B"acklund Transformation for the BC-Type Toda Lattice

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 3 (2007), 080, 17 pages B¨ ac klund T ransformation for the BC-T yp e T o da Lattice ⋆ V adim KUZNETSOV † and Evgeny SKL Y ANIN ‡ † De c e ase d ‡ Dep artment of Mathematics, Uni v ersity of Y ork, Y ork YO10 5DD, UK E-mail: eks2@york.ac.uk Received July 13, 20 07; Publishe d online July 25, 2 007 Original article is av a ila ble at http:// www.e mis.de/journals/SIGMA/2007/080/ Abstract. W e study an int egrable cas e of n -pa r ticle T o da lattice: op e n chain with b oundar y terms co nt aining 4 parameters. F or this m o del we construct a B¨ acklund transformation and prov e its bas ic prope rties: canonicity , comm utativity and sp ectra lity . The B¨ ac klund transformatio n ca n b e als o viewed a s a discr etized time dynamics. Two Lax matrices a re used: of order 2 and of orde r 2 n + 2, whic h ar e m utually dual, sharing the s ame spe ctral curve. Key wor ds: B¨ acklund transforma tio n; T o da lattice; in tegrability; b oundar y conditions; clas- sical Lie alge br as 2000 Mathematics S ubje ct Classific ation: 7 0H06 1 In tro duction In the present p ap er w e study the Hamiltonian s y s tem of n one-dimensional particles with co ordinates x j and canonica l momen ta X j , j = 1 , . . . , n : { X j , X k } = { x j , x k } = 0 , { X j , x k } = δ j k , (1.1) c haracterize d by the Hamiltonia n H = n X j =1 1 2 X 2 j + n − 1 X j =1 e x j +1 − x j + α 1 e x 1 + 1 2 β 1 e 2 x 1 + α n e − x n + 1 2 β n e − 2 x n (1.2) con taining 4 arbitrary parameters: α 1 , β 1 , α n , β n . The m o del w as missin g from th e early lists of in tegrable cases of the T o da lattice [1, 2] b ased on Dynkin diagrams for simple af f ine Lie algebras. Its in tegrabilit y was p ro v ed f irst in [3, 4, 5]. As f or the m ore recent classif ications, in [6] the mo del is enlisted as the case (i). In [7, 8] particular cases of the Hamiltonian (1.2 ) are assigned to the C (1) n case with ‘Morse terms ’. F or brevit y , we refer to the mo del as ‘BC-T o da lattice’ emphasising the fact that eac h b oun d ary term is a linear combinatio n of the term ∼ α corresp onding to the ro ot system B and of the term ∼ β corr esp onding to th e ro ot system C , see [1, 2, 7, 8]. In s ection 2 we r eview br ief ly the kno wn facts ab out the in tegrabilit y of the mo del u sing the approac h d ev eloped in [3, 4] and based on th e Lax matrix L ( u ) of order 2 and the corre- sp ond ing quadr atic r -matrix algebra. In particula r, w e construct explicitly a generating f unction of the complete set of comm u ting Hamiltonians H j ( j = 1 , . . . , n ) whic h includes the physica l Hamiltonian H (1.2 ). ⋆ This p ap er is a contribution to the V adim Ku znetsov Memori al Issue ‘Integrable Systems and Related T opics’. The full collection is a v ailable at http://www.emi s.de/journals/SIGMA/kuznetso v.html 2 V. Kuznetso v and E. S kly anin In Section 3 we describ e the main result of our p ap er: constru ction of a B¨ ac klund transf or- mation (BT) for our mo d el as a one-parametric family of maps B λ : ( X x ) 7→ ( Y y ) from the v ariables ( X x ) to the v ariables ( Y y ). W e construct the BT c hoosing an appropriate gauge (or Darb oux) transformation of th e lo c al Lax matrices. In S ection 4, adopting the Hamiltonian p oint of view devel op ed in [9, 10], w e prov e the basic prop erties of the BT: 1. Pr eserv ation of the comm uting Hamiltonians B λ : H j ( X, x ) 7→ H j ( Y , y ). 2. Can on icity: preserv ation of the Po isson brac k et (1.1 ). 3. Commutativit y: B λ 1 ◦ B λ 2 = B λ 2 ◦ B λ 1 . 4. S p ectralit y: the f act that the graph of the BT is a Lagrangian manifold on which the 2-form Ω ≡ n X j =1  d X j ∧ d x j − d Y j ∧ d y j  − d ln Λ ∧ d λ (1.3) v anishes. Here Λ is an eigenv alue of th e matrix L ( λ ). In other w ords, the p arameter λ of the BT and its exp onen tia ted canonical conjugate Λ lie on the sp ectral curve of L ( u ): det  Λ − L ( λ )  = 0 . (1.4) W e also p ro v e the follo wing expansion of B λ in λ − 1 B λ : f 7→ f − 2 λ − 1 { H, f } + O ( λ − 2 ) , λ → ∞ . (1.5) whic h allo ws to in terpret the BT as a d iscrete time d ynamics appro ximating th e con tin uous-time dynamics generate d by the Hamiltonian (1.2). In S ection 5 w e constru ct for our system an alternativ e Lax matrix L ( v ) . The new Lax matrix of ord er 2 n + 2 is du al to the matrix L ( u ) of order 2 in th e sense that they share the same sp ectral cur v e with the parameters u and v ha ving b een swa pp ed : det  v − L ( u )  = ( − 1) n +1 v det  u − L ( v )  . (1.6) In the same section w e pr o vide an in terpretation of the BT in terms of th e ‘big’ Lax mat- rix L ( v ) and establish a remark able factoriza tion formula for λ 2 − L 2 ( v ). The concluding Section 6 con tains a sum mary and a d iscussion. All the tec hnical pro ofs and tedious calcula tions are remo v ed to the Ap p end ices. 2 In tegra bilit y of th e mo del In demonstr ating the in tegrabilit y of the mo del we f ollo w the app roac h to the integrable c hains with b ound ary conditions dev elo p ed in [3, 4] and use the notation of [9, 10]. The L ax matrix L ( u ) for the BC-T o da lattice is constru cted as the pro duct L ( u ) = K − ( u ) T t ( − u ) K + ( u ) T ( u ) (2.1) of the follo wing matrices ( T t stands for the matrix transp osition). The m ono dromy matrix T ( u ) is itself the pr o duct T ( u ) = ℓ n ( u ) · · · ℓ 1 ( u ) (2.2) of the lo cal Lax matrices ℓ j ( u ) ≡ ℓ ( u ; X j , x j ) =  u + X j − e x j e − x j 0  , (2.3) B¨ ac klund T ransformation for the BC-T yp e T o da Lattice 3 eac h con taining on ly the v ariables X j , x j describing a single particle. Note th at tr T ( u ) is the generating function for the Hamilto nians of the p erio dic T o da latti ce. The m atrices K ± ( u ) conta ining the information ab out the b oun d ary in teractio ns are def ined as [3, 4] K − ( u ) =  u − α 1 α 1 β 1 u  , K + ( u ) =  u − α n α n β n u  . (2.4) The signif icance of the Lax matrix L ( u ) is th at its sp ectrum is inv arian t under th e d ynamics generated by the Hamiltonian (1.2), th e corresp on d ing equations of motion d G/ d t ≡ ˙ G = { H , G } for an observ able G b eing ˙ x j = X j , j = 1 , . . . , n (2.5) and ˙ X j = e x j +1 − x j − e x j − x j − 1 , j = 2 , . . . , n − 1 , (2.6a) ˙ X 1 = e x 2 − x 1 − α 1 e x 1 − β 1 e 2 x 1 , (2.6b) ˙ X n = − e x n − x n − 1 + α n e − x n + β n e − 2 x n . (2.6c) T o pro v e the inv ariance of the sp ectrum of L ( u ) w e introdu ce the matrices A j ( u ) A j ( u ) =  − u e x j − e − x j − 1 0  , j = 2 , . . . , n − 1 , (2.7) A 1 ( u ) =  − u e x 1 − α 1 − β 1 e x 1 0  , A n +1 ( u ) =  − u α n + β n e − x n − e − x n 0  , (2.8) whic h satisfy the easily verif ied identiti es ˙ ℓ j = A j +1 ℓ j − ℓ j A j , j = 1 , . . . , n, (2.9) − ˙ K + = 0 = K + A n +1 ( u ) + A t n +1 ( − u ) K + , (2.10a ) ˙ K − = 0 = A 1 ( u ) K − + K − A t 1 ( − u ) . (2.10b) F rom (2.2) and (2.9) it follo ws immediately that ˙ T ( u ) = A n +1 ( u ) T ( u ) − T ( u ) A 1 ( u ) . (2.11) Then, using (2.1) and (2.1 0), w e obtain the equalit y ˙ L ( u ) =  A 1 ( u ) , L ( u )  (2.12) implying th at th e sp ectrum of L ( u ) is preserved by the d ynamics. There are only tw o sp ectral inv arian ts of a 2 × 2 matrix: the trace and the determin ant. F rom (2.3) it follo ws that det ℓ ( u ) = 1 and, resp ectiv ely , det T ( u ) = 1, so, by (2.1), the determi- nan t of L ( u ) d ( u ) ≡ det L ( u ) = det K − ( u ) det K + ( u ) = ( α 2 1 + β 1 u 2 )( α 2 n + β n u 2 ) (2.13) con tains no dyn amical v ariables X x . Th e trace t ( u ) ≡ tr L ( u ) = tr K − ( u ) T t ( − u ) K + ( u ) T ( u ) , (2.14) 4 V. Kuznetso v and E. S kly anin ho w ev er, d o es con tain d ynamical v ariables and th erefore can b e used as a generating fun ction of the in tegrals of motion, which can b e c hosen as the co ef f icien ts of the p olynomial t ( u ) of d egree 2 n + 2 in u . Not e that t ( − u ) = t ( u ) due to the symmetry K t ± ( − u ) = − K ± ( u ) . (2.15) The leading co ef f icien t of t ( u ) at u 2 n +2 is a constan t ( − 1) n . Same is true for its free term t (0) = tr K + (0) K − (0) = − 2 α n α 1 (2.16) due to the iden tit y M K ± (0) M t = d et M · K ± (0) , (2.17) whic h h olds f or an y matrix M . W e are left then with n nontrivia l coef f icien ts H j t ( u ) = ( − 1) n u 2 n +2 − 2 α n α 1 + n X j =1 H j u 2 j (2.18) whic h are inte grals of motion ˙ H j = 0 since ˙ t ( u ) = 0 due to (2.12). The conserved quan tities H j are ob viously p olynomial in X , e ± x . Their in dep end ence can easily b e established b y setting e ± x = 0 in (2.3) and analysing the resulting p olynomials in X . It is also easy to v erify th at th e physical Hamiltonian (1.2) is expressed as H = ( − 1) n +1 2 H n . (2.19) The q u an tities H j are also in in v olutio n { H j , H k } = 0 (2.20) with r esp ect to the Po isson b rac k et (1.1). T ogether with the indep endence of H j , it constitutes the Liouville inte grabilit y of our system. The commutativit y (2.2 0) of H j or, equiv alen tly , of t ( u ) { t ( u 1 ) , t ( u 2 ) } = 0 (2.21) is pro v ed in the stand ard wa y u sing the r -matrix tec h nique [3, 4 ]. Let 1 b e the unit m atrix of order 2 and for any matrix L def ine 1 L ≡ L ⊗ 1 , 2 L ≡ 1 ⊗ L . (2.22) W e ha v e then the quadratic Poisson br ac k et s [10, 11 ] { 1 ℓ ( u 1 ) , 2 ℓ ( u 2 ) } = [ r ( u 1 − u 2 ) , 1 ℓ ( u 1 ) 2 ℓ ( u 2 )] , (2.23) and, as a consequence, { 1 T ( u 1 ) , 2 T ( u 2 ) } = [ r ( u 1 − u 2 ) , 1 T ( u 1 ) 2 T ( u 2 )] , (2.24) with th e r -matrix r ( u ) = P u , (2.25) where P is the p ermuta tion matrix P a ⊗ b = b ⊗ a . B¨ ac klund T ransformation for the BC-T yp e T o da Lattice 5 Let e r ( u ) = r t 1 ( u ) = r t 2 ( u ) , (2.26) t 1 and t 2 b eing, resp ectiv ely , transp osition with resp ect to the f irst and second comp onent of the tensor pro duct C 2 ⊗ C 2 . Then for b oth T ( u ) = T ( u ) K − ( u ) T t ( − u ) and T ( u ) = T t ( − u ) K + ( u ) T ( u ) w e ob tain th e same P oisson algebra [3, 4] { 1 T ( u 1 ) , 2 T ( u 2 ) } = r ( u 1 − u 2 ) 1 T ( u 1 ) 2 T ( u 2 ) − 1 T ( u 1 ) 2 T ( u 2 ) r ( u 1 − u 2 ) − 1 T ( u 1 ) e r ( u 1 + u 2 ) 2 T ( u 2 ) + 2 T ( u 2 ) e r ( u 1 + u 2 ) 1 T ( u 1 ) , (2.27) whic h ens ures the comm utativit y (2.21) of t ( u ). 3 Describing B¨ ac klund transformation In this section we shall construct a B¨ ac klund transform ation (BT) for our mo d el. W e shall sta y in the fr amew ork of the Hamiltonian approac h prop osed in [9] and follo w closely our pr evious treatmen t of the p erio dic T o d a latt ice [9, 10], with the necessary mo dif ications taking in to accoun t the b oundary conditions. W e are lo oking thus for a one-parametric family of m aps B λ : ( X x ) 7→ ( Y y ) fr om the v ariables ( X x ) to the v ariables ( Y y ) c haracterised by the pr op erties enlisted in the Intro d uction: Invarianc e of Hamiltonians, Canonicity, Commutativity and Sp e ctr ality. The inv ariance of the comm uting Hamiltonians H j , or of their generating p olynomial t ( u ) = tr L ( u ) w ill b e ensured if we f ind an in v er tib le matrix M 1 ( u, λ ) intert win in g the matrices L ( u ) dep end ing on the v ariables X x and Y y : M 1 ( u, λ ) L ( u ; Y , y ) = L ( u ; X , x ) M 1 ( u, λ ) . (3.1) T o f ind M 1 ( u, λ ) let u s lo ok for a gauge transformation M j +1 ( u, λ ) ℓ ( u ; Y j , y j ) = ℓ ( u ; X j , x j ) M j ( u, λ ) , j = 1 , . . . , n, (3.2) implying th at d et M j do es not dep en d on j . F rom (3.2) and (2.2) w e obtain M n +1 ( u, λ ) T ( u ; Y , y ) = T ( u ; X , x ) M 1 ( u, λ ) . (3.3) Let J b e the the stand ard ske w-symmetric matrix of order 2 J =  0 1 − 1 0  , J t = − J, J 2 = − 1 , (3.4) and def ine the an tip o d e M a as M a ≡ − J M J (3.5) for any matrix M of order 2. It is easy to see that M t M a = M a M t = d et M . (3.6) T ransp osing (3.3) and u sing (3.6) toget her with the the fact that det M j is ind ep endent of j w e obtain the relatio n T t ( − u ; X , x ) M a n +1 ( − u, λ ) = M a 1 ( − u, λ ) T t ( − u ; Y , y ) . (3.7) 6 V. Kuznetso v and E. S kly anin W e shall b e able to obtain (3.1 ) if we imp ose tw o add itional r elations K − ( u ) M a 1 ( − u, λ ) = M 1 ( u, λ ) K − ( u ) , (3.8a) K + ( u ) M n +1 ( u, λ ) = M a n +1 ( − u, λ ) K + ( u ) . (3.8b) Then, starting with the righ t-hand side L ( u ; X , x ) M 1 ( u, λ ) of (3.1 ) and using (2.1) and (3.3) w e obtain L ( u ; X , x ) M 1 ( u, λ ) = K − ( u ) T t ( − u ; X , x ) K + ( u ) T ( u ; X , x ) M 1 ( u, λ ) = K − ( u ) T t ( − u ; X , x ) K + ( u ) M n +1 ( u, λ ) T ( u ; Y , y ) (3.9) Using then (3.8b) to mov e M n +1 ( u, λ ) through K + ( u ), then us ing (3.7) and f inally (3.8a ) we get, step by step, L ( u ; X , x ) M 1 ( u, λ ) = K − ( u ) T t ( − u ; X , x ) M a n +1 ( − u, λ ) K + ( u ) T ( u ; Y , y ) = K − ( u ) M a 1 ( − u, λ ) T t ( − u ; Y , y ) K + ( u ) T ( u ; Y , y ) = M 1 ( u, λ ) K − ( u ) T t ( − u ; Y , y ) K + ( u ) T ( u ; Y , y ) = M 1 ( u, λ ) L ( u ; Y , y ) (3.10) arriving f inally at (3.1). W e ha v e thus to f ind a set of matrices M j ( u, λ ), j = 1 , . . . , n + 1 compatible w ith the conditions (3.2) and (3.8). A quic k calculation sho ws that the so called DS T -ansatz for M j used in [9 , 10] for the p erio dic T o da lattice cont radicts the conditions (3.8). The philosophy ad vocated in [10] requ ir es that the ansatz for the gauge matrix M j ( u ) b e c hosen in the form of a Lax matrix satisfying th e r -matrix Po isson brac k et (2.23) with the same r -matrix (2.25) as the Lax op erator ℓ ( u ). It was sh o wn in [10] that the so-call ed DST-ansatz M DST j ( u, λ ) =  u − λ + s j S j − s j S j − 1  (3.11) serv es well for the p erio dic T o da case. The ab o v e ansatz is how ever n ot compatible with the b ound ary conditions (3.8) and w e ha v e to use a more complicated ansatz for M j in th e form of the Lax matrix for the isotropic Heisen berg magnet (XXX-model): M j ( u, λ ) =  u − λ + s j S j s 2 j S j − 2 λs j S j − u − λ + s j S j  , det M j ( u, λ ) = λ 2 − u 2 . (3.12) The same gauge transformation wa s used in [12] for constructing a Q -op erator for th e quan- tum XXX-magnet. Substituting (3.12) into (3.2) w e obtain the relatio ns X j = − λ + s − 1 j e x j + s j +1 e − x j , (3.13a ) Y j = λ − s − 1 j e y j − s j +1 e − y j , (3.13b) S j = 2 λs − 1 j − s − 2 j e x j − s − 2 j e y j , (3.13c ) S j +1 = e − x j + e − y j , (3.13d) for j = 1 , . . . , n , and from (3.8), resp ectiv ely , S 1 = 2( α 1 + β 1 λs 1 ) 1 + β 1 s 2 1 , S n +1 = 2( λs n +1 − α n ) β n + s 2 n +1 . (3.14) B¨ ac klund T ransformation for the BC-T yp e T o da Lattice 7 Eliminating the v ariables S j , w e arrive to the equations def inin g the BT ( j = 1 , . . . , n ): X j = − λ + s − 1 j e x j + s j +1 e − x j , (3.15a ) Y j = λ − s − 1 j e y j − s j +1 e − y j . (3.15b) The v aria bles s j , j = 1 , . . . , n + 1 in (3.15) are implicitly def ined as fun ctions of x , y and λ from th e quadratic equations (e − x j − 1 + e − y j − 1 ) s 2 j − 2 λs j + (e x j + e y j ) = 0 , j = 2 , . . . , n (3.16a ) (2 α 1 + β 1 e x 1 + β 1 e y 1 ) s 2 1 − 2 λs 1 + (e x 1 + e y 1 ) = 0 , (3.16b) (e − x n + e − y n ) s 2 n +1 − 2 λs n +1 + (2 α n + β n e − x n + β n e − y n ) = 0 . (3.16c ) Lik e in the p erio dic case [9, 10], the BT map B λ : ( X x ) 7→ ( Y y ) is describ ed implicitly by the equations (3.15). Unlik e the p erio d ic case, w e hav e extra v ariables s j . It is more conv enient not to exp r ess s j from equations (3.16) and to substitute them in to (3.15) but rather def in e the BT by the whole set of equations (3.15) and (3.16). Equations (3.15) and (3.16) are algebraic equations and therefore def ine ( Y y ) as m ultiv alued functions of ( X x ), whic h is a common situ ation with in tegrable maps [13]. In this pap er, to a v oid the complications of the real algebraic geometry w e allo w all our v ariables to b e complex. 4 Prop erties of the B¨ ac k lund transformation Ha ving def ined th e map B λ : ( X x ) 7→ ( Y y ) in the previous section, we pro ceed to establish its prop erties from the list given in the In trod uction. 4.1 P reserv ation of Hamiltonians The equalit y H j ( X, x ) = H j ( Y , y ) ∀ λ , or, equ iv alen tly , t ( u ; X, x ) = t ( u ; Y , y ) holds by construc- tion, b eing a direct consequence of (3.1). 4.2 Canonicity The canonicit y of the BT means that the v aria bles Y ( X, x ; λ ) and y ( X , x ; λ ) ha v e the same canonical P oisson brac k et s (1.1) as ( X x ). An equiv alen t formula tion can b e giv en in terms of symplectic spaces and Lagrangian manifolds. Consid er the 4 n -dimensional symplectic sp ace V 4 n with co ord inates X xY y and symplectic 2-form Ω 4 n ≡ n X j =1  d X j ∧ d x j − d Y j ∧ d y j  . (4.1) Equations (3.15) and (3.16) def in e a 2 n -dimensional s u bmanifold Γ 2 n ⊂ V 4 n whic h can b e considered as the graph Y = Y ( X , x ; λ ), y = y ( X , x ; λ ) of the BT (the p arameter λ is assumed here to b e a constant). The canonicit y of the BT is then equ iv alen t to the fact that the manifold Γ 2 n is L agr angian , meaning that: (a) it is isotr opic , that is nullif ies the form Ω 4 n Ω 4 n | Γ 2 n = 0 , (4.2) and (b) it has maximal p ossible dimens ion for an isotropic manifold: dim Γ 2 n = 1 2 dim V 4 n . 8 V. Kuznetso v and E. S kly anin One w a y of p r o ving the canonicit y is to present explicitly the generating function Φ λ ( y ; x ) of the canonical transf orm ation, su c h that X j = ∂ Φ λ ∂ x j , Y j = − ∂ Φ λ ∂ y j . (4.3) The r equired function is giv en b y the expression Φ λ ( y ; x ) = n X j =1 f λ ( y j , s j +1 ; x j , s j ) + ϕ (0) λ ( s 1 ) + ϕ ( n +1) λ ( s n +1 ) , (4.4) where f λ ( y j , s j +1 ; x j , s j ) = λ (2 ln s j − x j − y j ) + s − 1 j (e x j + e y j ) − s j +1 (e − x j + e − y j ) , (4.5a) ϕ (0) λ ( s 1 ) = − λ ln  1 + β 1 s 2 1 ) − 2 α 1 √ β 1 arctan  p β 1 s 1  , (4 .5b) ϕ ( n +1) λ ( s n +1 ) = λ ln  β n + s 2 n +1  − 2 α n √ β n arctan  s n +1 √ β n  , (4.5c) and s j ( x, y ; λ ) are def ined implicitly th r ough (3.16). Equalities (4.3 ) can b e ve rif ied by a direct, though tedious, compu tation. Another, more elegan t, w a y is to use the argument from [10 ] b ased on imp osing a set of co nstraint s in an extended p hase space, see App end ix A. 4.3 Commutativit y The comm utativit y B λ 1 ◦ B λ 2 = B λ 2 ◦ B λ 1 of the BT follo ws f r om the preserv a tion of th e complete set of Hamiltonians and th e canonicit y by the standard argument [9, 10] based on V eselo v ’s theorem [13] ab out the action-a ngle repr esen tatio n of in tegrable maps . 4.4 Sp ectralit y The sp ectralit y p rop erty f orm ulated f irst in [9] generalises the canonicit y by allo w ing the pa- rameter λ of the BT to b e a dynamical v ariable lik e x and y . Let us extend the symplectic space V 4 n from section 4.2 to a (4 n + 2)-dimensional sp ace V 4 n +2 b y adding tw o more co ordinates λ , µ and d ef ining the extension Ω 4 n +2 of symplectic form Ω 4 n (4.1) as Ω 4 n +2 ≡ Ω 4 n − d µ ∧ d λ = n X j =1  d X j ∧ d x j − d Y j ∧ d y j  − d µ ∧ d λ. (4.6) Def ine the extended graph Γ 2 n +1 of the BT by equations (3.15) and a new equation µ = − ∂ ∂ λ Φ λ ( y ; x ) . (4.7) The 2-form Ω 4 n +2 ob viously v anishes on Γ 2 n +1 , and the manifold Γ 2 n +1 is lagrangia n. An amazing fact is th at e µ is pr op ortional to an eigen v alue of th e matrix L ( λ ), see (1.4). I n fact, the t w o eigen v alues of L ( λ ) can b e found explicitly to b e Λ = ( α 2 n + β n λ 2 ) 1 + β 1 s 2 1 β n + s 2 n +1 n Y j =1  − s − 2 j e x j + y j  , (4.8a) B¨ ac klund T ransformation for the BC-T yp e T o da Lattice 9 e Λ = ( α 2 1 + β 1 λ 2 ) β n + s 2 n +1 1 + β 1 s 2 1 n Y j =1  − s 2 j e − x j − y j  , (4.8b) see Ap p end ix B for the pro of. Ha ving the explicit form ulae (4.8a ) for Λ and (4.4) for Φ λ ( y ; x ) one can easily v erify that Λ = ( − 1) n ( α 2 n + β n λ 2 ) e µ . (4.9) 4.5 B¨ ac klund transformation as discret e time dynamics One of applications of a BT is that it might provide a d iscrete-time app ro ximation of a con tin uous- time int egrable sy s tem [14, 15]. Indeed, iterations of the canonical map B λ generate a discrete time dyn amics. F u r thermore, if we f ind a p oin t λ = λ 0 that (a) the m ap B λ 0 b ecomes the iden- tit y map, and (b) in a neigh b ourh o o d of λ 0 the inf initesimal map B λ 0 + ε ∼ ε { H , ·} repro du ces the Hamiltonian f low w ith the Hamiltonian (1.2), we can claim that B λ is a d iscrete time ap- pro ximation of the BC-T o da lattice. An attractiv e feature of this approximat ion is that, unlik e some others [14], the d iscrete-time sys tem and the con tin u ous-time one share the same integ rals of motion. In our case λ 0 = ∞ . Letting ε = λ − 1 and assuming the ansatz y j = x j + O ( ε ) , j = 1 , . . . , n (4.10) w e obtain from (3.16a) and (3.16b) the expansion s j = ε e x j + O ( ε 2 ) , j = 1 , . . . , n (4.11a ) and from (3.16c) the expansion s n +1 = ε ( α n + β n e − x n ) + O ( ε 2 ) . (4.11b) Substituting then expans ions (4.10) in to equation (3.13d) we obtain S j = 2e − x j − 1 + O ( ε ) , j = 2 , . . . , n + 1 (4.12a ) and substituting expansion (4.11) for s 1 in to form ula (3.14) for S 1 w e obtain S 1 = (2 α 1 + β 1 e x 1 ) + O ( ε ) . (4.12b) Then from (3.12) w e h a v e − εM j = 1 + ε  u 1 + 2 A j  + O ( ε 2 ) , j = 1 , . . . , n + 1 , (4.13) where A j coincides with the matrix (giv en b y (2.7) and (2.8)) whic h describ es the contin u ous- time d ynamics of the Lax matrix. F rom (3.2) w e obtain then ℓ ( u ; Y j , y j ) = ℓ ( u ; X j , x j ) − 2 ε  A j +1 ℓ ( u ; X j , x j ) − ℓ ( u ; X j , x j ) A j  + O ( ε 2 ) , (4.14) for j = 1 , . . . , n + 1. Comp aring the result to (2.9) w e get the expansion (1.5). 10 V. Kuznetso v and E. S kly anin 5 Dual Lax matrix Man y integ rable systems p ossess a pair of Lax matrices sharing the same sp ectral curve with the parameters u and v sw app ed lik e in (1.6), see [16] for a list of examples and a discussion. In p articular, the p erio dic n -particle T o da lattice has t w o L ax matrices: th e ‘small’ one, of order 2 [11], and the ‘b ig’ one, of order n [17]. F or v arious degenerate cases of th e BC-T o da lattice ‘big’ Lax matrices are also kno wn [2, 7, 8 , 17]. In this section w e pr esen t a new Lax matrix of ord er 2 n + 2 for the most general, 4-parametric BC-T o da lattice. Here w e describ e the result, remo ving the detailed deriv atio n to Ap p end ix C. Let E j k b e the square matrix of order 2 n + 2 w ith the only nonzero entry ( E j k ) j k = 1. The Lax matrix L ( v ) is then d escrib ed f or the generic case n ≥ 3 as L ( v ) = n X j,k =1 L j k E j k = n X j =2 e x j − x j − 1 E j,j − 1 + n X j =1  − X j E j j + E j,j +1  − n − 1 X j =1 e x j +1 − x j E 2 n +2 − j, 2 n +1 − j + n X j =1  X j E 2 n +2 − j, 2 n +2 − j − E 2 n +2 − j, 2 n +3 − j  +  α n e − x n + β n 2 e − 2 x n   E n +1 ,n − E n +2 ,n +1  + β n 2 e − x n − x n − 1  E n +3 ,n − E n +2 ,n − 1  − E n +1 ,n +2 −  α 1 e x 1 + β 1 2 e 2 x 1   E 2 n +2 , 2 n +1 + v − 1 E 1 , 2 n +2  + β 1 2 v e x 1 + x 2  E 2 , 2 n +1 − E 1 , 2 n  − v E 2 n +2 , 1 (5.1) and consists of a bulk ‘Jacobian’ strip (the main diagonal and t w o adjacen t diagonals) whic h repro du ces the Lax matrix for th e op en T o d a lattice together with b oundary b lo c ks contai ning parameters α 1 β 1 α n β n . W e do n ot consider here the sp ecial case of small dimensions n = 1 , 2 when the t w o b oun dary blo c ks in terfere with eac h other and the s tr ucture of the Lax matrices b ecomes more complicated T o help visualise th e m atrix L ( v ) we pr esent an illustration f or the case n = 3, using the shorthand notati on ξ j ≡ e x j , η j ≡ e y j : L ( v ) =               − X 1 1 0 0 0 − β 1 2 v ξ 1 ξ 2 0 α 1 v ξ 1 + β 1 2 v ξ 2 1 ξ 2 ξ 1 − X 2 1 0 0 0 β 1 2 v ξ 1 ξ 2 0 0 ξ 3 ξ 2 − X 3 1 0 0 0 0 0 0 α 3 ξ 3 + β 3 2 ξ 2 3 0 − 1 0 0 0 0 − β 3 2 ξ 2 ξ 3 0 − α 3 ξ 3 − β 3 2 ξ 2 3 X 3 − 1 0 0 0 0 β 3 2 ξ 2 ξ 3 0 − ξ 3 ξ 2 X 2 − 1 0 0 0 0 0 0 − ξ 2 ξ 1 X 1 − 1 − v 0 0 0 0 0 − α 1 ξ 1 − β 1 2 ξ 2 1 0               . (5.2) The m atrix L ( v ) p ossesses the symmetry L ( v ) = − C v L t ( v ) C − 1 v , (5.3) B¨ ac klund T ransformation for the BC-T yp e T o da Lattice 11 where C v = − v E 2 n +2 , 2 n +2 + 2 n +1 X j =1 E j, 2 n +2 − j =         0 0 . . . 0 1 0 0 0 . . . 1 0 0 . . . . . . . . . . . . . . . . . . 0 1 . . . 0 0 0 1 0 . . . 0 0 0 0 0 . . . 0 0 − v         (5.4) (note that C − 1 v = C v − 1 ). The matrix L ( v ) s h ares the same sp ectral cur v e w ith the ‘small’ Lax op erator L ( u ) satisfying the determinan tal identit y (1.6 ) and th us generates the same comm uting Hamilto nians H j . The Lax m atrix L ( v ) of order 2 n + 2 s eems to b e n ew. When one or more of the constan ts α 1 β 1 α n β n v anish it degenerates (with a drop of dimension) in to kn o wn Lax matrices f or simp le af f ine Lie algebras [2, 7, 8, 17]. F or the general 4-parametric case a Lie-algebraic int erpretation of L ( v ) is still unkno wn. In particular, it is an in teresting qu estion whether L ( v ) satisf ies a kind of r -matrices Poisso n algebra. Inozem tsev [5] presen ted a d if ferent Lax matrix for th e BC-T o da lattic e, of order 2 n instead of 2 n + 2 and with a more complicated dep end ence on the sp ectral parameter. Th e relatio n of these t wo Lax matrices is yet to b e inv estig ated. F or the dynamics (2.5), (2.6) w e ha v e an analog of the Lax equation (2.12): ˙ L ( v ) = [ A ( v ) , L ( v )] (5.5) with A ( v ) def ined as A ( v ) = n X j =1  X j E j j − E j,j +1 − X j E 2 n +2 − j, 2 n +2 − j + E 2 n +2 − j, 2 n +3 − j  + E n +1 ,n +1 + v E 2 n +2 , 1 − β 1 2 e 2 x 1  E 2 n +2 , 2 n +1 + v − 1 E 1 , 2 n +2  + β n 2 e − 2 x n  E n +1 ,n − E n +2 ,n  (5.6) and satisfying A ( v ) C v + C v A t ( v ) = 0 . (5.7) The an alog of the formula (3.1) for the B¨ ac klund transformation is M ( v , λ ) L ( v ; Y , y ) = L ( v ; X , x ) M ( v , λ ) , (5.8a) f M ( v , λ ) L ( v ; X , x ) = L ( v ; Y , y ) f M ( v , λ ) , (5.8b) where M ( v ) is giv en by M ( v ) = n X j,k =1 M j k E j k = − n X j =2 ξ j η j − 1 E j,j − 1 + n X j =1  s j +1 η j − ξ j s j  E j j + E j,j +1 (5.9) + n − 1 X j =1 η j +1 ξ j E 2 n +2 − j, 2 n +1 − j + n X j =1  s j +1 ξ j − η j s j  E 2 n +2 − j, 2 n +2 − j − E 2 n +2 − j, 2 n +3 − j +  α n ξ n + β n 2 ξ 2 n   E n +1 ,n − E n +2 ,n +1  + β n 2 ξ n ξ n − 1  E n +3 ,n − E n +2 ,n − 1  − E n +1 ,n +2 −  α 1 ξ 1 + β 1 ξ 2 1 2   E 2 n +2 , 2 n +1 + v − 1 E 1 , 2 n +2  + β 1 ξ 1 ξ 2 2 v  E 2 , 2 n +1 − E 1 , 2 n  − v E 2 n +2 , 1 , 12 V. Kuznetso v and E. S kly anin (using again the notatio n ξ j ≡ e x j , η j ≡ e y j ) and f M ( v ) is d ef ined as f M ( v ) ≡ C v M t ( v ) C − 1 v . (5.10) One of common w a ys to obtain a B¨ ac klund transform ation is from factorising a Lax matrix in tw o dif f er ent w a ys, see [18] for T o da lattices and [13] for other integrable mo dels. F or our mo del w e also ha v e a r emark able factorisation, only instead of L ( v ) we h a v e to tak e its square: λ 2 − L 2 ( v ; X , x ) = M ( v , λ ) f M ( v , λ ) , (5.11a ) λ 2 − L 2 ( v ; Y , y ) = f M ( v , λ ) M ( v , λ ) . (5.11b) 6 Discussion The metho d for constructing a B¨ ac klund transf orm ation presen ted in this pap er seems to b e quite general and app licable as well to other integ rable sl (2)-t yp e c hains with the b oundary conditions treata ble within the framework develo p ed in [3, 4]. There is little doubt that a similar BT can b e constructed for the D -t yp e T o da latti ce and a m ore general Inozem tsev’s T o da lattice [5] with the b ound ary terms like a 1 sinh 2 x 1 2 + b 1 sinh 2 x 1 + a n sinh 2 x n 2 + b n sinh 2 x n since those, as sho wn in [20], can also b e describ ed in the formalism b ased on the b oundary K matrices (2.1) and the Poisson algebra (2.27). The ‘big’ Lax matrix L ( v ) s till aw aits a prop er Lie-algebraic in terpretatio n. Ob taining a BT from th e factorisation of λ 2 − L 2 lik e in (5.11) might prov e to b e useful for other integ rable systems r elated to classical Lie algebras. It is we ll kno wn that the quant um analog of a BT is the s o-called Q-op erator [21], see also [9]. Examples of Q -op erators for quantum in tegrable chains with a b oundary hav e b een constructed recen tly for the XXX magnet [12 ] and f or the T o da lattices of B, C and D t yp es [22]. O ur results f or the BC-T o da lattice agree with those of [22], the generating fu n ction of the BT b eing a classical limit of the ke rnel of the Q -op erator. Hop efully , our results will help to construct the Q -op erator for the general 4-parametric quantum BC-T o da lattice. A Pro of of canonicit y Here we adapt to the BC-T o da case the argument from [10] d ev eloped originally for the p erio dic case. The tric k is to obtain the graph Γ 2 n of the BT as a pro jection of another manifold in a b igger s y m plectic sp ace, the men tioned manifold b eing Lagrangian for trivial reason. Consider the 8-dimens ional symplectic space W 8 with co ord inates X xY y S sT t and the sym- plectic form ω 8 ≡ d X ∧ d x + d S ∧ d s − d Y ∧ d y − d T ∧ d t. (A.1) The m atrix relatio n M ( u, λ ; T , t ) ℓ ( u ; Y , y ) = ℓ ( u ; X, x ) M ( u, λ ; S, s ) (A.2) is equiv alen t to 4 relations X = − λ + s − 1 e x + t e − x , (A.3a) B¨ ac klund T ransformation for the BC-T yp e T o da Lattice 13 Y = λ − s − 1 e y − t e − y , (A.3b) S = 2 λs − 1 − s − 2 e x − s − 2 e y , (A.3 c) T = e − x + e − y , (A.3d) def ining a 4-dimensional submanif old G 4 ⊂ W 8 . T he fact that G 4 is Lagrangian, that is ω 8 | G 4 = 0, is pro v ed by presen ting explicitly the generating function f λ ( y , t ; x, s ) = λ (2 ln s − x − y ) + s − 1 (e x + e y ) − t (e − x + e − y ) , (A.4) suc h th at X = ∂ f λ ∂ x , S = ∂ f λ ∂ s , Y = − ∂ f λ ∂ y , T = − ∂ f λ ∂ t . (A.5) An alternativ e pr o of [10] is based on the fact that ℓ ( u ) and M ( u, λ ) are s ymplectic lea ves of the same P oisson alge bra (2.23). Relation (A.2) def ines thus a canonical transformation fr om X xS s to Y y T t . Let us tak e n copies W ( j ) 8 of W 8 decorating the v ariables X xY y S sT t with the in dices j = 1 , . . . , n and imp ose on them n matrix r elations obtained from (A.2) by addin g sub script j to all v ariables. W e obtain then a Lagrangian manifold G 4 n = ⊗ n j =1 G ( j ) 4 in the 8 n -dimensional sym- plectic space W 8 n = ⊕ n j =1 W ( j ) 8 with the symplectic form ω 8 n = n P j =1 ω ( j ) 8 and th e corresp ondin g canonical transformation with the generating fun ction n P j =1 f λ ( y j , t j ; x j , s j ). Let us also introd uce 4 add itional v ariables T 0 , t 0 and S n +1 , s n +1 serving as co ordinates in the 4-dimensional sy m plectic space W 4 with the s y m plectic form ω 4 ≡ d S n +1 ∧ d s n +1 − d T 0 ∧ d t 0 . The r elations T 0 = 2( α 1 + β 1 λt 0 ) 1 + β 1 t 2 0 , S n +1 = 2( λs n +1 − α n ) β n + s 2 n +1 (A.6) def ine then a 2-dimensional Lagrangian submanifold G 2 ⊂ W 4 c haracterised by the generat- ing fun ction ϕ = ϕ (0) λ ( t 0 ) + ϕ ( n +1) λ ( s n +1 ) with ϕ (0) λ and ϕ ( n +1) λ def ined by (4.5b) and (4.5c), resp ectiv ely: T 0 = − ∂ ϕ λ ∂ t 0 , S n +1 = ∂ ϕ λ ∂ s n +1 . (A.7) W e end up with the (8 n + 4)-dimensional sym plectic space W 8 n +4 = W 8 n + W 4 , symp lectic form ω 8 n +4 = ω 8 n + ω 4 , and the (4 n + 2)-dimensional Lagrangian submanifold G 4 n +2 = G 4 n × G 2 ⊂ W 8 n +4 def ined b y the generating function F λ = ϕ (0) λ ( t 0 ) + ϕ ( n +1) λ ( s n +1 ) + n X j =1 f λ ( y j , t j ; x j , s j ) . (A.8) The f inal step is to imp ose 2 n + 2 constr aints T j = S j +1 , t j = s j +1 , j = 0 , . . . , n , (A.9) whic h def in e a su bspace W 6 n +2 ⊂ W 8 n +4 of dimension (8 n + 4) − (2 n + 2) = 6 n + 2 and resp ectiv e 2 n -dimensional submanifold G 2 n = G 4 n +2 ∩ W 6 n +2 . Constrain ts (A.9) allo w to eliminate the v ariables T t . The sp ace W 6 n +2 splits then in to the direct sum W 6 n +2 = V 4 n + W 2 n +2 of the space W 4 n with co ordinates X j x j Y j y j ( j = 1 , . . . , n ) 14 V. Kuznetso v and E. S kly anin and W 2 n +2 with co ordinates S j s j ( j = 1 , . . . , n + 1). Using (A.9) w e obtain that d T j ∧ d t j − d S j +1 ∧ d s j +1 = 0 and therefore the symplectic form ω 8 n +4 restricted on W 6 n +2 ω 8 n +4 | W 6 n +2 = n X j =1  d X j ∧ d x j − d Y j ∧ y j  , (A.10) degenerates: it v anishes on W 2 n +2 and remains nond egenerate on V 4 n . In fact, on V 4 n the form ω 8 n +4 coincides with the standard symplectic form (4.1). ω 8 n +4 | V 4 n = Ω 4 n . (A.11) After the elimination of the v ariables T t fr om equations (A.3) and (A.6), th e resulting s et of equations def inin g the submanifold G 2 n = G 4 n +2 ∩ W 6 n +2 ⊂ W 6 n +2 coincides with equa- tions (3.1 3) and (3.14 ) def ining the BT. As w e ha v e seen in Section 3, the v a riables S j s j can also b e eliminated lea ving a 2 n dimen- sional su bmanifold Γ 2 n ⊂ V 4 n coinciding with th e graph of the BT discussed in S ection 4.2. By construction, Γ 2 n is the p ro jection of G 2 n from W 6 n +2 on to V 4 n parallel to W 2 n +2 . F urther- more, Γ 2 n is Lagrangian since ω 8 n +4 v anishes on G 4 n +2 , therefore on G 2 n = G 4 n +2 ∩ W 6 n +2 , and therefore on Γ 2 n . The canonicit y of the BT is thus established geometrica lly , w ithout tedious calculatio ns. The same argument as in [10 ] sho ws that the generating fun ction Φ λ of the Lagrangian submanifold Γ 2 n is obtained by setting t j = s j +1 in (A.8), which pro du ces form ula (4.4). B Pro of of sp ectralit y Here w e pro vide the pro of of formulae (4.8 ) for the eige n v alues of L ( λ ). F or th e p ro of w e use an observ ation from [10] and sho w that the eigen v ec tors of L ( λ ) are give n by null-v ectors of M 1 ( ± λ, λ ). After setting u = − λ in (3.12) the m atrix M j b ecomes a pro j ector M j ( − λ, λ ) =  − 2 λ + s j S j s 2 j S j − 2 λs j S j s j S j  =  − 2 λ + s j S j S j  (1 s j ) (B.1) with th e n ull-v ec tor σ j ≡  − s j 1  , M j ( − λ, λ ) σ j = 0 . (B.2) Let us set u = − λ in the matrix equalit y (3.1) and apply it to the v ecto r σ 1 . By (B.2), the righ t-hand side giv es 0. Therefore, L ( − λ ) σ 1 should b e p rop ortional to the same null-v ec tor σ 1 of M j ( − λ, λ ), and σ 1 is an eigen v ecto r of L ( − λ ). T o f ind the corresp on d ing eigen v alue Λ, use the factorised expression (2.1) of L ( − λ ) and apply it to σ 1 . Using (2.3) we obtain ℓ ( − λ ; Y j , y j ) σ j = − s j e − y j σ j +1 , (B.3) hence T ( u ; Y , y ) σ 1 = σ n +1 n Y j =1  − s j e − y j  . (B.4) F rom (3.5) and (3.12) w e ob tain M a j ( u, λ ) =  − u − λ + s j S j − S j 2 λs j − s 2 j S j u − λ + s j S j  , (B .5) B¨ ac klund T ransformation for the BC-T yp e T o da Lattice 15 hence M a j ( λ, λ ) =  − 2 λ + s j S j − S j (2 λs j − s 2 j ) S j s j S j  =  − 1 s j  (2 λ − s j S j S j ) , (B .6) the corresp onding null-v ec tor b eing e σ j ≡  S j s j S j − 2 λ  , M a j e σ j = 0 . (B.7) A direct calculation using (2.3) and (3.13 d) yields ℓ t j ( λ ; Y j , y j ) e σ j +1 = s j e − x j e σ j (B.8) and, consequ ently , T t ( λ ; Y , y ) e σ n +1 = e σ 1 n Y j =1  s j e − x j  . (B.9) F rom (2.4) we get, resp ective ly , the iden tities K + ( − λ ) σ n +1 = 1 2 ( β n + s 2 n +1 ) e σ n +1 , K − ( − λ ) e σ 1 = 2 α 2 1 + β 1 λ 2 1 + β 1 s 2 1 e σ 1 . (B.10) Using the ab o v e formulae w e are able to mo v e σ 1 through all the factors constituting L ( − λ ) and obtain the equalit y L ( − λ ; Y , y ) σ 1 = Λ σ 1 , (B.11) where Λ is giv en b y (4.8a). Note that Λ is an eige n v alue of L ( λ ) as w ell since Λ( λ ) = Λ( − λ ). The s econd eigen v alue e Λ (4.8b) of L ( λ ) is obtained from Λ e Λ = det L ( λ ) ≡ d ( λ ) = ( α 2 n + β n λ 2 )( α 2 1 + β 1 λ 2 ) , (B.12) see (2.13 ). C Deriv ation of the dual Lax matrix T o construct the ‘b ig’ Lax op erator L ( v ) fr om the ‘small’ one L ( u ) w e u se the tec hnique dev elop ed for the p erio dic the p erio dic T o da lattice [10, 19], with the necessary corrections to accommo date the b oundary conditions. Let θ 1 b e an eigen v ecto r of L ( u ) with th e eigen v alue v : L ( u ) θ 1 = v θ 1 , θ 1 =  ϕ 1 ψ 1  . (C.1) Reading off the factors constituting the pro duct L ( u ), see (2.1), (2.2), def ine r ecur siv ely the v ectors θ j θ j =  ϕ j ψ j  , j = 1 , . . . 2 n + 2 , (C.2) b y the relations θ j +1 = ℓ ( u ; X j , x j ) θ j , j = 1 , . . . , n , (C.3a) 16 V. Kuznetso v and E. S kly anin θ n +2 = K + ( u ) θ n +1 , (C.3b) θ n + j +3 = ℓ t ( − u ; X n − j , x n − j ) θ n + j +2 , j = 0 , . . . , n − 1 , (C.3c) and close the circuit with the equation v θ 1 = K − ( u ) θ 2 n +2 , (C.3d) whic h is equiv ale n t to (C.1). A recursiv e elimination of ψ j results in the equations uϕ 1 = ϕ 2 − X 1 ϕ 1 +  α 1 v e x 1 + β 1 v e 2 x 1  ϕ 2 n +2 − β 1 v e 2 x 1 X 1 ϕ 2 n +1 + β 1 v e x 1 + x 2 ϕ 2 n , (C.4a) uϕ j = ϕ j +1 − X j ϕ j + e x j − x j − 1 ϕ n − 1 , j = 2 , . . . , n (C.4b) uϕ n +1 = ϕ n +2 + α n e − x n ϕ n , (C.4c) uϕ n +2 = − ϕ n +3 + X n ϕ n +2 + ( α n e − x n + β n e − 2 x n ) ϕ n +1 − β n e − 2 x n X n ϕ n + β n e − x n − x n − 1 ϕ n − 1 , (C.4d) uϕ j = − ϕ j +1 + X 2 n +2 − j ϕ j − e x j − 3 − x j − 4 ϕ j − 1 , j = n + 3 , . . . , 2 n + 1 , (C.4e) uϕ 2 n +2 = v ϕ 1 − α 1 e x 1 ϕ 2 n +1 . (C.4f ) In order to simplify the 6-terms relations (C.4a) and (C.4d) and mak e the matrix L ( v ) more symmetric w e p erform an additional rev ersible c hange of v ariables ϕ 1 = e ϕ 1 + β 1 2 v e 2 x 1 e ϕ 2 n +1 , (C.5a) ϕ j = e ϕ j , j = 2 , . . . , n + 1 , (C.5b) ϕ n +2 = e ϕ n +2 + β n 2 e − 2 x n e ϕ n , (C.5c) ϕ j = − e ϕ j , j = n + 3 , . . . , 2 n + 2 . (C.5d) The r esulting equations for e ϕ j read u e ϕ 1 = e ϕ 2 − X 1 e ϕ 1 −  α 1 v e x 1 + β 1 2 v e 2 x 1  e ϕ 2 n +2 − β 1 2 v e x 1 + x 2 e ϕ 2 n , (C.6a) u e ϕ 2 = e ϕ 3 − X 2 e ϕ 2 + e x 2 − x 1 e ϕ 1 + β 1 2 v e 2 x 1 e ϕ 2 n +1 , (C.6b) u e ϕ j = e ϕ j +1 − X j e ϕ j + e x j − x j − 1 e ϕ n − 1 , j = 3 , . . . , n (C.6c) u e ϕ n +1 = − e ϕ n +2 +  α n e − x n + β n 2 e − 2 x n  e ϕ n , (C.6d) u e ϕ n +2 = − e ϕ n +3 + X n e ϕ n +2 −  α n e − x n + β n 2 e − 2 x n  e ϕ n +1 − β n 2 e − x n − x n − 1 e ϕ n − 1 , (C.6e) u e ϕ n +3 = − e ϕ n +4 + X n − 1 e ϕ n +3 − e x n − x n − 1 e ϕ n +2 + β n 2 e − 2 x n , (C.6f ) u e ϕ j = − e ϕ j +1 + X 2 n +2 − j e ϕ j − e x j − 3 − x j − 4 e ϕ j − 1 , j = n + 4 , . . . , 2 n + 1 , (C.6g) u e ϕ 2 n +2 = − v e ϕ 1 −  α 1 e x 1 + β 1 2 e 2 x 1  e ϕ 2 n +1 . 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