X − m.l and Y − m,l . Let us denote C + = q 1 2 ( C m + C l ), C − = 1 2 ( m − l )( m + l + 1) = 1 2 ( C m − C l ), and deﬁne a function W = −  ln dn C + x  ′ . (4.23) 22 Then (4.2 2) can b e rewritten equiv alently as H ± m,l − (1 + k ′ 2 ) C 2 + = − D 2 + W 2 ± C − C + W ′ . (4.24) Eq. (4.24) is reminiscen t o f supersymmetric quan tum mec hanics r epresen tation (3.7). This is not just a coincidence. In the case m − l = 1, t he system H − m,m − 1 is c haracterized b y the presence of only one p erio dic singlet Ψ 0 = dn m x , whic h is the ground state with energy corresp onding to a subtracted constan t term on the left hand side of Eq. (4.24). In this case the ﬁrst order sup ercharge X m,m − 1 reduces to one of the ﬁrst order sup erc harges (3 .8) of N = 2 sup ersymmetry , (4.23) tak es a form of a usual represen tation of a sup erp oten tial in terms of the g round state, and (4 .2 4) reduces to (3.7). There exists a simple g eneralization of a classical mo del for sup ersymmetric quan tum mec hanics to the case of nonlinear sup ersymmetry of order n > 1 [32]. It consists in the c ha nge of the b oson-fermion coupling term θ + θ − W ′ in classical Hamiltonia n for nθ + θ − W ′ , where θ + θ − is a classical analog for σ 3 and θ ± are G rassmann v ariables describing fermion degrees of freedom. Ho w ev er, unlik e the linear case n = 1, for n > 1 suc h a generalization suﬀers a pro blem o f the quan tum anomaly , whic h can b e solv ed in a general form only for n = 2 [3 3]. One can sho w tha t for m − l = 2, when the sup erc harge X m,m − 2 is the diﬀeren tial op erator of the second order, represen tation (4.24) is in corresp ondence with the solution of the quantum anomaly problem for n = 2. Let us stress that f or m − l > 1 the argumen t of logarithm in the deﬁnition of the sup erp oten tial-lik e function (4.23) do es not corresp ond to a ground state o f t he system 8 . It is j ust the appropr ia tely rescaled loga rithmic deriv ativ e of the W ro nskian (4 .13), W = − C + C − (ln W m,l ) ′ . 4.3 Some examples Due to the general result presen ted in Section 3, there exist 2 m − 1 isosp ectral extensions of an m -g ap asso ciated Lam´ e Hamiltonian. The new systems can b e obta ined fr o m the initial system b y sequen t use of Darb oux tra nsformation and the tra nsformation of Crum of the second order. Moreo ve r, the complete set o f 2 m isosp ectral systems can b e sorted out into 2 m − 1 self-isosp ectral tri-sup ersymmetric pairs. W e presen t here an example of Lam´ e system to illustrate this picture explicitly . First, w e explain brieﬂy some subtleties related to the sequen t use of the transformation of Crum. Let us consider a tw o-g a p asso ciated Lam´ e system represen ted b y Hamiltonian H . There are three admissible separations of the singlet states: sorting out t he states at the edges of the ﬁrst g a p, or of the second gap, or of b ot h of them. Let us denote the sup erc harg es asso ciated with these separations as Q ± ,j , j = 1 , 2 , 3. Then w e can construct three isosp ectral sup er-partner Hamiltonians H ( j ) satisfying H ( j ) Q ± ,j = Q ± ,j H . (4.25) W e can rep eat the pro cedure with a n yone of the new systems, obtaining another isosp ectral Hamiltonian H ( k, j ) ( j refers to the system H ( j ) , k denotes the next ch oice of separation). 8 F or the explicit form of the gro und states of the a sso ciated Lam´ e system with some v alues of m and l see [62]. 23 Although we can make this pro cedure rep eatedly , it will generate only limited num b er of new systems. This is due to simple rules following the sequen t use of transformatio ns of Crum. The ﬁrst rule: when w e c ho ose sequen tly t w o iden tical separations, w e return to the initial system. In our tw o-gap setting, let us start with t he ﬁrst separation and construct the new Hamiltonian H 1 with the help of op erator Q + , 1 . The op erator which annihilates the states at the edges o f the ﬁr st gap of H 1 coincides with Q † + , 1 . So, rep eating the pro cedure with H 1 with the same separation, w e obtain a new Hamiltonian H (1 , 1) whic h is related with initial H by the in tertw ining relation H (1 , 1) Q † ± , 1 Q ± , 1 = Q † ± , 1 Q ± , 1 H . But the o p erator Q † ± , 1 Q ± , 1 is a p olynomial in H , and w e get H (1 , 1) = H . In g eneral, there holds H ( j,j ) = H . The other rule tells that the c hoice of sequen t separations is “comm utativ e”. Con- struct t he Hamiltonian H 1 using the sup erc harge Q + , 1 . Then, c ho osing the second sep- aration, w e construct H (2 , 1) with help of op erator ˜ Q + , 2 whic h annihilates the states at the edges o f the second gap of the system H 1 . Hamiltonian H (2 , 1) satisﬁes a r elat io n H (2 , 1) ˜ Q ± , 2 Q ± , 1 = ˜ Q ± , 2 Q ± , 1 H . The op erato r Q + , 3 = ˜ Q ± , 2 Q ± , 1 is of the fo urth order and can b e factorized in diﬀerent w ay s. F or instance , its alternativ e factorization is Q + , 3 = ˜ Q ± , 1 Q ± , 2 , that corresp onds to the interc hanged c hoices of the separations. Sp eaking in general, there holds H ( k, j ) = H ( j,k ) . Let us denote shortly ( j ) (or ( j, k )) a transforma t ion of Crum of the second (or of the fourth) order whic h annihilates singlet stat es a t the edges of the j -t h gap (or of the b oth j -th a nd k -th ga ps). In this no t a tion, D arb oux transformat io n asso ciated with the ground state is represen ted b y (0). Coheren tly , w e denote H ( j ) or H ( j,k ) the Hamiltonians obtained b y these transformations. Then the rules for sequen t use of Da rb oux-Crum transformations can b e depicted by the following sc hemes H H ( j ) H ( j,j ) = H ( j ) ( j ) I d. ✻ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✸ ✲ H ( j ) H H ( j,k ) H ( k ) ( k ) ( k ) ( j ) ( j,k )=( k, j ) ( j ) ❄ ❄ ✲ ✲ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ s where I d. represen ts an identit y op erato r. In the t w o-ga p case, we can ﬁnd three new isosp ectral Ha milto nians in this w a y . Let us presen t isosp ectral transformations of the ( m = 3 , l = 0) La m ´ e Hamiltonian with their relation to the orig ina l syste m. Sev en new isospectral Hamiltonains are fo und a nd four self- isosp ectral pairs can b e formed. In the following sche me any of the new systems can b e reac hed by sequen t application of Da r b oux (0) or second-order Crum’s transformation ( k ) on the initial Hamiltonian H . The v ertical lines corresp ond to tra nsformation represen ting the natural separation so tha t the Hamiltonians related by these lines f o rm self-isosp ectral pairs. 24 three-gap H (1 , 3) H H (0) H (2) H (2 , 3) H (1 , 2) H (3) H (1) H H (1 , 3) (2) (1) (2) (3) (2) (1) (2) (3) (1 , 3) (1 , 3) (1 , 3) (1 , 3) (1 , 3) ❄ ❄ ❄ ❄ ❄ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ As it can b e observ ed in Fig. 2, the nature of the p ot entials of the obtained systems is distinct from t he original ones. How ev er, the sp ectrum of the corresp onding Hamiltonians is completely iden tical. The p ot entials can b e tuned with mo dular parameter k that broadens the applicabilit y of these syste ms. The inﬁnite p erio d limit of the self-isospectral extension of the asso ciat ed Lam´ e systems is discussed in detail in the f o rthcoming section. 5 Sup erextend ed P¨ osc hl-T elle r sys tem and inﬁ nite p e- rio d limit of tri-s up e rsymmetry In this section w e analyze in detail the inﬁnite p erio d limit of the tri-sup ersymmetric self- isosp ectral extension (4.18)–(4.21) of the asso ciated Lam ´ e system. W e explain ho w t he struc- ture of tri-sup ersymmetry is mo diﬁed and study its implications, in particular the restoration of the un broken tr i- sup ersymmetry . Inﬁnite p erio d limit corresp onds to k → 1 ( k ′ → 0). In this limit the asso ciated Lam´ e Hamiltonian c hanges to the energy op erator of the P¨ osc hl-T eller system. The band structure transforms in the follow ing wa y . The states of the conduction band are transfor med in to the states o f the scattering sector of the sp ectrum, a nd the singlet edge-state of the conduction band is tr ansformed in to the ﬁrst (low est) singlet state o f the contin uous sp ectrum. The v alence bands shrink, t w o band-edge states corresponding to the same p ermitted band con- v erge smo othly in a unique w av e function. This wa v efunction can b e non-phys ical in one of the limit systems . 5.1 Self-isosp ectral sup ersymmetry in the inﬁnite-p erio d limit When k tends to one, the Jacobi elliptic functions cease t o b e doubly p erio dic as their real p erio d extends to inﬁnity while the complex p erio d ta kes a ﬁnite v alue 2 iK ′ = 2 iπ ; they transfor m into the hyperb olic functions, dn x → sec h x, cn x → sec h x, sn x → tanh x. In this limit, the sup erextended setting describ ed by the tw o m utually shifted p erio dic Hamiltonians (4.1 8) acquires the following form H m,l =  H + m,l 0 0 H − m,l  − → k =1 H P T m,l =  ˆ H + l 0 0 ˆ H − m  , (5.1) where the r esulting op erators r epresen t t w o systems with the P¨ osc hl-T eller p oten tial of dif- feren t in teraction strengths sp eciﬁed by the integers m and l , ˆ H − m = − d 2 dx 2 − C m sec h 2 x, ˆ H + l = − d 2 dx 2 − C l sec h 2 x. (5.2 ) 25 The system k eeps the parit y-in v ariance, [ R, H P T m,l ] = 0. As we deal with in t eger v alues of m and l , Hamiltonians (5.2 ) are reﬂectionless, i.e the transmission co eﬃcien ts are equal to o ne. The Hamiltonian ˆ H − m ( ˆ H + l ) p ossesses m + 1 ( l + 1) singlet states, m ( l ) of them are b ound stat es, the remaining one corresp onds to the low est stat e in the scattering sector. Hamiltonians (5.2) are almost-isosp ectral. Their sp ectra coincide in the contin uous part E ∈ [0 , ∞ ) and just in l + 1 singlet states. The Hamiltonian ˆ H − m has additional m − l lo w er energy lev els. This indicates what happ ens with the sp ectral structure o f the exten ded Hamiltonian H m,l when w e stretc h the real p erio d in to the inﬁnit y . The 2 m + 1 band-edge states of H − m,l transform into m + 1 ph ysical states of ˆ H − m , while in the case of the sup er- partner system H + m,l only 2 l + 1 band-edge states of highest excitations con v erg e to t he ph ysical wa v e f unctions, the rest is ph ysically unacceptable. Th us the doublets of band-edge states and quadruplets of the quasi-p erio dic states c hange into m − l singlets and l + 1 doublet states, a nd into the quadruplets of the scattering states (see Fig. 3). The presence o f the singlet states gives a taste of a diﬀeren t na ture of the tri-sup ersymmetry whic h w e discuss in what follo ws. The Hamilto nia ns (5.2) admit the follo wing represen tation ˆ H − m = −D − m D m − m 2 , ˆ H + l = −D − l D l − l 2 , (5.3) where the deﬁnition and the basic prop erties o f the o p erator D n are D n = D + n tanh x, D † n = −D − n , D − n D n = D n +1 D − n − 1 + (2 n + 1 ) . (5.4) W e shall fo cus to the prop erties of tri- sup ersymmetry and of the sup erc harg es (4.1 9)–(4.21) in particular. T aking the limit k = 1, the non-diago nal integrals (4.20), (4.21) transform as follo ws X m,l = i m − l  0 X − m,l X + m,l 0  − → k =1 X P T m,l = i m − l 0 ˆ X − m,l ˆ X + m,l 0 ! , (5.5) Y m,l = i m + l +1  0 Y − m,l Y + m,l 0  − → k =1 Y P T m,l = i m + l +1 0 ˆ Y − m,l ˆ Y + m,l 0 ! , (5.6) where the non-diagonal compo nents are ˆ Y − m,l = D − l D − l +1 . . . D m − 1 D m , ˆ Y + m,l = D − m D − m +1 . . . D l − 1 D l , (5.7) ˆ X − m,l = D l +1 D l +2 . . . D m − 1 D m , ˆ X + m,l = D − m D − m +1 . . . D − l − 2 D − l − 1 , (5.8) and ˆ X + m,l = ( − 1) m − l ( ˆ X − m,l ) † , ˆ Y + m,l = ( − 1) m + l +1 ( ˆ Y − m,l ) † . The limit do es no t violate comm uta tion relations (tri- sup ersymmetry is main tained) and we ha ve [ H P T m,l , X P T m,l ] = [ H P T m,l , Y P T m,l ] = [ X P T m,l , Y P T m,l ] = 0 . (5.9) The comp o nen ts of t he squares of the non-diag onal sup erc harges, ( X P T m,l ) 2 = ˆ X − m,l ˆ X + m,l 0 0 ˆ X + m,l ˆ X − m,l ! , ( Y P T m,l ) 2 = ˆ Y − m,l ˆ Y + m,l 0 0 ˆ Y + m,l ˆ Y − m,l ! , (5.10) 26 corresp ond to the in tegrals of motion o f individual P¨ osc hl-T eller subsystems . The results of the previous section suggest that these will b e prop ortional to certain sp ectral t yp e p olyno- mials in Hamiltonian. This is the case indeed. How ev er, the situation c hanges signiﬁcan tly comparing with the p erio dic system. With the sequen t use of the iden tity in (5.4), w e can deriv e ( X P T m,l ) 2 = m − l − 1 Y j = 0 ( H P T m,l − E m,j ) = P P T X ( H P T m,l ) . (5.11) Here E m,j = − ( m − j ) 2 , j = 0 , . . . , m − l − 1 , corresp ond to the m − l singlet states of the sup erextended system ( m − l b ound-states corresp ond to the low est energies of ˆ H − m or, equiv alen tly to the m − l nonph ysical states of ˆ H + l ). The square of the second non-diagonal sup erc harg e can b e factorized with use of P P T X ( Y P T m,l ) 2 = ( H P T m,l − E m,m ) m − 1 Y j = m − l ( H P T m,l − E m,j ) 2 P P T X ( H P T m,l ) , (5.12) Apart from the r o ots shared with P P T X , there also app ear t he l + 1 doubly-degenerate energies of the sup erextended system (5 .1). The l b ound-states energies E m,j = − ( m − j ) 2 , j = m − l , . . . , m − 1 , are the double ro ots, and the energy of t he lo wes t state of the con tin uous sp ectrum, E m,m = 0, is a simple ro ot. Considering the limit case of the dia g onal sup erch arge (4.19), w e encoun ter an interesting situation. In [6 4] it w as observ ed tha t for a r eﬂe c tionless P¨ osc hl-T eller (PT) system, there exists a hidden b osonized supersymmetry . If the sys tem has n b ound states (and henc e n + 1 singlet states), there is a parity-o dd in tegral of motion of order 2 n + 1, A 2 n +1 = D − n D − n +1 . . . D 0 . . . D n − 1 D n , (5.13) whic h annihilates all the singlet states and some non-physic al states, whose origin w as clar- iﬁed in [65] from the p oin t of view of t he Lam ´ e equation a nd its hidden b osonized sup er- symmetry 9 . The subsystems ˆ H − m and ˆ H + l ha v e o dd-integrals of mot io n A 2 m +1 and A 2 l +1 of orders 2 m + 1 and 2 l + 1 , resp ectiv ely . On the o ther hand, w e know that the diago na l comp onen ts of in tegral Z P T are the parit y-o dd integrals of order 2 m + 1 fo r each subsys- tem. P articularly , ˆ H + l w o uld hav e tw o parity-o dd symmetries o f diﬀerent o rders. Let us explain the picture and sho w ho w the in tegrals A 2 m +1 an A 2 l +1 manifest their presence in the tri- sup ersymmetric sc heme. In the limit Z m,l − → k =1 Z P T m,l , w e can tra ce the presence of t he integrals A 2 m +1 and A 2 l +1 in the diagonal comp onents Z − m,l − → k =1 ˆ Z − m,l = ˆ X + m,l ˆ Y − m,l = A 2 m +1 = D − m D − m +1 . . . D 0 . . . D m − 1 D m , (5.14) Z + m,l − → k =1 ˆ Z + m,l = ˆ X − m,l ˆ Y + m,l = ˆ X − m,l ˆ X + m,l A 2 l +1 . ( 5 .15) 9 Earlier , higher o r der diﬀerential op erator s o f this nature were discussed in the context of super symmetric quantum mechanics in [48, 66, 67]. How ever, their sense and the intimate relation with the algebro-g e o metric po ten tials were not understo o d, see also foo tno te 7 . 27 Eac h of these op erators (it is w orth to not e a gain that they hav e the same order) is the in tegral for the correspo nding subsystem, and tog ether they annihilate the singlet and doublet states of the super- extended system. ( Z P T m,l ) 2 pro duces a p olynomial of the fo rm ( Z P T m,l ) 2 = ( H P T m,l − E m,m ) m − 1 Y j = 0 ( H P T m,l − E m,j ) 2 , (5.16) whic h can b e related with a de gener ate sp ectral h yp erelliptic curv e of g enus m , in con trary to the m -gap system (4.1 8), whose sp ectral p olynomial (3.22) corresp onds to a non-degenerate h yp erelliptic curv e of the same gen us. It reﬂects the fact that the band structure disapp eared; ev ery t w o band- edge states of the same band transform in to a single b ound state whic h ends up in the degeneracy in the sp ectral p olynomial. Accordingly , the degeneracy do es not app ear for the low est state of the contin uous sp ectrum. The comp o nen ts of t he in tegral Y P T m,l can b e rewritten in the following w a y ˆ Y − m,l = A 2 l +1 D l +1 . . . D m − 1 D m = A 2 l +1 ˆ X − m , ˆ Y + m,l = ˆ X + l A 2 l +1 . (5.17) The relatio n (5.12) can b e expres sed also as ( Y P T m,l ) 2 = A 2 2 l +1 ( H P T m,l ) P P T X ( H P T m,l ) , (5.18) where in corr espo ndence with (5.13) A 2 l +1 = D − l D − l +1 . . . D 0 . . . D l − 1 D l . These formulas pro vide an alternativ e insigh t in t o the k ernel of the sup erc harge Y P T m,l and its comm uta- tion relation with H P T m,l . The later one can b e deriv ed indep endently just with use o f the comm uta tion relatio ns [ H P T m,l , X P T m,l ] = 0 and [ ˆ H + l , A 2 l +1 ] = 0. In par t icular, w e ha v e ˆ Y − m,l ˆ H − m = A 2 l +1 ˆ X − m,l ˆ H − m = A 2 l +1 ˆ H + l ˆ X − m,l = ˆ H + l A 2 l +1 ˆ X − m,l = ˆ H + l ˆ Y − m,l (5.19) In the inﬁnite-p erio d limit, the tigh t relation of the non-diagonal in tegrals (5.5) and (5.6) is manifested b y means o f the pa rit y-o dd in tegr a l A 2 l +1 of ˆ H + l , see Eq. ( 5 .17), whic h w as not presen ted in the p erio dic case. As a conseque nce, this integral app ears a lso in the structure of the diagonal inte gral ˆ Z + m,l , see (5.15). 5.2 Sup erc harges action and relation of nonphysical with physical solutions In the case of asso ciated Lam ´ e system, the action of the o p erators X − m,l and Y − m,l w a s quite clear. Eac h of them annihilated t w o disjoin t subsets of the whole family of 2 m + 1 singlet states o f the Hamiltonian H − m,l . In the limit case, the situation ceases to b e so transparen t. The systems described by (5.2) diﬀer in the num b er of the singlet states, the Hamiltonian ˆ H − m ( ˆ H + l ) has m + 1 ( l + 1) b ound-states. How ev er, the order of the sup erc harges is not aﬀected by the limit. There arises a natural question: what kind of functions is annihilated additionally by the non tr ivial inte grals. W e clarify here this in t ricate situation. W e in tro duced the op erator D n (see (5.4)) whic h pro v ed to b e useful in factorizatio n of b oth the Hamiltonians (5.3 ) and the sup erc harges (5.5), (5.6). W e can in terpret this op erator as a Darb oux tranfo rmation whic h satisﬁes D m ˆ H − m = ˆ H − m − 1 D m , (5.20) 28 where the new Ha milto nian ˆ H m − 1 has m − 1 b ound states. The op erator D m annihilates the gr ound state of ˆ H − m so tha t the corresp onding energy lev el is missing in the sp ectrum of ˆ H m − 1 . The Hamiltonians ˆ H m and ˆ H m − 1 related by D m are of t he same nature but with a shifted parameter. This phenomenon, mediated b y D arb oux transformatio n, is called a shap e invarian c e . W e can a pply the transformation o f Darb oux rep eatedly , annihilating the lo w est b o und state in eac h step. This pro cedure induces the following sequence of Hamiltonians ˆ H − m → ˆ H − m − 1 → ˆ H − m − 2 → .... → ˆ H − l +1 → ˆ H − l = ˆ H + l , (5.21) The resulting op erato r whic h relates ˆ H − m and ˆ H + l can b e interpre ted as a Crum-Darb o ux transformation of o r der m − l . This transformation coincides with that pro duced by the op erator ˆ X − m,l ˆ X − m,l ˆ H − m = ˆ H + l ˆ X − m,l , ˆ X − m,l = D l +1 . . . D m . (5.22) Therefore, ˆ X − m,l annihilates m − l b ound states of ˆ H − m . In t he sense of the sup erextended Hamiltonian H P T m,l , these b ound-states are single ts . The energy lev els corresp onding to the states annihilat ed b y ˆ X − m,l are absen t in t he spectrum of ˆ H + l (that mak es the systems almost- isosp ectral). Let us denote the corresp onding m − l b ound-state energies of ˆ H − m as E m . W e denote l + 1 remaining energies of singlet states of ˆ H − m as E l . These l + 1 lev els are shared b y the singlet stat es o f ˆ H + l . So, the energies of singlet states of ˆ H − m are formed b y E m and E l . The op erato r ˆ X + m,l annihilates a non-phy sical eigenstate of ˆ H + l corresp onding to the energy E m ˆ X + m,l ˜ ψ m = 0 , ˆ H + l ˜ ψ m = E m ˜ ψ m Acting with the op erator ˆ X + m,l on the l + 1 ph ys- ical states ( l b ound states a nd the low est state of the contin uous sp ectrum) ˜ ψ l of ˆ H + l , ˆ H + l ˜ ψ l = E l ˜ ψ l , w e get the b ound states o f ˆ H − m corresp onding to the energy E l ˆ X + m,l ˜ ψ l = ψ l , ˆ H − m ψ l = E l ψ l . The op erator ˆ Y + m,l annihilates the ph ysical eigenstates ˜ ψ l , ˆ Y + m,l ˜ ψ l = 0 , and also annihilates the second, non-phys ical, solution ˜ η m of the Hamiltonian ˆ H + l corresp ond- ing to t he eigen v alue E m , ˆ Y + m,l ˜ η m = 0 , ˆ H + l ˜ η m = E m ˜ η m , ˜ η m = ˜ ψ m R x ˜ ψ − 2 m dx. The function ˜ η m is mapp ed b y ˆ X + m,l to a phy sical stat e ψ m of ˆ H − m with energy E m . The same result is obtained when w e act b y ˆ Y + m,l on the state ˜ ψ m ∈ Ker ˆ X + m,l , ˆ Y + m,l ˜ ψ m = ψ m , ˆ X + m,l ˜ η m ∝ ψ m , ˆ H − m ψ m = E m ψ m . T o g et an insigh t into the kernel of the op erato r ˆ Y − m,l , it is conv enien t t o rewrite this op erator in the f a ctorized form ˆ Y − m,l = D − l D − l +1 . . . D − 1 D 0 D 1 . . . D m − 1 D m | {z } annihilates s inglets . (5.23) The indicated rig h t part of the op erator annihilates all the m + 1 singlet states of ˆ H − m . D ue to t he remaining part, there ar e a dditional a nnihilat ed l f unctions φ j , j = m − l , . . . , m − 1. These additional f unctions need not to b e solutions of the Sc hr¨ odinger equation cor r esponding to ˆ H − m . Indeed, due to represen tatio n ˆ Y + m,l ˆ Y − m,l = ( ˆ H − m − E m,m ) m − 1 Y j = m − l ( ˆ H − m − E m,j ) 2 P P T X ( ˆ H − m ) , (5.24) 29 the f unctions φ j satisfy ( ˆ H − m − E m,j ) φ j = ψ j , ( ˆ H − m − E m,j ) ψ j = 0 . Hence, the Ha milto nian ˆ H − m restricted on the k ernel of ˆ Y − m,l con tains Jordan blo ck s asso ciated with the energies E l . D ue to the similar structure of the sp ectral p olynomial of ˆ Z − m,l (5.16), the no n- ph ysical states annihilated by the sup ercharge of the hidden b osonized sup ersymmetry are of the same nature. It is instructiv e to discuss the case of l = 0 in detail. In this case ˆ H + 0 corresp onds to the free particle. The comm uta tion relations (5.9 ) tells that t he free part icle energy op erator is related with ˆ H − m via the follo wing in tertwining r elat io ns ˆ X + m, 0 ˆ H + 0 = ˆ H − m ˆ X + m, 0 , ˆ Y + m, 0 ˆ H + 0 = ˆ H − m ˆ Y + m, 0 , (5.25) ˆ X − m, 0 ˆ H − m = ˆ H + 0 ˆ X − m, 0 , ˆ Y − m, 0 ˆ H − m = ˆ H + 0 ˆ Y − m, 0 . (5.26) The ﬁrst relation of (5.26) is mediated by ˆ X − m, 0 = D 1 ... D m . Keeping in mind (5.22), the sup erc harg e ˆ X − m, 0 annihilates all t he b ound-states of ˆ H − m . In the second relation of (5.26), the Hamiltonians are in tertw ined by ˆ Y − m, 0 = D 0 D 1 ... D m . Apparently , this o p erator mak es the same job as ˆ X − m, 0 but ˆ Y − m, 0 annihilates additionally the ﬁrst scattering state of ˆ H − m . The op erator ˆ X − m, 0 transforms the low est scattering state of ˆ H − m in to a constan t f unction, the scattering state of a free particle corresp onding to the lo w est energy . This function is annihilated by ˆ Y + m, 0 while applying ˆ X + m, 0 w e get the initial ﬁrst scattering state of ˆ H − m . In general, the op era t o rs ˆ X + m, 0 and ˆ Y + m, 0 transform solutions of Sc hr¨ odinger eq uation corresp ond- ing to the free particle H + 0 in to the (for mal) eigenstates of ˆ H − m . They can b e employ ed in reconstruction of the scattering states of the Hamiltonian ˆ H − m from the plane w a v e stat es of a free particle, ψ ± κ = ˆ X + m, 0 e ± iκx = D − m D − m +1 . . . D − 2 D − 1 e ± iκx ∝ ˆ Y + m, 0 e ± iκx = D − m D − m +1 . . . D − 1 D 0 e ± iκx , ( 5 .27) where ˜ ψ ± κ = e ± iκx satisﬁes ˆ H + 0 ˜ ψ ± κ = E κ ˜ ψ ± κ , ˆ H − m ψ ± κ = E κ ψ ± κ , E k = κ 2 . (5.28) Let us summarize the obtained results. Extending the real p erio d of the self-isosp ectral extension o f asso ciated Lam´ e Hamilto nia n to inﬁnit y , the a sso ciat ed sup eralgebraic structure w a s mo diﬁed. The squares of the sup erch arges Z P T m,l and Y P T m,l turned out to b e degenerated p olynomials in H P T m,l . This mo diﬁes t he structure of the underlying sup eralgebra of the b osonic op erators G ( ± ) a in the dep endence o n a chos en grading op erator Γ ∗ . Comparing with the p erio dic case, the b osonic o p erators form the a lgebra u (1) ⊕ e (2) ⊕ e (2) for the singlet energy lev els. Recall that the p erio dic e xtende d tri- sup ersymmetric system do es not ha v e singlet states in its sp ectrum. The op erators ˆ X − m,l and ˆ Y − m,l annihilate the m − l low est b o und states of the Hamilto- nian ˆ H − m . The eigen v ectors of ˆ H + l corresp onding to these energies cease to b e ph ysically acceptable. Conseq uen tly , isosp ectralit y of the initial system is brok en. Sp eaking in terms 30 of the extended system, the sup erc harge X P T m,l annihilates the singlet states. The o p erator Y P T m annihilates b ot h doublets and singlets whic h are annihilated b y the diagonal op erat o r Z P T m as we ll. F rom this p oint of view, the sp on taneously (or, dynamically) partially brok en tri-sup ersymmetry of the p erio dic system is recov ered in the inﬁnite p erio d limit. 6 Conclud ing remarks and ou tlo ok In the particular case of asso ciated Lam´ e systems, the results of the presen t pap er should b e understo o d in a broader context of t he existing literature. Dunne and F ein b erg considered the class l = m − 1 of asso ciated Lam ´ e Hamiltonia ns (4.2) as an example of t he self-isosp ectral extension pro vided b y Darb oux transformation [25]. Khare and Sh ukhatme [27] found that this transformat ion prov ides a self-isosp ectral extension of pure Lam´ e systems just in the one-gap case while for the other setting the extension pro ved to b e of a completely diﬀeren t nature. On the other ha nd, F ern´ andez et al rev ealed self-isosp ectralit y o f t w o-gap Lam´ e Hamiltonian when the second-order transformation w as applied [30]. In the ligh t of the presen ted results, w e can understand those ﬁndings just as pieces of the mosaic, whic h w as fully unfolded b y the structure of the tri-sup ersymmetry and esp ecially b y the self-isosp ectral supersymmetry of the asso ciated Lam´ e system. In particular, the system considered b y D unne and F ein b erg is the self-isosp ectral extension H m,m − 1 of the asso ciated Lam ´ e Hamiltonian, see (4.21). Besides the ﬁrst-order sup erc harge X m,m − 1 , the list of its lo cal in tegrals of motions should b e completed b y the other non-diagonal supercharge Y m,m − 1 and diagonal integral Z m,m − 1 whic h play s the role of the central c harge of the resulting extended N = 4 nonlinear sup ersymmetry . Although b oth X m,m − 1 and Z m,m − 1 annihilate the doublet of ground states, the tri- sup ersymmetry is sp ontaneous ly partially brok en since the doublet of ground states do es not v anish under the action of the supercharge Y m,m − 1 . This suggests that the sup ersymmetry breaking should b e analyzed having in mind the complete set of non trivial lo cal in tegrals, whic h a re Z , Q ( a ) + and Q ( a ) − in the case of the studied general class of ﬁnite-gap systems. On our w ay to the presen ted results w e left untouc hed v arious app ealing questions and problems. F or instance, the self-isosp ectralit y conjecture could b e tested on the ﬁnite-ga p systems with missing an ti-p erio dic states. Since these should b e prev en ted from the self- isosp ectral extensions, the structure o f the tri-sup ersymmetry could exhibit p eculiarities in this case. Besides, the exact pro of of the conjecture should b e prov ided. The inﬁnite-p erio d limit could b e an eﬀectiv e tec hnique in pro duction of the tri- sup ersymmetric sy stems with non-p erio dic p ot en tia ls. In the limit case of the self-isospectral extension of the a sso ciated Lam ´ e Hamiltonian, the isospectrality w a s brok en follo w ed by the reco v ery of the exact t ri-sup ersymmetry . There app ears a natura l question whether this is the common feature or there exist isosp ectral extens ions of non-p erio dic systems with brok en tri-sup ersymmetry . The limit o f other isosp ectral extensions of associat ed Lam´ e sys tem could pro vide an insigh t in to the general situatio n. The relation of the tri-sup ersymmetry and the represen tations of Lie algebras might giv e an in teresting insigh t in to inv olv ed ph ysical models as we ll. Our construction of the tri-sup ersymmetric extensions was ba sed on the sp eciﬁc factor- ization of the o dd-order integral of motion. Relaxing the smo othness of the p otential, the formal construction should b e applicable o n the bro ad family of algebro- geometric p oten- 31 tials where the presence of the parit y-o dd diagona l integral Z is guarantee d. It is worth to men tio n the T reibic h-V erdier family of p oten tials [68] in this con text. Besides the asso ciated Lam ´ e systems , this family con tains singular p o ten tia ls, whic h could b e conv enien t examples to study the tri-sup ersymmetry in singular systems. Regular Crum-Dar b oux transformations with zero mo des in the prohibited bands can pro duce self-isosp ectral p otentials with a generic shift of t he co ordinate, or sup erpartners with p erio dicity defects. The particular results of this type w ere obtained in [28, 29, 4 4 , 45, 46] with making use of the ﬁrst- and the second-order tra nsforma t ions applied to one- a nd t w o-gap Lam ´ e equation. It w ould b e interes ting to analyse suc h a class o f systems o n the presence of the tri- sup ersymmetric structure. The reve aled sup ersymmetric structure w as based on the internal prop erties of the in te- gral of motion Z , related with the KdV hierarc h y . This indication of the tri-sup ersymmetry and self-isosp ectralit y in the con text of nonlinear in tegrable systems should b e follow ed and analyzed. A sp ecial atten tion should b e paid to p o ssible manifestations of the tri- sup ersymmetry in ph ysical systems [34]. The work has b een partia lly supp orted by CONICYT, DICYT (USA CH) a nd b y F ONDE- CYT under g r a n ts 1050001 a nd 3085013. W e are grateful to V. Enolskii, V. Spiridono v, A. T reibic h, R. W eik ard and A. Zabro din for v aluable comm unications. Our sp ecial thanks are to B. Dubro vin fo r man y detailed explanations on the theory o f ﬁnite-gap systems. App endix A Higher-order diﬀeren tial op erators play the k ey role in the construction of the tri- sup ersymmetry since they mediate intert wining of the sup er-partner Hamiltonia ns. As w e explained in the section on the Crum-Darb oux tr a nsformations, prop erties of these op erators are determining for the ph ysical c har a cteristics of the sup erpartner syste ms. W e presen t here a short resume of t he relev an t facts r eferring for t he details to [38], [40]. Consider a diﬀeren tial op era t or of o rder n whic h annihilates n functions ψ i , i = 1 , . . . , n , A n = D n + n X j = 1 c A j ( x ) D n − j , A n ψ i = 0 , i = 1 , ..., n. (A.1) Its co eﬃcien ts are determined b y the functions ψ i . F or instance, the co eﬃcien t c A 1 ( x ) can b e giv en in terms of the W ronskian of the n functions ψ i , c A 1 ( x ) = − d dx ln W ( ψ 1 , . . . , ψ n ) , where W ( ψ 1 , . . . , ψ n ) = W = det B , B i,j = d j − 1 ψ i dx j − 1 , i, j, = 1 , . . . , n. This is in accordance with the general form fo r the co eﬃcien ts c A j ( x ) = − W j W , j = 1 , . . . , n, where W j is the determinan t of the matrix B mo diﬁed b y replacing the line ψ ( n − j ) 1 , . . . , ψ ( n − j ) n b y ψ ( n ) 1 , . . . , ψ ( n ) n . In this notation, W 0 ≡ W . The operat or A n can b e f a ctorized in terms of the ﬁrst order diﬀeren tial op erator s. There follo w equiv alent represen t a tions of A n whic h provide a b etter insigh t in to the prop erties of the op era t o r, see [3 8], A n = ( − 1) n W n W n − 1 D W 2 n − 1 W n W n − 2 D . . . D W 2 1 W 2 W 0 D W 0 W 1 , (A.2) 32 W e can write equiv alen tly A n = L n L n − 1 . . . L 2 L 1 , L j = D − α j , α j = d dx ln W j W j − 1 , j = 1 , . . . , n. (A.3) The op erator can also b e expres sed as a determinan t A n = W − 1 ( ψ 1 , . . . , ψ n )            ψ 1 ψ 2 · · · ψ n 1 ψ ′ 1 ψ ′ 2 · · · ψ ′ n D . . . . . . . . . . . . . . . ψ ( n − 1) 1 ψ ( n − 1) 2 · · · ψ ( n − 1) n D n − 1 ψ ( n ) 1 ψ ( n ) 2 · · · ψ ( n ) n D n            , (A.4) where the m ultiplicative factor ﬁxes the co eﬃcien t of D n to b e equal to one. He re, the determinant of the op era t o r-v alued ( n + 1) × ( n + 1) matrix is deﬁned as det C = P σ ∈ G n +1 sg n ( σ ) C σ (1) , 1 C σ (2) , 2 . . . C σ ( n +1) ,n +1 , where G n +1 is a set o f all p o ssible p ermutations of the in tegers { 1 , . . . , n + 1 } . P a rticularly , when ψ i are p erio dic functions except ev en nu m b er of an tip erio dic ones, the W ronskian W is p erio dic. Since the deriv ativ es do not change the p erio d of the functions, W i are p erio dic as we ll. The formulas ab ov e then justify p erio dicit y of the op erator A n . Finally , let us make a few commen ts on the sup erpartner Hamiltonians H and ˜ H in ter- t wined b y op erat or A n (see (3.10)) whic h annihilates a part of the ph ysical states of H . Let the p oten tial of H b e smo o th and the W ro nskian W computed on the k ernel of A n b e a no deless function. Then the p o ten tia l of ˜ H is smo oth as we ll. The op erat or A n can b e used in reconstruction o f the eigenstates ˜ ψ of ˜ H corresp onding t o the eingenstates ψ 6 = ψ i of H with the same eigen v alue, ˜ H ˜ ψ = E ˜ ψ , H ψ = E ψ , ˜ ψ = A n ψ . These w a v e functions ˜ ψ can b e also represen ted a s ˜ ψ = A n ψ = W ( ψ 1 , . . . , ψ n , ψ ) W ( ψ 1 , . . . , ψ n ) . (A.5) This receipt fails in the reconstruction of the states ˜ ψ i , whic h corresp ond to the same eigen- v alue as ψ i , ˜ H ˜ ψ i = E i ˜ ψ i , H ψ i = E i ψ i , where ψ i is annihilated by A n . These functions ˜ ψ i , annihilated by A † n , are giv en b y ˜ ψ i = W ( ψ 1 , .., ˆ ψ i , .., ψ n , ) W ( ψ 1 , . . . , ψ n ) , i = 1 , ..., n, A † n ˜ ψ i = 0 , (A.6) where the en tr y b elo w a sym b ol “ ˆ ” is omitted. App endix B Grading Γ ∗ = σ 3 In this case, whic h corresponds to the usual c hoice of the grading op erator, the non-dia gonal sup erc harg es Q ± are fermionic op erators, {Q ± , σ 3 } = 0, whereas the diagonal in tegral Z is a b osonic generato r . T able 1 represen ts the explicit iden tiﬁcatio n of the b osonic and fermionic generators, and corresp onding p o lynomials a pp earing in the (anti)comm utation relations. 33 F ermionic F 1 = Q − F 2 = − Rσ 3 Q − F 3 = − iR Q − F 4 = iσ 3 Q − in t egrals F 5 = R Q + F 6 = −Q + F 7 = iσ 3 Q + F 8 = − iRσ 3 Q + Bosonic H Σ 1 = − R Γ ∗ = σ 3 Σ 2 = − Rσ 3 in t egrals B 1 = − iRσ 3 Z B 2 = − σ 3 Z B 3 = − iR Z B 4 = −Z P olynomials P 22 = P 2 B = P − ( H ) P 11 = P 1 B = P + ( H ) P 12 = 1 P B = P Z ( H ) T able 1: In t egra ls of motion and structure p olynomials, grading Γ ∗ = σ 3 . Grading Γ ∗ = R F or the choice Γ ∗ = R , the parity-o dd diagona l, Z , a nd non-diag onal, Q − , in tegrals are fermionic supercharges. The non-diagonal parity-ev en in tegral Q + is iden tiﬁed as a b osonic generator. The iden tiﬁcatio n of all t he generators and structure p olynomials ar e giv en b y T able 2. F ermionic F 1 = Q − F 2 = Rσ 3 Q − F 3 = iσ 3 Q − F 4 = iR Q − in t egrals F 5 = iR Z F 6 = iRσ 3 Z F 7 = σ 3 Z F 8 = Z Bosonic H Γ ∗ = R Σ 1 = σ 3 Σ 2 = Rσ 3 in t egrals B 1 = Q + B 2 = iσ 3 Q + B 3 = R Q + B 4 = iRσ 3 Q + P olynomials P 22 = P 12 = P − ( H ) P 11 = P Z ( H ) P 2 B = 1 P B = P 1 B = P + ( H ) T able 2: In t egra ls of motion and structure p olynomials, grading Γ ∗ = R . Grading Γ ∗ = R σ 3 With this choice of the grading o p erator, in tegrals Z and Q + are identiﬁe d as fermionic sup erc harg es, integral Q − is a b osonic generator. Complete iden tiﬁcation of the generators and structure p olynomials are represen ted b y T able 3. F ermionic F 1 = Z F 2 = − σ 3 Z F 3 = − iR Z F 4 = iRσ 3 Z in t egrals F 5 = R Q + F 6 = −Q + F 7 = iRσ 3 Q + F 8 = − iσ 3 Q + Bosonic H Σ 1 = − R Σ 2 = − σ 3 Γ ∗ = Rσ 3 in t egrals B 1 = − iσ 3 Q − B 2 = − Rσ 3 Q − B 3 = − iR Q − B 4 = −Q − P olynomials P 22 = P Z ( H ) P 11 = P 12 = P + ( H ) P 1 B = 1 P B = P 2 B = P − ( H ) T able 3: Inte grals of mot ion and structure p olynomials, grading Γ ∗ = Rσ 3 . The a nti-comm utatio n relations b et w een the fermionic op erators are giv en in T able 4 , while T able 5 provide s the b oson-fermion comm utation relations. 34 F 1 F 2 F 3 F 4 F 5 F 6 F 7 F 8 F 1 P 22 Σ 2 P 22 0 0 0 B 4 P 12 0 B 1 P 12 F 2 Σ 2 P 22 P 22 0 0 − B 2 P 12 0 − B 3 P 12 0 F 3 0 0 P 22 Σ 2 P 2 0 − B 3 P 12 0 B 2 P 12 F 4 0 0 Σ 2 P 22 P 22 B 1 P 12 0 − B 4 P 12 0 F 5 0 − B 2 P 12 0 B 1 P 12 P 11 Σ 1 P 11 0 0 F 6 B 4 P 12 0 − B 3 P 12 0 Σ 1 P 11 P 11 0 0 F 7 0 − B 3 P 12 0 − B 4 P 12 0 0 P 11 Σ 1 P 11 F 8 B 1 P 12 0 B 2 P 12 0 0 0 Σ 1 P 11 P 11 T able 4 : F ermion-fermion an ti-commutation relations. Here the ov erall m ultiplicativ e fac- tor 2 is omitt ed. T o get an t i- comm uta tor, the corres p onding en try should b e m ultiplied b y 2, for instance, { F 1 , F 1 } = 2 P 22 . F 1 F 2 F 3 F 4 F 5 F 6 F 7 F 8 Γ ∗ − iF 4 − iF 3 iF 2 iF 1 iF 8 iF 7 − iF 6 − iF 5 Σ 1 − iF 3 − iF 4 iF 1 iF 2 0 0 0 0 Σ 2 0 0 0 0 iF 7 iF 8 − iF 5 − iF 6 B 1 0 iF 6 P 2 B − iF 7 P 2 B 0 0 − iF 2 P 1 B iF 3 P 1 B 0 B 2 iF 7 P 2 B 0 0 iF 6 P 2 B 0 − iF 4 P 1 B − iF 1 P 1 B 0 B 3 − iF 5 P 2 B 0 0 iF 8 P 2 B iF 1 P 1 B 0 0 − iF 4 P 1 B B 4 0 − iF 8 P 2 B − iF 5 P 2 B 0 iF 3 P 1 B 0 0 iF 2 P 1 B T able 5: Bo son- fermion comm utation relatio ns. The ov erall multiplicativ e factor 2 is omit- ted. T o get comm utator, the corresp onding en t ry should b e m ultiplied by 2, for instance, [Γ ∗ , F 1 ] = − 2 iF 4 . App endix C In the treatmen t of the section 4 . 1, w e left untouc hed the system described b y Lam ´ e asso ci- ated Hamiltonian with m = l . D ue to a n iden tity dn ( x + K ) = k ′ / dn x , in con trary to the other members of the family , its p erio d is K . This fact explains wh y the algebraic metho ds applied to other mem b ers o f the family in t his case sa y that the dimension of sl (2 , R ) repre- sen tatio n realized on an tip erio dic in the p erio d 2 K singlet states is equal to m − l = 0. This is just b ecause singlet states with suc h a p erio d do es not exist. The place of this system in t he mosaic of the tri- sup ersymmetric self-isosp ectral systems is clariﬁed by its in timate relation t o pure Lam´ e system, mediated by Landen transformation [69, 70]. Landen’s t r a nsformation o f the elliptic functions can b e written as sn ( x, k ) = α sn  x α , κ  cn  x α , κ  dn  x α , κ  , cn ( x, k ) = 1 − α sn 2 ( x α , κ ) dn ( x α , κ ) , dn ( x, k ) = κ ′ + (2 − α ) cn 2 ( x α , κ ) dn ( x α , κ ) (C.1) where α = 2 1+ k , κ 2 = 4 k (1+ k ) 2 , k = 1 − κ ′ 1+ κ ′ , α = 1 + κ ′ . T o a v oid confusions, let us denote explicitly the dep endence of the complete elliptic integral K on the mo dular parameter suc h tha t w e 35 will write K ( k ) or K ( κ ). Since K ( κ ) = (1 + k ) K ( k ), Landen’s transformatio n divides in t w o the p erio d 2 K ( k ) of the elliptic f unctions in the sense tha t the p erio d of the resulting expression is K ( κ ). Using the identities ( C.1), we can rewrite the Lam´ e Hamiltonian in terms of the el- liptic functions of a new v ariable y = x α and the mo dular parameter κ , H − m, 0 ( x, k ) = 1 α 2 [ H m,m ( y , κ )] + const. Th e displacemen t K ( k ) of the pure Lam´ e tri-sup ersymmetric partner c ha nges to K ( κ ) 2 in the case m = l . It is in accordance with our r esult on the general case m 6 = l where the superpar t ner p otential w as displaced in the half of the real p erio d. Thu s, w e obtain ﬁnally the relation  H − m, 0 ( x + K ( k ) , k ) 0 0 H − m, 0 ( x, k )  = 1 α 2 H − m,m  y + K ( κ ) 2 , κ  0 0 H − m,m ( y , κ ) ! + c, (C.2) where c is a constan t term. It suggests directly the form of the tri- sup ersymmetry in the sp ecial case of m = l Lam ´ e asso ciated systems; all the op erato rs comm ut ing with H m, 0 comm ute with H m,m as we ll. T o get their explicit form for the systems describ ed by H m,m w e just hav e t o rescale the v ariable and apply the iden tities (C.1) in the form ulas (4.21) for X m, 0 ( x, k ) , Y m, 0 ( x, k ) and Z m, 0 ( x, k ). Then we can write immediately X m,m ( y , κ ) = X m, 0 ( αy , k ( κ )) = X m, 0  (1 + κ ′ ) y , 1 − κ ′ 1 + κ ′  , Y m,m ( y , κ ) = Y m, 0 ( αy , k ( κ )) = Y m, 0  (1 + κ ′ ) y , 1 − κ ′ 1 + κ ′  , Z m,m ( y , κ ) = Z m, 0 ( αy , k ( κ )) = Z m, 0  (1 + κ ′ ) y , 1 − κ ′ 1 + κ ′  . (C.3) Naturally , the algebraic relations b et w een the op erators remain unchanged. 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The mo dular para meter is set k = 0 . 99. -2 0 2 4 -25 -20 -15 -10 -5 0 -2 0 2 4 -25 -20 -15 -10 -5 0 m = 5 , l = 0 -2 0 2 4 -25 -20 -15 -10 -5 0 -2 0 2 4 -25 -20 -15 -10 -5 0 m = 5 , l = 3 Figure 3: P oten tials on the left corresp ond to H − m,l (dashed thic k line) and H + m,l (solid thick line); k 2 = 0 . 99. On the right, t he limit k → 1 of these p otential functions is show n. The solid thin lines represen t the shared b ound-states and the low est scattering state. Dashed thin lines represen t m − l = 2 singlet states 40

Finite-gap systems, tri-supersymmetry and self-isospectrality

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