Global action-angle variables for the periodic Toda lattice

Global action-angle v ariables for the p erio dic T o da lattice Andreas Henrici Thomas Kapp eler ∗ No v ember 7, 20 21 Abstract In this paper we construct glob al action-angle v ariables for the perio dic T oda lattice. 1 1 In tro duction Consider the T oda lattice w ith per iod N ( N ≥ 2), ˙ q n = ∂ p n H, ˙ p n = − ∂ q n H for n ∈ Z , where the (rea l) co or dina tes ( q n , p n ) n ∈ Z satisfy ( q n + N , p n + N ) = ( q n , p n ) for any n ∈ Z and the Hamiltonia n H T oda is given b y H T oda = 1 2 N X n =1 p 2 n + α 2 N X n =1 e q n − q n +1 (1) where α is a p ositive par ameter, α > 0. F o r the sta nda rd T oda lattice, α = 1 . The T oda lattice was in tro duced by T o da [19] and studied extensively in the sequel. It is an imp ortant model for an integrable system of N particles in one space dimension with nearest neighbor interaction and belo ngs to the family of lattices introduce and n umerically inv estigated b y F ermi, Pasta, and Ulam in their semina l pap er [5]. T o prov e the integrability of the T o da lattice, Fla sc hk a int ro duced in [3] the (noncano nical) co or dinates b n := − p n ∈ R , a n := αe 1 2 ( q n − q n +1 ) ∈ R > 0 ( n ∈ Z ) . These co ordinates describ e the motion of the T o da lattice r e la tiv e to the cen ter of mass. Note that the total momentum is co nserved by the T o da ﬂow, hence any tra jectory of the center o f mass is a straight line. ∗ Supported i n part by the Swi s s N ational Science F oundation, the pr ogramme SPECT, and the Europ ean Communit y through the FP6 Marie Cur ie R TN EN IGMA (MR TN- CT- 2004-5652) 1 2000 Mathematics Sub ject Classiﬁcation: 37J35, 39A12, 39A70, 70H06 1 2 1 INTR ODUCTION In these co ordina tes the Hamiltonian H T oda takes the simple form H = 1 2 N X n =1 b 2 n + N X n =1 a 2 n , and the equations of mo tion ar e  ˙ b n = a 2 n − a 2 n − 1 ˙ a n = 1 2 a n ( b n +1 − b n ) ( n ∈ Z ) . (2) Note that ( b n + N , a n + N ) = ( b n , a n ) for any n ∈ Z , and Q N n =1 a n = α N . Hence we can identify the s e quences ( b n ) n ∈ Z and ( a n ) n ∈ Z with the v ectors ( b n ) 1 ≤ n ≤ N ∈ R N and ( a n ) 1 ≤ n ≤ N ∈ R N > 0 . Our aim is to study the normal form of the system of equations (2) on the phas e spa c e M := R N × R N > 0 . This s y stem is Hamiltonia n with resp ect to the nonsta ndard Poisson structure J ≡ J b,a , deﬁned at a p oint ( b, a ) = (( b n , a n ) 1 ≤ n ≤ N by J =  0 A − t A 0  , (3) where A is the b -indep endent N × N -matrix A = 1 2          a 1 0 . . . 0 − a N − a 1 a 2 0 . . . 0 0 − a 2 a 3 . . . . . . . . . . . . . . . . . . 0 0 . . . 0 − a N − 1 a N          . (4) The Poisson bracket co rresp onding to (3) is then given by { F , G } J ( b, a ) = h ( ∇ b F, ∇ a F ) , J ( ∇ b G, ∇ a G ) i R 2 N = h∇ b F, A ∇ a G i R N − h∇ a F, A t ∇ b G i R N . (5) where F , G ∈ C 1 ( M ) and wher e ∇ b and ∇ a denote the gradients with res pect to the N - v ectors b = ( b 1 , . . . , b N ) and a = ( a 1 , . . . , a N ), resp ectively . Ther efore, equations (2) can alter nativ ely be written as ˙ b n = { b n , H } J , ˙ a n = { a n , H } J (1 ≤ n ≤ N ). F ur ther no te that { b n , a n } J = a n 2 ; { b n +1 , a n } J = − a n 2 , (6) while { b n , a k } J = 0 for any n, k with n / ∈ { k , k + 1 } . 3 Since the ma tr ix A deﬁned by (4) ha s r a nk N − 1, the Poisson structure J is degenerate. It admits the tw o Casimir functions 2 C 1 := − 1 N N X n =1 b n and C 2 := N Y n =1 a n ! 1 N (7) whose g radients ∇ b,a C i = ( ∇ b C i , ∇ a C i ) ( i = 1 , 2 ), given b y ∇ b C 1 = − 1 N (1 , . . . , 1) , ∇ a C 1 = 0 , (8) ∇ b C 2 = 0 , ∇ a C 2 = C 2 N  1 a 1 , . . . , 1 a N  , (9) are linearly indep endent at each po in t ( b, a ) of M . The main result o f this pap er ist the following one: Theorem 1 .1. The p erio dic T o da lattic e admits glob al ly deﬁn e d action-angle variables. Mor e pr e cisely: (i) T her e exist r e al analytic functions ( I n ) 1 ≤ n ≤ N − 1 on M which ar e p airwise in involution and which Poisson c ommut e with the T o da Hamiltonian H and the two Casimir functions C 1 , C 2 , i.e. for any 1 ≤ m, n ≤ N − 1 , i = 1 , 2 , { I m , I n } J = 0 on M and { H , I n } J = 0 and { C i , I n } J = 0 on M . (ii) F or any 1 ≤ n ≤ N − 1 ther e exist a r e al analytic submanifold D n of c o dimension 2 and a function θ n : M \ D n → R , deﬁne d mo d 2 π and r e al analytic whe n c onsider e d mo d π , so that on M \ S N − 1 n =1 D n , ( θ n ) 1 ≤ n ≤ N − 1 and ( I n ) 1 ≤ n ≤ N − 1 ar e c onjugate varia bles. Mor e pr e cisely, fo r any 1 ≤ m, n ≤ N − 1 , i = 1 , 2 { I m , θ n } J = δ mn and { C i , θ n } J = 0 on M \ D n and { θ m , θ n } J = 0 on M \ ( D m ∪ D n ) . Let M β ,α := { ( b, a ) ∈ R 2 N : ( C 1 , C 2 ) = ( β , α ) } denote the level set of ( C 1 , C 2 ) for ( β , α ) ∈ R × R > 0 . Note that ( − β 1 N , α 1 N ) ∈ M β ,α where 1 N = (1 , . . . , 1 ) ∈ R N . As the g radients ∇ b,a C 1 and ∇ b,a C 2 are linea rly indep enden t everywhere on M , the sets M β ,α are (real analytic) submanifolds o f M of co di- mension tw o. F urthermore the Poisson structur e J , restricted to M β ,α , b ecomes nondegenera te everywhere on M β ,α and therefore induces a symplectic structure ν β ,α on M β ,α . In this wa y , we obtain a symplectic foliation of M with M β ,α being the symplectic leaves. 2 A smo oth function C : M → R i s a Casimi r function for J if { C, ·} J ≡ 0. 4 1 INTR ODUCTION Corollary 1.2 . On e ach symple ctic le af M β ,α , the action variables ( I n ) 1 ≤ n ≤ N − 1 ar e a max imal set of functional ly indep endent inte gr als in involution of the p e- rio dic T o da lattic e. In subsequent work [9], we will use Theorem 1.1 to construct glob al Bir khoﬀ co ordinates for the per iodic T o da lattice. More precisely , we in tro duce the mo del space P := R 2( N − 1) × R × R > 0 endow ed with the degener ate Poisson structur e J 0 whose symplectic leav es are R 2( N − 1) × { β } × { α } endow ed with the standard Poisson structure, a nd prove the following theorem: Theorem 1. 3. Ther e ex ists a re al analytic, c anonic al diﬀe omorphism Ω : ( M , J ) → ( P , J 0 ) ( b, a ) 7→ (( x n , y n ) 1 ≤ n ≤ N − 1 , C 1 , C 2 ) such that t he c o or dinates ( x n , y n ) 1 ≤ n ≤ N − 1 , C 1 , C 2 ar e glob al Birkhoﬀ c o or dinates for the p erio dic T o da lattic e, i.e. ( x n , y n ) 1 ≤ n ≤ N − 1 ar e c anonic al c o or dinates, C 1 , C 2 ar e the Casimirs and the t r ansforme d T o da Hamiltonian ˆ H = H ◦ Ω − 1 is a function of t he actions ( I n ) 1 ≤ n ≤ N − 1 and C 1 , C 2 alone. In [10 ] we used Theor em 1.3 to obtain a KAM theorem for Hamiltonian per turbations of the p erio dic T o da lattice. R elate d work: T heo rem 1 .1 and Theorem 1 .3 improv e on earlier w ork on the normal for m of the p erio dic T o da lattice in [1, 2]. In particular, w e c onstruct global Bir khoﬀ co ordinates on a ll of M instead of a single sy mplectic leaf a nd show that techniques recent ly dev elo ped for treating the KdV equation (cf. [11, 12]) and the defocusing NLS equation (cf. [8, 16]) can a ls o b e a pplied for the T o da lattice. Outline of the p ap er: In section 2 we review the Lax pair of the p erio dic T o da lattice and colle ct some auxiliary results on the sp ectrum of the Jaco bi matrix L ( b, a ) asso ciated to an element ( b, a ) ∈ M . In section 3 we study the action v aria bles ( I n ) 1 ≤ n ≤ N − 1 , and in section 4 we deﬁne the angle v ariables ( θ n ) 1 ≤ n ≤ N − 1 on M \ ∪ n n =1 D n using ho lomorphic diﬀerentials deﬁned on the hyperelliptic Riemann surfa ce ass ocia ted to the sp ectrum of L ( b, a ). In sections 5 and 6 we establish formulas of the gradients o f the actio ns a nd angle s in terms of pro ducts of fundamen tal s olutions a nd pr o ve orthogona lit y relations b et ween such pro ducts which a re then us e d in section 7 to show tha t ( I n ) 1 ≤ n ≤ N − 1 and ( θ n ) 1 ≤ n ≤ N − 1 are canonical v ariables a nd to prov e Theore m 1.1 a nd Cor ollary 1.2. 5 2 Preliminaries It is well known (cf. e.g. [19]) that the system (2) admits a Lax pair formulation ˙ L = ∂ L ∂ t = [ B , L ], where L ≡ L + ( b, a ) is the p erio dic Jaco bi matrix deﬁned by L ± ( b, a ) :=          b 1 a 1 0 . . . ± a N a 1 b 2 a 2 . . . . . . 0 a 2 b 3 . . . 0 . . . . . . . . . . . . a N − 1 ± a N . . . 0 a N − 1 b N          , (10) and B the skew-symmetric matrix B =          0 a 1 0 . . . − a N − a 1 0 a 2 . . . . . . 0 − a 2 . . . . . . 0 . . . . . . . . . . . . a N − 1 a N . . . 0 − a N − 1 0          . Hence the ﬂow of ˙ L = [ B , L ] is isos pectra l. Prop osition 2.1. F or a solution  b ( t ) , a ( t )  of the p erio dic T o da lattic e (2), the eigenvalues ( λ + j ) 1 ≤ j ≤ N of L  b ( t ) , a ( t )  ar e c onserve d quantities. Let us now collect a few res ults from [1 7] and [19] of the sp ectral theory of Jacobi matrices ne e ded in the sequel. Denote by M C the co mplexiﬁcation of the phase space M , M C = { ( b, a ) ∈ C 2 N : Re a j > 0 ∀ 1 ≤ j ≤ N } . F or ( b, a ) ∈ M C we cons ider for any c o mplex num b er λ the diﬀerence equatio n ( R b,a y )( k ) = λy ( k ) ( k ∈ Z ) (11) where y ( · ) = y ( k ) k ∈ Z ∈ C Z and R b,a is the diﬀerence op erato r R b,a = a k − 1 S − 1 + b k S 0 + a k S 1 (12) with S m denoting the shift o pera tor of or der m ∈ Z , i.e. ( S m y )( k ) = y ( k + m ) for k ∈ Z . F u n damental solutions: The tw o fundamen tal solutions y 1 ( · , λ ) a nd y 2 ( · , λ ) of (11) ar e deﬁned by the standa r d initial co nditio ns y 1 (0 , λ ) = 1, y 1 (1 , λ ) = 0 and y 2 (0 , λ ) = 0, y 2 (1 , λ ) = 1. They s a tisfy the Wr onskian identity W ( n ) := y 1 ( n, λ ) y 2 ( n + 1 , λ ) − y 1 ( n + 1 , λ ) y 2 ( n, λ ) = a N a n . (13) 6 2 PREL IMINARIES Note that for n = N one gets W ( N ) = 1 . (14) F or each k ∈ N , y i ( k , λ, b, a ), i = 1 , 2, is a p olynomial in λ of degree at mos t k − 1 and dep e nds r eal analytically on ( b , a ) (see [17]). In par ticular, one ea sily veriﬁes that y 2 ( N + 1 , λ, b, a ) is a p olynomial in λ of deg ree N with lea ding co eﬃcien t α − N . Wr onskian: More genera lly , one deﬁnes for any t wo sequences ( v ( n )) n ∈ Z and ( w ( n )) n ∈ Z the W rons k ian sequence ( W ( n )) n ∈ Z = ( W ( v , w )( n )) n ∈ Z by W ( n ) := v ( n ) w ( n + 1) − v ( n + 1) w ( n ) . Let us recall the following pr o perties of the W ronskian, which can be easily veriﬁed. Lemma 2. 2. (i) If y and z ar e s olut ions of (11) for λ = λ 1 and λ = λ 2 , r esp e ctively, t hen W = W ( y , z ) satisﬁes for any k ∈ Z a k W ( k ) = a k − 1 W ( k − 1 ) + ( λ 2 − λ 1 ) y ( k ) z ( k ) . (15) (ii) If y ( · , λ ) is a 1 -p ar ameter-family of solutions of (11) which is c ontinuously diﬀer entiable with r esp e ct to the p ar ameter λ and ˙ y ( k , λ ) := ∂ ∂ λ y ( k , λ ) , then W = W ( y , ˙ y ) satisﬁes for any k ∈ Z a k W ( k ) = a k − 1 W ( k − 1) + y ( k , λ ) 2 . (16) Discriminant: W e denote by ∆( λ ) ≡ ∆( λ, b, a ) the discriminant of (11), deﬁned by ∆( λ ) := y 1 ( N , λ ) + y 2 ( N + 1 , λ ) . (17) In the sequel, we will often write ∆ λ for ∆( λ ). Note that y 2 ( N + 1 , λ ) is a p olynomial in λ of degree N with leading term α − N λ N , whereas y 1 ( N , λ ) is a p olynomial in λ of degree less than N , hence ∆( λ, b, a ) is a poly nomial in λ o f degree N with leading term α − N λ N , and it dep ends real a nalytically on ( b, a ) (see e.g. [19]). According to Floquet’s Theo r em (see e.g. [18]), for λ ∈ C g iven, (1 1) admits a p erio dic or antiperio dic solution of p erio d N if the discriminant ∆( λ ) sa tisﬁes ∆( λ ) = +2 or ∆( λ ) = − 2, resp e ctiv ely . (These solutions corre spond to eigenvectors o f L + or L − , res p ectively , with L ± deﬁned by (10).) It turns out to b e more con venien t to combine these t wo cases by 7 l 1 l 2 l 3 l 4 l 5 l 6 l 7 l 8 (a) N = 4 l 1 l 2 l 3 l 4 l 5 l 6 l 7 l 8 l 9 l 10 (b) N = 5 Figure 1: Ex amples of the discriminant ∆( λ ) considering the p erio dic J acobi matrix Q ≡ Q ( b, a ) of s ize 2 N deﬁned by Q =                   b 1 a 1 . . . 0 0 . . . 0 a N a 1 b 2 . . . . . . 0 . . . 0 . . . . . . . . . a N − 1 . . . . . . 0 . . . a N − 1 b N a N . . . 0 0 0 . . . 0 a N b 1 a 1 . . . 0 0 . . . 0 a 1 b 2 . . . . . . . . . . . . . . . . . . . . . a N − 1 a N . . . 0 0 0 . . . a N − 1 b N                   . Then the s pectrum of the matr ix Q is the union of the sp ectra of the ma trices L + and L − and ther efore the zero s et of the p olynomial ∆ 2 λ − 4. The function ∆ 2 λ − 4 is a p olynomial in λ o f degr ee 2 N and a dmits a pro duct r epresentation ∆ 2 λ − 4 = α − 2 N 2 N Y j =1 ( λ − λ j ) . (18) The factor α − 2 N in (1 8) comes from the ab ov e mentioned fact that the leading term of ∆( λ ) is α − N λ N . F or any ( b, a ) ∈ M , the matrix Q is symmetric and hence the eigenv alues ( λ j ) 1 ≤ j ≤ 2 N of Q ar e r eal. When listed in increa sing o rder and w ith their alge- braic m ultiplicities, they satisfy the following rela tions (cf. [1 7]) λ 1 < λ 2 ≤ λ 3 < λ 4 ≤ λ 5 < . . . λ 2 N − 2 ≤ λ 2 N − 1 < λ 2 N . As ex pla ined a bove, the λ j are per iodic or ant ip erio dic eige nv alues o f L and th us eigenv alues of L + or L − according to whether ∆( λ ) = 2 o r ∆( λ ) = − 2. One has (cf. [17]) ∆( λ 1 ) = ( − 1 ) N · 2 , ∆( λ 2 n ) = ∆( λ 2 n +1 ) = ( − 1 ) n + N · 2 , ∆( λ 2 N ) = 2 . (19) 8 2 PREL IMINARIES Since ∆ λ is a p olynomial of degree N , ˙ ∆ λ ≡ ˙ ∆( λ ) = d dλ ∆( λ ) is a p olynomia l of degree N − 1 , a nd it admits a pro duct repres en tation o f the form ˙ ∆ λ = N α − N N − 1 Y k =1 ( λ − ˙ λ k ) . (20) The zero es ( ˙ λ n ) 1 ≤ n ≤ N − 1 of ˙ ∆ λ satisfy λ 2 n ≤ ˙ λ n ≤ λ 2 n +1 for any 1 ≤ n ≤ N − 1. The in terv a ls ( λ 2 n , λ 2 n +1 ) a re referre d to as the n -th sp e ctr al gap and γ n := λ 2 n +1 − λ 2 n as the n -th gap length . Note that | ∆( λ ) | > 2 on the sp ectral gaps. W e say that the n -th gap is op en if γ n > 0 and c ol lapse d otherwise. The set of elements ( b, a ) ∈ M for which the n -th gap is co llapsed is denoted b y D n , D n := { ( b, a ) ∈ M : γ n = 0 } . (21) By writing the condition γ n = 0 as γ 2 n = 0 and exploiting the fact that γ 2 n (unlik e γ n ) is a real analytic function on M , it ca n b e shown as in [12] that D n is a real analytic submanifold of M of co dimension 2. Isolating neigh b orho o ds: Let ( b, a ) ∈ M b e given. The stric t inequality λ 2 n − 1 < λ 2 n guarantees the existence o f a family o f mutually disjoin t op en subsets ( U n ) 1 ≤ n ≤ N − 1 of C s o that for any 1 ≤ n ≤ N − 1 , U n is a neig h bo rho od of the clo sed interv al [ λ 2 n , λ 2 n +1 ]. Suc h a family o f neig h bo rho ods is referred to as a family of isolating neighb orho o ds (for ( b, a )). In the case where ( b, a ) ∈ M C , we list the eigenv alues ( λ j ) 1 ≤ j ≤ 2 N in lexico - graphic ordering 3 λ 1 ≺ λ 2 ≺ λ 3 ≺ . . . ≺ λ 2 N . W e then extend the gap lenghts γ n to all of M C by γ n := λ 2 n +1 − λ 2 n (1 ≤ n ≤ N − 1 ) and deﬁne D C n := { ( b, a ) ∈ W : γ n = 0 } . (23) In the sequel, we will omit the sup erscript and alwa ys write D n for D C n . Similarly , we do this for the zero es ( ˙ λ n ) 1 ≤ n ≤ N − 1 of ˙ ∆ λ . The λ i ’s and ˙ λ i ’s no lo ng er dep end con tinuously on ( b, a ) ∈ M C . Howev er , if w e choo se a small enough complex neighborho o d W of M in M C , then for any ( b, a ) ∈ W the closed int erv a ls G n ⊆ C (1 ≤ n ≤ N − 1) deﬁned by G n := { (1 − t ) λ 2 n + tλ 2 n +1 : 0 ≤ t ≤ 1 } (24) are pairwise disjoint, and hence, as in the real case, ther e exists a family of isolating neighborho o ds ( U n ) 1 ≤ n ≤ N − 1 . 3 The lexicographic ordering a ≺ b for complex n umbers a and b is deﬁned by a ≺ b : ⇐ ⇒ 8 < : Re a < Re b or Re a = Re b and Im a ≤ Im b. (22) 9 Lemma 2.3. Ther e exists a neighb orho o d W of M in M C such that for any ( b, a ) ∈ W , ther e ar e neighb orho o ds U n of G n in C ( 1 ≤ n ≤ N − 1 ) which ar e p airwise disjoint. Remark 2. 4. In the se quel, we wil l have to shrink the c omplex neighb orho o d W sever al times, but c ontinu e t o denote it by the same letter. Contours Γ n : F or any ( b, a ) ∈ W and any 1 ≤ n ≤ N − 1, we denote by Γ n a circuit in U n around G n with counterclo ckwise or ien tation. Isosp e ct ra l set: F or ( b, a ) ∈ M , the set Iso( b, a ) of all elements ( b ′ , a ′ ) ∈ M so tha t Q ( b ′ , a ′ ) has the same sp ectrum a s Q ( b, a ) is describ ed with the help of the Dirichlet eigenv alues µ 1 < µ 2 < . . . < µ N − 1 of (11) deﬁned by y 1 ( N + 1 , µ n ) = 0 . (25) They co incide with the eig en v alues of the ( N − 1) × ( N − 1 )-matrix L 2 = L 2 ( b, a ) given b y          b 2 a 2 0 . . . 0 a 2 . . . . . . . . . . . . 0 . . . . . . . . . 0 . . . . . . . . . . . . a N − 1 0 . . . 0 a N − 1 b N          . In the seq uel, w e will also refer to µ 1 , . . . , µ N − 1 as the Dirichlet eigenv alues of L ( b, a ). Ev alua ting the W ronsk ian iden tit y (1 3) at λ = µ n one sees that µ n lies in the closure of the n -th sp ectral gap. More precisely , substituting y 1 ( N + 1 , µ n ) = 0 in the identit y (13) with λ = µ n yields y 1 ( N , µ n ) y 2 ( N + 1 , µ n ) = 1 . (26) Hence the v alue o f the discr iminan t at µ n is given by ∆( µ n ) = y 2 ( N + 1 , µ n ) + 1 y 2 ( N + 1 , µ n ) (27) and | ∆( µ n ) | ≥ 2. By Lemma 2.6 below, given the p oint ( b, a ) with b 1 = . . . = b N = β and a 1 = . . . = a N = α , one has λ 2 n = λ 2 n +1 and hence µ n = λ 2 n for an y 1 ≤ n ≤ N − 1 . It then follows from a straightforw ard deformation argument that λ 2 n ≤ µ n ≤ λ 2 n +1 everywhere in the rea l spa c e M . Conv e rsely , accor ding to v an Mo erb eke [17], given any (re a l) Jacobi matrix Q with s pectrum λ 1 < λ 2 ≤ λ 3 < λ 4 ≤ λ 5 < . . . λ 2 N − 2 ≤ λ 2 N − 1 < λ 2 N and any sequence ( µ n ) 1 ≤ n ≤ N − 1 with λ 2 n ≤ µ n ≤ λ 2 n +1 for n = 1 , . . . , N − 1, there are exactly 2 r N -p erio dic Jacobi ma trices Q with s pectrum ( λ n ) 1 ≤ n ≤ 2 N and Dirichlet sp ectrum ( µ n ) 1 ≤ n ≤ N − 1 , where r is the num ber o f n ’s with λ 2 n < µ n < λ 2 n +1 . In the ca se where ( b , a ) ∈ M C , we contin ue to deﬁne the Dirichlet eigenv alues ( µ n ) 1 ≤ n ≤ N − 1 by (25), and we list them in lexicographic or dering µ 1 ≺ µ 2 ≺ 10 2 PREL IMINARIES . . . ≺ µ N − 1 . Then the µ i ’s no long er dep end contin uously on ( b, a ) ∈ M C . How ever, if we choose the complex neighborho o d W of M in M C of Lemma 2.3 small eno ugh, then fo r any ( b, a ) ∈ W and 1 ≤ n ≤ N − 1 , µ n is co n tained in the neighborho o d U n of G n (but not necessa r ily in G n itself ). Rie mann su rfac e Σ b,a : Denote by Σ b,a the Riemann sur fa ce obtained a s the compactiﬁcation of the aﬃne curve C b,a deﬁned by { ( λ, z ) ∈ C 2 : z 2 = ∆ 2 ( λ, b , a ) − 4 } . (28) Note that C b,a and Σ b,a are spectr al inv ariants. (Strictly spea king, Σ b,a is a Riemann surface only if the sp ectrum of Q ( b, a ) is simple - s ee e.g. App endix A in [18] for details in this ca se. If the spectr um of Q ( b, a ) is not simple, Σ( b, a ) bec omes a Riemann surface after doubling the multiple e ig en v alues - see e.g. section 2 of [13].) Dirichlet divisors: T o the Dirichlet eige nv alue µ n (1 ≤ n ≤ N − 1 ) w e asso ciate the po in t µ ∗ n on the surface Σ b,a , µ ∗ n :=  µ n , ∗ q ∆ 2 µ n − 4  with ∗ q ∆ 2 µ n − 4 := y 1 ( N , µ n ) − y 2 ( N + 1 , µ n ) , (29) where we used that, in view of (26) and the W r onskian identiy (14), ∆ 2 µ n − 4 =  y 1 ( N , µ n ) − y 2 ( N + 1 , µ n )  2 . Standar d r o ot: The s ta ndard ro ot or s -ro ot for s ho rt, s √ 1 − λ 2 , is deﬁned for λ ∈ C \ [ − 1 , 1 ] by s p 1 − λ 2 := iλ + p 1 − λ − 2 . (30) More g e nerally , we deﬁne for λ ∈ C \ { ta + (1 − t ) b | 0 ≤ t ≤ 1 } the s -ro ot of a radicand of the form ( b − λ )( λ − a ) with a ≺ b, a 6 = b by s p ( b − λ )( λ − a ) := γ 2 s p 1 − w 2 , (31) where γ := b − a , τ := b + a 2 and w := λ − τ γ / 2 . Canonic al she et and c anonic al r o ot: F or ( b, a ) ∈ M the canonical sheet of Σ b,a is g iv en by the set o f p oint s ( λ, c p ∆ 2 λ − 4) in C b,a , where the c -ro ot c p ∆ 2 λ − 4 is deﬁned on C \ S N n =0 ( λ 2 n , λ 2 n +1 ) (where λ 0 := −∞ and λ 2 N +1 := ∞ ) a nd determined by the sig n co nditio n − i c q ∆ 2 λ − 4 > 0 for λ 2 N − 1 < λ < λ 2 N . (32) As a consequence one ha s for a n y 1 ≤ n ≤ N sign c q ∆ 2 λ − i 0 − 4 = ( − 1) N + n − 1 for λ 2 n < λ < λ 2 n +1 . (33) The deﬁnition o f the ca nonical sheet and the c -ro ot can be extended to the neighborho o d W o f M in M C of Lemma 2.3. 11 The s - ro ot and the c -ro ot will b e used together in the following way: By the pro duct r epresentations (20) and (18) of ˙ ∆ λ and ∆ 2 λ − 4, res p ectively , one sees that for any ( b, a ) in W \ D n with 1 ≤ n ≤ N − 1, ˙ ∆ λ c p ∆ 2 λ − 4 = N ( λ − ˙ λ n ) s p ( λ 2 n +1 − λ )( λ − λ 2 n ) χ n ( λ ) ∀ λ ∈ Γ n (34) where χ n ( λ ) = ( − 1) N + n − 1 + p ( λ − λ 1 )( λ 2 N − λ ) Y m 6 = n λ − ˙ λ m + p ( λ − λ 2 m +1 )( λ − λ 2 m ) . (35) Note that the principal branches of the squa r e ro ots in (35) are well deﬁned for λ near G n and tha t the function χ n is a nalytic and non v anishing on U n . In addition, for ( b, a ) rea l, χ n is nonnega tiv e on the interv al ( λ 2 n , λ 2 n +1 ). Ab elian diﬀer entials: Let ( b, a ) ∈ M and 1 ≤ n ≤ N − 1. Then ther e exists a unique p olynomial ψ n ( λ ) of degree at most N − 2 such that for any 1 ≤ k ≤ N − 1 1 2 π Z c k ψ n ( λ ) p ∆ 2 λ − 4 dλ = δ kn . (36) Here, for any 1 ≤ k ≤ N − 1, c k denotes the lift of the contour Γ k to the canonical sheet of Σ b,a . F or any k 6 = n with λ 2 k 6 = λ 2 k +1 , it fo llo ws from (3 6) that 1 π Z λ 2 k +1 λ 2 k ψ n ( λ ) + p ∆ 2 λ − 4 dλ = 0 . (37) Hence in every gap ( λ 2 k , λ 2 k +1 ) with k 6 = n the p olynomial ψ n has a z ero which we deno te by σ n k . If λ 2 k = λ 2 k +1 then it follows fro m (36) and Cauch y’s theorem that σ n k = λ 2 k = λ 2 k +1 . As ψ n ( λ ) is a p o lynomial o f degr ee at most N − 2, one has ψ n ( λ ) = M n Y 1 ≤ k ≤ N − 1 k 6 = n ( λ − σ n k ) , (38) where M n ≡ M n ( b, a ) 6 = 0. In a straightforw ard w ay one can prov e that ther e ex is ts a neighborho o d W of M in M C , so that for any ( b , a ) ∈ W a nd any 1 ≤ n ≤ N − 1 , there is a unique po lynomial ψ n ( λ ) of degree a t most N − 2 satisfying (36) fo r a n y 1 ≤ k ≤ N − 1 as well as the pro duct r e present ation (3 8), and so that the zero es ar e analytic functions on W . Lemma 2.5. L et 1 ≤ n ≤ N − 1 b e ﬁxe d. Then the zer o es ( ˙ λ k ) 1 ≤ k ≤ N − 1 of ˙ ∆( λ ) and ( σ n k ) 1 ≤ k ≤ N − 1 ,k 6 = n of ψ n ( λ ) satisfy the estimates ˙ λ k − τ k = O ( γ 2 k ) , (39) σ n k − τ k = O ( γ 2 k ) . (40) ne ar any given p oint ( b , a ) ∈ W , wher e τ k = 1 2 ( λ 2 k +1 + λ 2 k ) . 12 2 PREL IMINARIES Pr o of. T o verify (39), wr ite ∆ 2 λ − 4 in the fo rm ∆ 2 λ − 4 = ( λ − λ 2 n )( λ 2 n +1 − λ ) p n ( λ ) (41) where p n is a p olynomial which do es not v anish for λ ∈ U n . Then (39) follows by diﬀerentiating (41) with resp ect to λ at ˙ λ n . Fix 1 ≤ k , n ≤ N − 1 with k 6 = n . In a ﬁrst s tep we prove that σ n k − τ k = O ( γ k ) near an y giv en p oint ( b, a ) ∈ W . If γ k = 0, then σ n k = τ k , and (40) is clearly fulﬁlled. Hence we as sume in the sequel tha t γ k 6 = 0. By the pro duct for m ulas (38) and (18) for ψ n ( λ ) and ∆ 2 λ − 4, resp ectively , we obtain, for λ nea r G k , ψ n ( λ ) c p ∆ 2 λ − 4 = λ − σ n k s p ( λ 2 k +1 − λ )( λ − λ 2 k ) ζ n k ( λ ) (42) where ζ n k ( λ ) = M ′ n ( b, a ) ( λ − σ n n ) + p ( λ − λ 1 )( λ 2 N − λ ) Y m 6 = k λ − σ n m + p ( λ 2 m +1 − λ )( λ 2 m − λ ) , (43) with σ n n := τ n and M ′ n ( b, a ) 6 = 0. The function ζ n k is analytic and nonv anishing in U k . Substituting (42) into (36) one gets 1 2 π Z Γ k λ − σ n k s p ( λ 2 k +1 − λ )( λ − λ 2 k ) ζ n k ( λ ) dλ = 0 . (44) W e now dro p the sup erscript n for the remainder o f this pro of and write ζ k as ζ k ( λ ) = ξ k + ( ζ k ( λ ) − ξ k ) with ξ k := ζ k ( τ k ) 6 = 0 . Note that 1 2 π Z Γ k λ − σ k s p ( λ 2 k +1 − λ )( λ − λ 2 k ) dλ = τ k − σ k and hence (44) b ecomes ( σ k − τ k ) ξ k = 1 2 π Z Γ k ( λ − σ k )( ζ k ( λ ) − ξ k ) s p ( λ 2 k +1 − λ )( λ − λ 2 k ) dλ. (45) T o es timate the integral on the rig h t ha nd side of (45), note that      1 2 π Z Γ k f ( λ ) s p ( λ 2 k +1 − λ )( λ − λ 2 k ) dλ      ≤ max λ ∈ G k | f ( λ ) | (46) for an a r bitrary function f a nalytic on U k . W e wan t to apply (46) fo r f ( λ ) = ( λ − σ k )( ζ k ( λ ) − ξ k ). Note that for λ ∈ G k , | ζ k ( λ ) − ξ k | = | ζ k ( λ ) − ζ k ( τ k ) | ≤ M | γ k | , where M = sup S 1 ≤ k ≤ N − 1 {| ζ k ( λ ) | : λ ∈ G k } . Hence (46) leads to | σ k − τ k || ξ k | = sup λ ∈ G k | λ − σ k | O ( γ k ) . 13 Dividing b y | ξ k | 6 = 0 , we g et | σ k − τ k | = sup λ ∈ G k | λ − σ k | O ( γ k ) ( 47) and in particular | σ k − τ k | = O ( γ k ). In a second step, we now improv e the es tima te (47). Note that sup λ ∈ G k | λ − σ k | ≤ | σ k − τ k | + sup λ ∈ G k | λ − τ k | = O ( γ k ) . ( 48) Combining (47) and (48), we obtain the clained estimate (40). F or later use, we compute the sp ectra of Q ( b, a ) and L 2 ( b, a ) in the sp e- cial case ( b, a ) = ( β 1 N , α 1 N ) with β ∈ R and α > 0. Here 1 N denotes the vector (1 , . . . , 1) ∈ R N . These p oin ts are the e q uilibrium po in ts (of the res tric- tions) of the T o da Hamiltonian vector ﬁeld (to the sy mplectic leaves M β ,α ). W e compute the sp ectrum ( λ j ) 1 ≤ j ≤ 2 N of the matrix Q ( β 1 N , α 1 N ) and the Dirich- let eigenv alues ( µ k ) 1 ≤ k ≤ N − 1 of L = L ( β 1 N , α 1 N ) together with a norma lized eigenv ector g l =  g l ( j )  1 ≤ j ≤ N of µ l , i.e. L g l = µ l g l , g l (1) = 0, and a vector h l =  h l ( j )  1 ≤ j ≤ N which is the nor malized solution o f Ly = µ l y orthogonal to g l satisfying W ( h l , g l )( N ) > 0 . Lemma 2. 6. The sp e ctru m ( λ j ) 1 ≤ j ≤ 2 N of Q ( β 1 N , α 1 N ) and the Dirichlet eigen- values ( µ l ) 1 ≤ l ≤ N − 1 of L ( β 1 N , α 1 N ) ar e given by λ 1 = β − 2 α, λ 2 l = λ 2 l +1 = µ l = β − 2 α c o s l π N (1 ≤ l ≤ N − 1) , λ 2 N = β + 2 α. In p articular, al l sp e ctra l gaps of Q ( β 1 N , α 1 N ) ar e c ol lapse d. F or any 1 ≤ l ≤ N − 1 , the ve ctors g l and h l deﬁne d by g l ( j ) = ( − 1) j +1 r 2 N sin ( j − 1) l π N (1 ≤ j ≤ N ) , (49) h l ( j ) = ( − 1) j r 2 N cos ( j − 1) l π N (1 ≤ j ≤ N ) (50) satisfy L y = µ l y and the normalization c onditions N X j =1 g l ( j ) 2 = N X j =1 h l ( j ) 2 = 1 , g l (0) > 0 , g l (1) = 0 ; W ( h l , g l )( N ) > 0 , h h l , g l i R N = 0 . 14 2 PREL IMINARIES Remark 2.7. F or ( b, a ) = ( β 1 N , α 1 N ) t he fundamental solutions y 1 and y 2 ar e given by y 1 ( j, λ ) = − sin( ρ ( j − 1)) sin ρ ( j ∈ Z ) (51) y 2 ( j, λ ) = sin( ρj ) sin ρ ( j ∈ Z ) (52) wher e π < ρ < 2 π is determine d by co s ρ = λ − β 2 α . Pr o of. F or any λ ∈ R , the diﬀerence equation (11) for ( β 1 N , α 1 N ) reads ( R β ,α y )( k ) := β y ( k ) + αy ( k − 1) + αy ( k + 1) = λy ( k ) (53) and can b e written as y ( k − 1) + y ( k + 1) = λ − β α y ( k ) . (54) Since we are lo oking for p erio dic solutions of (54), we make the ansatz y ( k ) = e ± iρk . This leads to the characteris tic equation 2 cos ρ ≡ e iρ + e − iρ = λ − β α . F or the solution to be 2 N -p e r iodic , it is re quired that ρ ∈ π N Z . T o put the eigenv a lues in asce nding or der, in tro duce ρ l = (1 + l N ) π with 0 ≤ l ≤ N . Then for any 1 ≤ j ≤ 2 N , ther e exis ts 0 ≤ l ≤ N such tha t λ j = β + 2 α cos ρ l = β − 2 α cos l π N . Note that for l = 0 , λ 1 = β − 2 α is a n eigenv a lue of Q ( β 1 N , α 1 N ) with eig en vector y ( k ) = e iπk = ( − 1) k . Similarly , for l = N , λ 2 N = β + 2 α is a n eigenv alue with eigenv ector y ( k ) ≡ 1. F or the eigenv a lue λ 2 l = β − 2 α cos lπ N (1 ≤ l ≤ N − 1), y ± ( k ) = e ± iρ l k are t wo linear ly independent eigenv e c to rs. As there are 2 N eigen v alues allto- gether, λ 2 l is double for a n y 1 ≤ l ≤ N − 1, and λ 1 and λ 2 N are b oth simple. It follows that all N − 1 gaps are colla psed and hence µ l = λ 2 l for a ll 1 ≤ l ≤ N − 1. T urning to the computatio n of g k and h k , o ne easily veriﬁes that for any r eal nu mber λ 6 = ± 2 α + β , the fundament al s olution y 1 ( · , λ ) of (54) with y 1 (0 , λ ) = 1 and y 1 (1 , λ ) = 0 is given by y 1 ( j, λ ) = − sin  ρ ( j − 1)  sin ρ ( j ∈ Z ) where π < ρ < 2 π is determined b y cos ρ = λ − β 2 α , thus proving (51). In the same wa y , one veriﬁes (52). F or λ = µ l = β − 2 α cos lπ N we then g et sin  ρ l ( j − 1)  = sin  (1 + l N ) π ( j − 1)  = ( − 1) j +1 sin ( j − 1) l π N . 15 In particular, sin  ρ l ( j − 1)  = 0 for j = 1 and j = N + 1. As N X j =1 sin 2 ( j − 1) l π N = N X j =1 cos 2 ( j − 1) l π N and these tw o sums a dd up to N , one sees that N X j =1 sin 2 ( j − 1) l π N = N 2 , (55) yielding the claimed formula (4 9) for g l . By the same ar g umen t one s ho ws that ˜ h l given by ( − 1) j q 2 N cos ( j − 1) lπ N (i.e. the rig ht side of (50)) s atisﬁes R β ,α ˜ h l = µ l ˜ h l and the normalizatio n condition P N j =1 ˜ h l ( j ) 2 = 1. Using standard trigonometric identities one veriﬁes that h g l , ˜ h l i = N X j =1 g l ( j ) ˜ h l ( j ) = 0 and W ( ˜ h l , g l )( N ) can b e computed to b e ˜ h l ( N ) g l ( N + 1 ) − ˜ h l ( N + 1) g l ( N ) = − ˜ h l ( N + 1) g l ( N ) = − ˜ h l (1) g l (0) > 0 . Hence ˜ h l is indee d the eigenvector with the required normalizatio n, i.e. h l = ˜ h l , th us proving (50). 3 Action V ariables In the next tw o sections, we deﬁne the candida tes for action-a ngle v ariables on the phase space M and in vestigate some of their pro perties . In this s e c - tion we intro duce glo bally deﬁned action v ar iables ( I n ) 1 ≤ n ≤ N − 1 as prop osed by Flaschk a -McLaughlin [4]. Deﬁnition 3. 1. L et ( b, a ) ∈ M . F or 1 ≤ n ≤ N − 1 , I n := 1 2 π Z Γ n λ ˙ ∆ λ c p ∆ 2 λ − 4 dλ (56) wher e ˙ ∆ λ = d dλ ∆ λ is the λ -derivative of the di scriminant ∆ λ = ∆( λ, b, a ) and the c ontour Γ n and t he c anonic al r o ot c √ · ar e given as in s e ction 2. Remark 3.2. The c ontours Γ n c an b e chosen lo c al ly indep endently of ( b, a ) . In view of t he fact that ∆ λ is a sp e ctr al invari ant of L ( b, a ) the actio ns I n ar e entir ely determine d by t he sp e ctrum of L ( b, a ) . In p articular, ( I n ) 1 ≤ n ≤ N − 1 ar e c onstants of motion, sinc e by Pr op osition 2.1 , t he T o da ﬂow is isosp e ctr al. 16 3 ACTION V ARIABLES Remark 3.3. The varia bles ( I n ) 1 ≤ n ≤ N − 1 c an also b e r epr esente d as inte gr als on the Riemann surfac e Σ b,a . If c n denotes the lift of Γ n to the c anonic al she et of Σ b,a , (56) b e c omes I n = 1 2 π Z c n λ ˙ ∆ λ p ∆ 2 λ − 4 dλ (1 ≤ n ≤ N − 1) . (57) F rom the deﬁnition (56), the following r e sult can be deduced: Prop osition 3.4. On the r e al sp ac e M , e ach funct ion I n is re al, nonne gative, and it vanish es if γ n = 0 . Pr o of. Since Z Γ n ˙ ∆ λ c p ∆ 2 λ − 4 dλ = 0 , it follows that I n = 1 2 π Z Γ n ( λ − ˙ λ n ) ˙ ∆ λ c p ∆ 2 λ − 4 dλ. (58) By shrinking the contour o f integration to the re a l interv a l, we get I n = 1 π Z λ 2 n +1 λ 2 n ( − 1) N + n − 1 ( λ − ˙ λ n ) ˙ ∆ λ + p ∆ 2 λ − 4 dλ by taking into acc oun t the deﬁnition (32) of the c - r oo t. Since sign( λ − ˙ λ n ) ˙ ∆ λ = ( − 1) N + n − 1 on [ λ 2 n , λ 2 n +1 ] \ { ˙ λ n } , the int egra nd is real and nonneg ativ e, hence I n is real and nonneg ativ e o n M , as claimed. If γ n = 0, then λ 2 n = λ 2 n +1 . Hence ˙ λ n = λ 2 n = λ 2 n +1 = τ n and λ − ˙ λ n = i s p ( λ 2 n +1 − λ )( λ − λ 2 n ) . Therefore the integrand in (5 6) is holomorphic in the interior of the contour Γ n , and b y Cauchy’s theorem the integral in (56 ) v a nishes. The action v ariables ( I n ) 1 ≤ n ≤ N − 1 can be extended in a straightforward wa y to a complex neighbo r hoo d W of M in M C . Theorem 3.5. T her e exists a c omplex neighb orho o d W of M in M C such that for al l 1 ≤ n ≤ N − 1 , t he functions I n deﬁne d by (56) ex t end analytic al ly to W , I n : W → C . Pr o of. Let W denote a neighborho o d of M in M C of Lemma 2.3 and deﬁne for any 1 ≤ n ≤ N − 1 the functions I n on W by the formula (56). Let ( b, a ) ∈ W be given. Then ther e exists a neighborho o d W b,a of ( b, a ) in W s o tha t the int egra tion co n tours Γ n in (56) can b e c hosen to be the same for any element in W b,a and ˙ ∆ λ / c p ∆ 2 λ − 4 is a na lytic on B ε (Γ n ) × W b,a , where B ε (Γ n ) := { λ ∈ C | dist ( λ, Γ n ) < ε } is the ε -neighborho o d of Γ n with ε suﬃciently small. This shows that I n is analytic on W . 17 Prop osition 3 .6. Ther e exists a c omplex neighb orho o d W of M in M C such that for any 1 ≤ n ≤ N − 1 , the quotient I n /γ 2 n extends analytic al ly fr om M \ D n to al l of W and has strictly p ositive re al p art on W . As a c onse qu enc e, ξ n = + p 2 I n /γ 2 n is an analytic and nonvanishing function on W , wher e + √ · is the princip al br anch of the squar e ro ot on C \ ( −∞ , 0] . Pr o of. Let W b e the complex neig h b orho od of Theorem 3 .5. Substituting (34) int o (58) leads to the following identit y o n W \ D n I n = N 2 π Z Γ n ( λ − ˙ λ n ) 2 s p ( λ 2 n +1 − λ )( λ − λ 2 n ) χ n ( λ ) dλ, where χ n is given by (35). Up on the substitution λ ( ζ ) = τ n + γ n 2 ζ , with τ n = 1 2 ( λ 2 n + λ 2 n +1 ) and δ n = 2( ˙ λ n − τ n ) γ n , one then obtains 2 I n γ 2 n = N 4 π Z Γ ′ n ( ζ − δ n ) 2 s p 1 − ζ 2 χ n ( τ n + γ n 2 ζ ) dζ , (59) where Γ ′ n is the pullback of Γ n under the substitution λ = λ ( ζ ), i.e. a cir cuit in C aro und [ − 1 , 1 ]. By (39), ˙ λ n − τ n = O ( γ 2 n ), and hence δ n → 0 as γ n → 0 . W e conclude that lim γ n → 0 2 I n γ 2 n = N 4 π Z Γ ′ n χ n ( τ n ) ζ 2 dζ s p 1 − ζ 2 = χ n ( τ n ) N 2 π Z 1 − 1 t 2 dt + √ 1 − t 2 = N 4 χ n ( τ n ) . By deﬁning 2 I n γ 2 n by N 4 χ n ( τ n ) o n W ∩ D n , it follows that 2 I n γ 2 n is a contin uous function on a ll of W . This extended function is ana lytic on W \ D n as is its restriction to W ∩ D n . By Theorem A.6 in [12] it then follows that 2 I n γ 2 n is ana lytic on all of W . By Lemma 3 .7 b elow, the quotient I n /γ 2 n can be b ounded a wa y from zer o on M , I n γ 2 n ≥ 1 3 π ( λ 2 N − λ 1 ) . By shrinking W , if nece s sary , it then follows that for any 1 ≤ n ≤ N − 1, the real part of I n /γ 2 n is po sitiv e a nd never v a nishes on W . Hence the pr incipal br anch of the square ro ot o f 2 I n /γ 2 n is w ell deﬁned o n W and ξ n has the claimed pr oper ties. T o show that + q 2 I n γ 2 n is well deﬁned o n W , we used in the pro of of Prop osition 3.6 the following auxilia ry res ult, which w e prov e in Appendix A: Lemma 3. 7. F or any ( b, a ) ∈ M and any 1 ≤ n ≤ N − 1 , γ 2 n ≤ 3 π ( λ 2 N − λ 1 ) I n . (60) F rom the deﬁnition (56 ), Prop osition 3.4, and the e s timate (60) one o btains Corollary 3.8. F or any ( b, a ) ∈ M and any 1 ≤ n ≤ N − 1 , I n = 0 if and only if γ n = 0 . 18 4 ANGLE V ARIABLES Actually , Lemma 3.7 can b e improved. W e ﬁnish this section with an a priori estimate of the g ap lengths γ n in ter ms o f the action v ar iables and the v alue of the Casimir C 2 alone, which will b e shown in Appendix B. Theorem 3. 9. F or any ( b, a ) ∈ M β ,α with β ∈ R , α > 0 arbitr ary, N − 1 X n =1 γ 2 n ≤ 12 π 2 α N − 1 X n =1 I n ! + 9 π 2 ( N − 1) N − 1 X n =1 I n ! 2 . (61) 4 Angle V ariables In this section, w e deﬁne and study the ang le co ordinates ( θ n ) 1 ≤ n ≤ N − 1 . Each θ n is deﬁned mo d 2 π on W \ D n , wher e W is a complex neighborho o d of M in M C as in Lemma 2.3 and D n is given by (23). Deﬁnition 4. 1. F or any 1 ≤ n ≤ N − 1 , t he function θ n is deﬁne d for ( b, a ) ∈ M \ D n by θ n := η n + N − 1 X n 6 = k =1 β n k mo d 2 π , (62) wher e for k 6 = n , β n k = Z µ ∗ k λ 2 k ψ n ( λ ) p ∆ 2 λ − 4 dλ, η n = Z µ ∗ n λ 2 n ψ n ( λ ) p ∆ 2 λ − 4 dλ ( mo d 2 π ) , (63) and wher e for 1 ≤ k ≤ N − 1 , µ ∗ k is the Dirichlet divisor deﬁn e d in (29), and λ 2 k is identiﬁe d with the r amiﬁc ation p oint ( λ 2 k , 0) on the Ri emann surfac e Σ b,a . The inte gr ation p aths on Σ b,a in (63) ar e r e quir e d to b e admissible in the sense that their image under t he pr oje ct ion π : Σ b,a → C on the ﬁ rst c omp onen t stays inside the isolating neighb orho o ds U k . Note that, in view o f the normalization conditio ns (3 6) of ψ n , the ab ov e restriction of the pa ths of integration in (63) implies that η n and hence θ n are well-deﬁned mo d 2 π . Theorem 4.2 . Le t W b e the c omplex neighb orho o d of M in M C intr o duc e d in L emma 2.3. Then for any 1 ≤ n ≤ N − 1 , the function θ n : W \ D n → C ( mo d π ) is analytic. Remark 4.3. As the lexic o gr aphic or dering of the eigenvalues of Q ( b, a ) is n ot c ontinuous on W , it fol lows that η n and henc e θ n ar e only c ontinuous mo d π on W . Pr o of of The or em 4.2. T o see that θ n : W \ D n → C (mo d π ) is ana lytic, deﬁne for any 1 ≤ k ≤ N − 1 the set E k := { ( b, a ) ∈ M C : µ k ( b, a ) ∈ { λ 2 k ( b, a ) , λ 2 k +1 ( b, a ) }} . 19 Below, we show that for any 1 ≤ k ≤ N − 1 with k 6 = n , β n k is a nalytic o n W \ ( D k ∪ E k ), that its restrictions to D k ∩ W and E k ∩ W ar e weakly analytic 4 , and that it is co n tin uous o n W . T ogether with the fact that E k ∩ W and D k ∩ W are a nalytic subv arieties of W it then follows that β n k is a nalytic on W - see Theorem A.6 in [12]. Similar r e sults can be shown for β n n = η n (mo d π ) on W \ D n , and one concludes that θ n (mo d π ) is analy tic on W \ D n . T o prov e that β n k , k 6 = n , is ana lytic on W \ ( D k ∪ E k ), note tha t since λ 2 k is a simple eigenv alue on W \ D k , it is analytic there. F urthermore, µ ∗ k is an analytic function on the (suﬃciently small) neighbor hoo d W of M in M C . On W \ ( D k ∪ E k ) we ca n use the subs titution λ = λ 2 k + z to get β n k = Z µ ∗ k λ 2 k ψ n ( λ ) p ∆ 2 λ − 4 dλ = Z µ ∗ k − λ 2 k 0 ψ n ( λ 2 k + z ) √ z p D ( z ) dz , where D ( z ) = ∆ 2 ( λ 2 k + z ) − 4 z is a nalytic nea r z = 0 and D (0) 6 = 0. Note that D ( z ) do es no t v anish for z on an admissible integration path not going throug h λ 2 k +1 . Such a path exists since ( b, a ) is in the complement of E k . F urthermo re ψ n ( λ 2 k + z ) and D ( z ) ar e analytic in z near s uc h a path and dep end analytically on ( b, a ) ∈ W \ ( D k ∪ E k ). Com bining these ar gumen ts shows that β n k is analy tic on W \ ( D k ∪ E k ). F or k 6 = n with λ 2 k 6 = λ 2 k +1 one has Z λ 2 k +1 λ 2 k ψ n ( λ ) p ∆ 2 λ − 4 dλ = 0 . (64) As σ n k = λ 2 k if λ 2 k = λ 2 k +1 one sees that (64) contin ues to hold for ( b, a ) ∈ E k ∩ W with λ 2 k = λ 2 k +1 and we hav e β n k | E k ∩ W ≡ 0. T o prov e the analy ticity of β n k | D k ∩ W consider the representation (42) of ψ n ( λ ) √ ∆ 2 λ − 4 . F o r ( b, a ) ∈ D k ∩ W , one has λ 2 k = λ 2 k +1 = τ k = σ n k , which implies that the factor λ − σ n k s √ ( λ 2 k +1 − λ )( λ − λ 2 k ) in (42) equals ± i . Hence we can write β n k = Z µ ∗ k λ 2 k ψ n ( λ ) p ∆ 2 λ − 4 dλ = ± i Z µ k τ k ζ n k ( λ ) dλ. As µ k is ana lytic on W , it then follo ws that β n k | D k ∩ W is ana lytic. T o see that β n k is contin uous on W , one sepa rately shows that β n k is contin uous at p oints in W \ ( D k ∪ E k ), E k ∩ W \ D k , D k ∩ W \ E k and D k ∩ E k ∩ W , where fo r the pro of of the contin uity of β n k at p oints in D k ∩ E k ∩ W we use (42) and the estimate σ n k − τ k = O ( γ 2 k ) of Lemma 2.5. By (64), η n v anishes mo d π on E n ∩ W \ D n . Arguing in a simila r wa y as for β n k one then concludes that η n (mo d π ) is analy tic on W \ D n . 4 Let E and F b e complex Banach spaces, and l et U ⊂ E b e op en. The map f : U → F is we akly analytic on U , if for each u ∈ U , h ∈ E and L ∈ F ∗ , the function z 7→ Lf ( u + z h ) is analytic in some neighborhoo d of the origin i n C . 20 5 GRADIENTS 5 Gradien ts In this se c tio n we establish formulas of the gra dien ts of I n , θ n (1 ≤ n ≤ N − 1) on M in terms o f pro ducts of the fundamental solutions y 1 and y 2 . Consider the discr iminan t for a ﬁxed v alue of λ as a function on M , ∆ λ ( b, a ) = y 1 ( N ) + y 2 ( N + 1) . Then ∆ λ is a real analytic function on M . T o o bta in a fo r m ula for the gra dien ts of y 1 ( N ) and y 2 ( N + 1) with resp ect to b , diﬀerentiate R b,a y i = λy i with resp ect to b in the dir e ction v ∈ R N to get ( R b,a − λ ) h∇ b y i , v i ( k ) = − v k y i ( k ) . (65) Diﬀerent iating R b,a y i = λy i with resp ect to a in the directio n u ∈ R N leads to ( R b,a − λ ) h∇ a y i , u i ( k ) = − u k − 1 y i ( k − 1) − u k y i ( k + 1) . (66) T aking the s um of (6 5) a nd (66) yields ( R b,a − λ ) ( h∇ b y i , v i + h∇ a y i , u i ) ( k ) = − ( R v, u y i )( k ) (67) which we ca n rewr ite as ( R b,a − λ ) h∇ b,a y i , ( v , u ) i ( k ) = − ( R v, u y i )( k ) , (68) where h· , ·i in (68) now denotes the sta ndard scalar pro duct in R 2 N , wherea s in (65), (66), and (6 7) it is the one in R N . The inhomo geneous Jac o bi diﬀerence equation (68) for the s equence h∇ b,a y i , ( v , u ) i ( k ) ca n b e integrated using the discrete ana logue of the metho d of the v ariatio n of constants used fo r inhomo- geneous diﬀerential eq uations. As h∇ b,a y i , ( v , u ) i (0) = h∇ b,a y i , ( v , u ) i (1) = 0, one obtains in this way for m ≥ 1 h∇ b,a y i , ( v , u ) i ( m ) = − y 2 ( m ) a N m X k =1 y 1 ( k )( R v, u y i )( k ) − y 1 ( m ) a N m X k =1 y 2 ( k )( R v, u y i )( k ) ! . (69) In the sequel, w e will use (69) to der iv e v ar ious formulas for the gradi- ent s. The c o mmon feature among these formulas is that they inv olve pro ducts betw een the fundamen tal so lutions y 1 and y 2 of (11). Wherea s the gra dien ts with resp ect to b = ( b 1 , . . . , b N ) inv olve pr oducts computed b y co mponent wise m ultiplication, the gradients with r espect to a = ( a 1 , . . . , a N ) inv olve pro ducts obtained b y multiplying shifted comp onents, reﬂecting the fact that the b j are the diagonal elements o f the symmetric matrix L ( b , a ), wher eas the a j are the oﬀ-diagona l elements of L ( b , a ). 21 T o simplify nota tion for the formulas in this section, we deﬁne for seq uences  v ( j ) j ∈ Z  ,  w ( j ) j ∈ Z  ⊆ C the N -vectors v · w := ( v ( j ) w ( j )) 1 ≤ j ≤ N , (70) v · S w := ( v ( j ) w ( j + 1)) 1 ≤ j ≤ N , (71) where S deno tes the shift op erator o f order 1 . Combining (70) a nd (71), we deﬁne the 2 N -vector v · s w := ( v · w , v · S w + w · S v ) . (72) In case v = w we also use the s ho rter nota tion v 2 := v · s v . (73 ) W ritten comp onent wise, v · s w is the 2 N -vector ( v · s w )( j ) =  v ( j ) w ( j ) (1 ≤ j ≤ N ) v ( j − N ) w ( j − N + 1 ) + v ( j − N + 1) w ( j − N ) ( N < j ≤ 2 N ) Prop osition 5.1. F or any ( b, a ) ∈ M , the gr adient ∇ b,a ∆ λ = ( ∇ b ∆ λ , ∇ a ∆ λ ) is given by − a N ∇ b ∆ λ = y 2 ( N ) y 1 · y 1 − y 1 ( N + 1 ) y 2 · y 2 +  y 2 ( N + 1 ) − y 1 ( N )  y 1 · y 2 (74) − a N ∇ a ∆ λ = 2 y 2 ( N ) y 1 · S y 1 − 2 y 1 ( N + 1) y 2 · S y 2 +  y 2 ( N + 1) − y 1 ( N )  y 1 · S y 2 + y 2 · S y 1  (75) or in the notation intr o duc e d ab ove ∇ b,a ∆ λ = − 1 a N  y 2 ( N ) y 2 1 − y 1 ( N + 1 ) y 2 2 + ( y 2 ( N + 1) − y 1 ( N )) y 1 · s y 2  . (76) The gr adients ∇ b ∆ λ and ∇ a ∆ λ admit t he r epr esentations (1 ≤ m ≤ N ) ∂ ∆ λ ∂ b m = − 1 a m y 2 ( N , λ, S m b, S m a ) , (77) ∂ ∆ λ ∂ a m = −  1 a m y 2 ( N + 1 , λ, S m b, S m a ) + 1 a m + 1 y 2 ( N − 1 , λ, S m +1 b, S m +1 a )  . (78) Pr o of. The cla imed formula (76) follows fro m the deﬁntion of ∆ λ and for m ula (69). Indeed, ev alua te (69) for i = 1 a nd m = N to get h∇ b,a y 1 , ( v ,u ) i ( N ) = − y 2 ( N ) a N N X k =1 y 1 ( k ) ( u k − 1 y 1 ( k − 1 ) + v k y 1 ( k ) + u k y 1 ( k + 1 )) + y 1 ( N ) a N N X k =1 y 2 ( k ) ( u k − 1 y 1 ( k − 1) + v k y 1 ( k ) + u k y 1 ( k + 1)) . (79) 22 5 GRADIENTS In order to identify these tw o sums with h y 2 1 , ( v , u ) i a nd h y 1 · s y 2 , ( v , u ) i , resp ec- tively , note tha t N X k =1 u k − 1 y 1 ( k ) y 1 ( k − 1) = N X k =1 u k y 1 ( k ) y 1 ( k + 1) + u N T 1 where T 1 := y 1 (0) y 1 (1) − y 1 ( N ) y 1 ( N + 1 ) . F or the s econd s um in (79), we g et an expressio n of the sa me type with a similar correctio n term T 2 := y 1 (0) y 2 (1) − y 1 ( N ) y 2 ( N + 1 ) . T aking into account the initial conditions of the fundamental so lutions and the W ronskia n identit y (13), one see s that y 2 ( N ) T 1 − y 1 ( N ) T 2 v anishes. Hence we hav e the formula h∇ b,a y 1 , ( v , u ) i ( N ) = − 1 a N  y 2 ( N ) h y 2 1 , ( v , u ) i − y 1 ( N ) h y 1 · s y 2 , ( v , u ) i  . (80) Similarly , ev a luating formula (69) for i = 2 and m = N + 1 leads to h∇ b,a y 2 , ( v , u ) i ( N + 1 ) = − 1 a N  y 2 ( N + 1) h y 1 · s y 2 , ( v , u ) i − y 1 ( N + 1) h y 2 2 , ( v , u ) i  . (81) Here w e used that the v alue of the right s ide of (69) do es not change when we omit the term for k = m = N + 1 in b oth sums. It remains to pr o ve the t wo formulas (77) and (78). W e ﬁrst note that y 2 ( n, λ, S m b, S m a ) = a m a N  y 2 ( n + m, λ, b, a ) y 1 ( m, λ, b , a ) − y 1 ( n + m, λ, b, a ) y 2 ( m, λ, b , a )  , (82) since b oth sides of (8 2) are so lutions of R S m b,S m a y = λy (for ﬁxed m ∈ Z ) with the same initial conditions a t n = 0 , 1 . F or n = 1 this follows from the W ronskia n identit y (13). Similarly , one shows that y 1 ( N + m, λ ) = y 1 ( N , λ ) y 1 ( m, λ ) + y 1 ( N + 1 , λ ) y 2 ( m, λ ) , (83) y 2 ( N + m, λ ) = y 2 ( N , λ ) y 1 ( m, λ ) + y 2 ( N + 1 , λ ) y 2 ( m, λ ) . (84) for any ( b, a ) ∈ M . Hence, suppres sing the v ariable λ , we get y 2 ( N , S m b, S m a ) = a m a N   y 2 ( N ) y 1 ( m ) + y 2 ( N + 1 ) y 2 ( m )  y 1 ( m ) −  y 1 ( N ) y 1 ( m ) + y 1 ( N + 1) y 2 ( m )  y 2 ( m )  = a m a N  y 2 ( N ) y 1 ( m ) 2 +  y 2 ( N + 1) − y 1 ( N )  y 1 ( m ) y 2 ( m ) − y 1 ( N + 1 ) y 2 ( m ) 2  . 23 By (74) this leads to y 2 ( N , S m b, S m a ) = − a m ∂ ∆ λ ∂ b m and formula (77) is es tablished. T o prove (78), w e ﬁrst conclude fro m (8 2) that a N a m +1 y 2 ( N − 1 , S m +1 b, S m +1 a ) = y 2 ( N + m, b , a ) y 1 ( m + 1 , b, a ) − y 1 ( N + m, b , a ) y 2 ( m + 1 , b, a ) (85) and a N a m y 2 ( N + 1 , S m b, S m a ) = y 2 ( N + m + 1 , b , a ) y 1 ( m, b, a ) − y 1 ( N + m + 1 , b , a ) y 2 ( m, b, a ) . (86) Now expand the right hand sides of (85) and (86) according to (83) and (84). By (75), the sum of (85) a nd (86) is − a N ∂ ∆ λ ∂ a m , thus proving (78 ). As a next step, we c o mpute the gra dien ts of the Dirichlet and per iodic eig en- v alues. In the following lemma, w e consider the fundamental solution y 1 ( · , µ ) as an N -vector y 1 ( j, µ ) 1 ≤ j ≤ N . Let k y 1 ( µ ) k 2 = P N j =1 y 1 ( j, µ ) 2 , and denote by ˙ the deriv ative with r espect to λ . Lemma 5. 2. If µ is a Dirichlet eigenvalue of L ( b, a ) , then a N y 1 ( N , µ ) ˙ y 1 ( N + 1 , µ ) = k y 1 ( µ ) k 2 > 0 . (87) In p articular, ˙ y 1 ( N + 1 , µ ) 6 = 0 , which implies that al l Dirichlet eigenvalues ar e simple. Pr o of. This follows from adding up the relations (16). As the Dirichlet eigenv a lues ( µ n ) 1 ≤ n ≤ N − 1 of L ( b, a ) coincide with the ro ots of y 1 ( N + 1 , µ ) and these ro ots a re simple, they are r eal analytic on M . Similarly , the eigenv alues λ 1 and λ 2 N are real analytic on M , whereas for an y 1 ≤ n ≤ N − 1 , λ 2 n and λ 2 n +1 are r eal analytic on M \ D n . Note that for ( b, a ) ∈ M \ D n and i ∈ { 2 n, 2 n + 1 } , we hav e ˙ ∆ λ i 6 = 0 as λ i is a simple eigenv alue. Prop osition 5 . 3. F or any 1 ≤ n ≤ N − 1 , the gr adients of the p erio dic eigen- values λ i ( i = 2 n , 2 n + 1 ) on M \ D n and of the D irich let eigenvalues µ n on M ar e given by ∇ b,a λ i = − ∇ b,a ∆ λ | λ = λ i ˙ ∆ λ i = f 2 i and ∇ b,a µ n = g 2 n , (88) wher e we denote by f i the eigenve ctor of L ( b, a ) asso ciate d to λ i , normalize d by N X j =1 f i ( j ) 2 = 1 and  f i (1) , f i (2)  ∈ ( R > 0 × R ) ∪ ( { 0 } × R > 0 ) , and wher e g n = ( g n ( j )) 1 ≤ j ≤ N is t he fundamental solution y 1 ( · , µ n ) n ormalize d so that P N j =1 g n ( j ) 2 = 1 . 24 5 GRADIENTS Pr o of. W e ﬁrst show the second formula in (88). Diﬀeren tiating y 1 ( N + 1 , µ n ) = 0 with resp ect to ( b, a ), one obta ins ∇ b,a µ n = − ∇ b,a y 1 ( N + 1 , λ ) | λ = µ n ˙ y 1 ( N + 1 , µ n ) . (89) Here w e used that ˙ y 1 ( N + 1 , µ n ) 6 = 0 by Lemma 5 .2. T o compute the g radient ∇ b,a y 1 ( N + 1 , λ ) | λ = µ n , w e ev a luate (69) for i = 1 a nd m = N + 1 . In view of y 1 ( N + 1 , µ n ) = 0 a nd taking int o account (26), o ne then g ets ∇ b,a µ n = y 2 1 ( µ n ) a N y 1 ( N , µ n ) ˙ y 1 ( N + 1 , µ n ) . (90) The cla imed formula ∇ b,a µ n = g 2 n now follows from Lemma 5.2. By diﬀeren ti- ating ∆ λ i = ± 2 with resp ect to ( b, a ), o ne obtains ∇ b,a λ i = −∇ b,a ∆ λ | λ = λ i / ˙ ∆ λ i in a s imilar fashion. T o see that ∇ b,a λ i = f 2 i , diﬀerentiate R b,a f i = λ i f i with resp ect to ( b, a ) in the directio n ( v , u ) ∈ R 2 N , R b,a h∇ b,a f i , ( v , u ) i ( k ) + ( R v, u f i )( k ) = h∇ b,a λ i , ( v , u ) i f i ( k ) + λ i h∇ b,a f i , ( v , u ) i ( k ) , where h· , ·i denotes the standard scalar pr oduct in R 2 N . T ake the scalar pr oduct (in R N ) of the ab ov e equa tion with f i . Now use that h∇ b,a f i ( v , u ) , R b,a f i i R N = λ i h∇ b,a f i ( v , u ) , f i i R N , h f i , f i i R N = 1, and h R v, u f i , f i i R N = h f 2 i , ( v , u ) i R 2 N , to conclude that ∇ b,a λ i = f 2 i holds. T o compute the Poisson brack ets in volving angle v ariables we need to esta b- lish some additional auxiliar y results. Recall from sectio n 3 that for 1 ≤ k , n ≤ N − 1 with k 6 = n and ( b, a ) ∈ M , β n k is given by β n k = Z µ ∗ k λ 2 k ψ n ( λ ) p ∆ 2 λ − 4 dλ, (91) whereas β n n := η n = Z µ ∗ n λ 2 n ψ n ( λ ) p ∆ 2 λ − 4 dλ (mo d 2 π ) . (92) By T heo rem 4.2, the functions β n k with k 6 = n ar e real analytic on M , wherea s β n n , when consider ed mo d π , is r e al a nalytic on M \ D n . Prop osition 5 . 4. L et 1 ≤ k ≤ N − 1 and ( b, a ) ∈ M . If γ k > 0 and λ 2 k = µ k , then for any 1 ≤ n ≤ N − 1 , ∇ b,a β n k = − ψ n ( µ k ) a N ˙ ∆ µ k g k · s h k , 25 wher e h k denotes the solution of R b,a y = µ k y ortho gonal to g k , i.e. N X j =1 g k ( j ) h k ( j ) = 0 , satisfying the normalization c ondition W ( h k , g k )( N ) = 1 . Pr o of. W e use a limiting pro cedure ﬁrst intro duced in [16] fo r the nonlinear Schr¨ odinger equation and subsequently used for the KdV equation in [11], [12]. W e approximate ( b, a ) ∈ M with λ 2 k ( b, a ) = µ k ( b, a ) < λ 2 k +1 ( b, a ) by ( b ′ , a ′ ) ∈ Iso( b, a ) satisfying λ 2 k ( b, a ) < µ k ( b ′ , a ′ ) < λ 2 k +1 ( b, a ). F or such ( b ′ , a ′ ), using the substitution λ = λ 2 k + z in the integral of (91), we o btain β n k ( b ′ , a ′ ) = Z µ ∗ k λ 2 k ψ n ( λ ) p ∆ 2 λ − 4 dλ = Z µ k − λ 2 k 0 ψ n ( λ 2 k + z ) √ z p D ( z ) dz , (93) where D ( z ) ≡ D ( λ 2 k , z ) := (∆ 2 ( λ 2 k + z ) − 4) / z . T aking the gradient, undoing the substitution, and rec alling the deﬁnition (29) of the star red square ro ot then leads to ∇ b,a β n k = ψ n ( µ k ) ∗ q ∆ 2 µ k − 4 ( ∇ b,a µ k − ∇ b,a λ 2 k ) + E ( b ′ , a ′ ) , (94) with the remainder term E ( b ′ , a ′ ) given by E ( b ′ , a ′ ) = Z µ k − λ 2 k 0 ∇ b ′ ,a ′ ψ n ( λ 2 k + z ) p D ( λ 2 k , z ) ! dz √ z . As the gradient in the latter in tegral is a b ounded function in z near z = 0, lo cally uniformly in ( b ′ , a ′ ), it follows by the dominated c o n vergence theor e m that lim ( b ′ ,a ′ ) → ( b,a ) E ( b ′ , a ′ ) = 0 . The gradient ∇ b,a β n k depe nds con tin uously on ( b, a ) ∈ M , hence we can conclude by (94) that it ca n be written as ∇ b,a β n k = lim ( b ′ ,a ′ ) → ( b,a ) ψ n ( µ k ) ∗ q ∆ 2 µ k − 4 ( ∇ b ′ ,a ′ µ k − ∇ b ′ ,a ′ λ 2 k ) . (9 5) The gradient of b oth sides of the W ronskian identit y (1 3), y 1 ( N , λ ) y 2 ( N + 1 , λ ) − y 1 ( N + 1 , λ ) y 2 ( N , λ ) = 1 , leads to y 1 ( N + 1 ) ∇ b,a y 2 ( N ) + y 2 ( N ) ∇ b,a y 1 ( N + 1 ) (96) = y 2 ( N + 1) ∇ b,a ∆ + ( y 1 ( N ) − y 2 ( N + 1)) ∇ b,a y 2 ( N + 1) , The λ -deriv ative ca n then b e computed to b e y 1 ( N + 1 ) ˙ y 2 ( N ) + ˙ y 1 ( N + 1 ) y 2 ( N ) = y 2 ( N + 1 ) ˙ ∆ + ( y 1 ( N ) − y 2 ( N + 1 )) ˙ y 2 ( N + 1) . (97) 26 5 GRADIENTS Using (96), (97), and y 1 ( N + 1 , µ k ) = 0 , formula (89) for ∇ b,a µ k leads to ∇ b,a µ k = − y 2 ( N + 1) ∇ b,a ∆ + ( y 1 ( N ) − y 2 ( N + 1 )) ∇ b,a y 2 ( N + 1 ) y 2 ( N + 1 ) ˙ ∆ + ( y 1 ( N ) − y 2 ( N + 1 )) ˙ y 2 ( N + 1)    µ k . (98) F urther by (88), ∇ b,a λ 2 k = − ∇ b,a ∆ ˙ ∆    λ = λ 2 k . (99) Now substitute (98) a nd (99) into ( ∇ b ′ ,a ′ µ k − ∇ b ′ ,a ′ λ 2 k ) and use that by (2 9), ∗ q ∆ 2 µ k − 4 = ( y 1 ( N ) − y 2 ( N + 1)) | µ k . W e claim that lim ( b ′ ,a ′ ) → ( b,a ) ∇ b,a µ k − ∇ b,a λ 2 k ∗ q ∆ 2 µ k − 4 = ˙ y 2 ( N + 1 ) ∇ b,a y 1 ( N ) − ˙ y 1 ( N ) ∇ b,a y 2 ( N + 1 ) ˙ ∆ ˙ y 1 ( N + 1 ) y 2 ( N )    λ 2 k . (10 0) Indeed, to obtain (100) a fter the above men tioned s ubstitutions, w e split the fraction 1 ∗ √ ∆ 2 µ k − 4 ( ∇ b ′ ,a ′ µ k − ∇ b ′ ,a ′ λ 2 k ) into t wo parts which a re trea ted sepa- rately . In the ﬁrst part we colle c t all terms in 1 ∗ √ ∆ 2 µ k − 4 ( ∇ b ′ ,a ′ µ k − ∇ b ′ ,a ′ λ 2 k ) which contain ( y 1 ( N ) − y 2 ( N + 1 )) | µ k in the nominator , I ( a ′ , b ′ ) := − ˙ ∆ | λ 2 k · ∇ b ′ ,a ′ y 2 ( N + 1 ) | µ k + ∇ b ′ ,a ′ ∆ | λ 2 k · ˙ y 2 ( N + 1 ) | µ k ˙ ∆ | λ 2 k · ( y 2 ( N + 1 ) ˙ ∆ + ( y 1 ( N ) − y 2 ( N + 1 )) ˙ y 2 ( N + 1)) | µ k . Using again (97) we then g et lim ( b ′ ,a ′ ) → ( b,a ) I ( b ′ , a ′ ) = ˙ y 2 ( N + 1) ∇ b,a y 1 ( N ) − ˙ y 1 ( N ) ∇ b,a y 2 ( N + 1) ˙ ∆ ˙ y 1 ( N + 1 ) y 2 ( N )    λ 2 k . The second term is then given b y I I ( b ′ , a ′ ) = y 2 ( N + 1) | µ k · ( ∇ b ′ ,a ′ ∆ | λ 2 k · ˙ ∆ | µ k − ˙ ∆ | λ 2 k · ∇ b ′ ,a ′ ∆ | µ k ) ˙ ∆ | λ 2 k · ( y 1 ( N ) − y 2 ( N + 1 )) | µ k · ( ˙ y 1 ( N + 1) y 2 ( N )) | µ k . Note that the nominator of I I ( b ′ , a ′ ) is of the o rder O ( µ k − λ 2 k ). In vie w of (29), we hav e ( y 1 ( N ) − y 2 ( N + 1)) | µ k = O ( p µ k − λ 2 k ) whereas the o ther ter ms in the denominator of I I ( b ′ , a ′ ) are b ounded aw ay fro m zero. Indee d, λ 2 k being a simple eigenv alue for ( b, a ) mea ns ˙ ∆ | λ 2 k 6 = 0 for ( b ′ , a ′ ) near ( b, a ). F urther, use a version of (97) in the ca se λ 2 k = µ k to conclude that ˙ y 1 ( N + 1 ) y 2 ( N ) = y 2 ( N + 1 ) ˙ ∆ λ 2 k . Hence ˙ y 1 ( N + 1) y 2 ( N ) | µ k 6 = 0 for ( b ′ , a ′ ) near ( b, a ) a nd I I ( b ′ , a ′ ) v anishes in the limit of µ k → λ 2 k . 27 Substituting (80) and (81) int o (100), we o btain ˙ y 2 ( N + 1) ∇ b,a y 1 ( N ) − ˙ y 1 ( N ) ∇ b,a y 2 ( N + 1 ) ˙ ∆ ˙ y 1 ( N + 1) y 2 ( N )    µ k = − 1 a N ˙ ∆ y 1 · s y 0 with y 0 = ˙ y 2 ( N +1) ˙ y 1 ( N +1) y 1 − y 2 . Hence ∇ b,a β n k = − ψ n ( µ k ) a N ˙ ∆( µ k ) y 1 · s y 0 . Since β n k is in v ar ian t under the tr a nslation b 7→ b + t (1 , . . . , 1), the scalar pr oduct h∇ b,a β n k , ( 1 , 0 ) i R 2 N v anishes. Hence 0 = N X j =1 ∂ β n k ∂ b j = − ψ n ( µ k ) a N ˙ ∆( µ k ) N X j =1 y 1 ( j ) y 0 ( j ) . It means that y 1 and y 0 are orthogona l to each other. Finally w e in tro duce h k := k y 1 k y 0 and verify that W ( h k , g k ) = W  k y 1 k y 0 , y 1 k y 1 k  = W ( y 0 , y 1 ) = W  ˙ y 2 ( N + 1) ˙ y 1 ( N + 1) y 1 − y 2 , y 1  . By (13), it then follows that W ( h k , g k ) = − W ( y 2 , y 1 ) = W ( y 1 , y 2 ) . Hence b y (14), W ( h k , g k )( N ) = W ( y 1 , y 2 )( N ) = 1 . This completes the pro of of Pro pos itio n 5.4. 6 Orthogonalit y relations In Prop ositions 5.1, 5.3, and 5.4, we have e xpressed the gra dien ts of ∆ λ , µ n , and, on a s ubset o f M , of β n k in terms of pro ducts o f fundamental so lutio ns of the diﬀerence equation (11). In this section we establish o rthogonality r elations betw een such pro ducts - see [2] for simila r computations. Recall that in (72) we hav e introduced for ar bitrary s equences ( v j ) j ∈ Z , ( w j ) j ∈ Z the 2 N -vector v · s w . Lemma 6.1. F or any ( b, a ) ∈ M , let v 1 , w 1 and v 2 , w 2 b e p airs of solutions of (11) for arbi tr arily given r e al numb ers µ and λ , r esp e ct ively . Then 2( λ − µ ) a 1 a N h v 1 · s w 1 , J ( v 2 · s w 2 ) i = V + B , (101) wher e V :=  W 1 · S W 2    N 0 +  S W 1 · W 2    N 0 (102) 28 6 OR THOGONALITY RE LA TIONS with W 1 and W 2 denoting the Wr onskians W 1 := W ( v 1 , w 2 ) , W 2 := W ( w 1 , v 2 ) , and wher e B is given by B := ( λ − µ ) a 1  ( v 1 · w 1 ) | N +1 1 ( v 2 · s w 2 )(2 N ) − ( v 2 · w 2 ) | N +1 1 ( v 1 · s w 1 )(2 N )  . (103) Pr o of. W e prov e (101) by a straightforw ard calculation, using the recur r ence prop erty (15) of the W ro nskian seq ue nc e s W 1 and W 2 . By the deﬁnition (3) of J w e can write 2 h v 1 · s w 1 , J ( v 2 · s w 2 ) i = E 1 + B 1 , where E 1 := N X k =1 a k  ( v 1 · s w 1 )( k )( v 2 · s w 2 )( N + k ) − ( v 1 · s w 1 )( k + 1)( v 2 · s w 2 )( N + k ) − ( v 1 · s w 1 )( N + k )( v 2 · s w 2 )( k ) + ( v 1 · s w 1 )( N + k )( v 2 · s w 2 )( k + 1)  and B 1 := a N  ( v 1 · s w 1 )( N + 1) − ( v 1 · s w 1 )(1)  ( v 2 · s w 2 )(2 N ) + a N  ( v 2 · s w 2 )(1) − ( v 2 · s w 2 )( N + 1)  ( v 1 · s w 1 )(2 N ) . Let us ﬁrst cons ide r E 1 . Calculating the pro ducts v j · s w j according to (72), w e obtain, after reg rouping, E 1 = N X k =1 a k  ( v 2 ( k ) w 1 ( k ) + v 2 ( k + 1) w 1 ( k + 1)) W 1 ( k ) +( v 1 ( k ) w 2 ( k ) + v 1 ( k + 1 ) w 2 ( k + 1 )) W 2 ( k )  + B 2 with B 2 := a N  v 1 ( N + 1) w 1 ( N + 1) − ( v 1 · s w 1 )( N + 1)  ( v 2 · s w 2 )(2 N ) + a N  ( v 2 · s w 2 )( N + 1) − v 2 ( N + 1) w 2 ( N + 1)  ( v 1 · s w 1 )(2 N ) . Multiply E 1 by ( λ − µ ) and use the recur rence rela tio n (15) to expres s ( λ − µ ) v 2 ( k ) w 1 ( k ), ( λ − µ ) v 2 ( k + 1) w 1 ( k + 1), ( λ − µ ) v 1 ( k ) w 2 ( k ), and ( λ − µ ) v 1 ( k + 1) w 2 ( k + 1) in terms o f the W ronskians W 1 and W 2 to get ( λ − µ ) E 1 = N X k =1 [ a k a k +1 ( W 1 ( k ) W 2 ( k + 1) + W 1 ( k + 1) W 2 ( k )) − a k − 1 a k ( W 1 ( k − 1 ) W 2 ( k ) + W 1 ( k ) W 2 ( k − 1))] +( λ − µ ) B 2 . The sum on the r igh t ha nd side of the latter identit y is a telescoping sum and equals the ter m a 1 a N V with V deﬁned in (102). In a straig h tforward w ay one sees that ( λ − µ ) a 1 a N ( B 1 + B 2 ) equals the expression B deﬁned b y (103), hence formula (101) is esta blished. 29 Corollary 6.2. F or any λ, µ ∈ C , { ∆ λ , ∆ µ } J = 0 . (104) Pr o of. By the formula (76) for the gradient of ∆ λ , { ∆ λ , ∆ µ } J = h∇ b,a ∆ λ , J ∇ b,a ∆ µ i is a linear co m bination of terms of the form h v 1 · s w 1 , J ( v 2 · s w 2 ) i for pairs of fundamen tal solutions v 1 , w 1 and v 2 , w 2 of (11) for µ and λ , resp ectively . In view of (77) a nd (7 8), ∇ b ∆ λ and ∇ a ∆ λ are b oth N -p erio dic. In the case λ 6 = µ we use Lemma 6 .1 and note that the b oundary terms (1 0 2) a nd (103) in Lemma 6.1 v anish, hence { ∆ λ , ∆ µ } J = 0. In the case λ = µ the iden tity (104) follows from the skew-symmetry o f {· , ·} J . Corollary 6.3. L et 1 ≤ k ≤ 2 N and λ ∈ C . On t he op en subset of M wher e λ k is a simple eigenvalue of Q ( b, a ) one has { λ k , ∆ λ } J = 0 . Pr o of. Using formula (88) for ∇ b,a λ k , w e conclude fro m Cor ollary 6.2 tha t { λ k , ∆ λ } J = − 1 ˙ ∆ λ k { ∆ µ , ∆ λ } J | µ = λ k = 0 . Corollary 6 .4. L et µ n b e the n - th Dirichlet eigenvalue of L ( b, a ) and λ 6 = µ n a r e al n u mb er. Then ( λ − µ n ) h y 2 1 ( µ n ) , J y 2 1 ( λ ) i =  a N y 1 ( N + 1 , λ ) y 2 ( N + 1 , µ n )  2 (105) ( λ − µ n ) h y 2 1 ( µ n ) , J y 1 ( λ ) · s y 2 ( λ ) i = a 2 N y 1 ( N + 1 , λ ) y 2 ( N + 1 , λ ) y 2 ( N + 1 , µ n ) 2 (106) ( λ − µ n ) h y 2 1 ( µ n ) , J y 2 2 ( λ ) i = a 2 N  y 2 ( N + 1 , λ ) y 2 ( N + 1 , µ n )  2 − 1 ! (107) Pr o of. The three stated iden tities follow fro m Lemma 6.1, us ing that y 1 ( N + 1 , µ n ) = 0 , y 1 (2 , µ n ) = − a N /a 1 , and, b y the W r o nskian identit y (26), y 1 ( N , µ n ) · y 2 ( N + 1 , µ n ) = 1 . Corollary 6 .5. L et µ n b e the n - th Dirichlet eigenvalue of L ( b, a ) and λ 6 = µ n a r e al n u mb er. Then { µ n , ∆ λ } J = y 1 ( N + 1 , λ ) ˙ y 1 ( N + 1 , µ n ) ∗ q ∆ 2 µ n − 4 λ − µ n . (108) 30 6 OR THOGONALITY RE LA TIONS Pr o of. By (90), combined with (2 6), we get { µ n , ∆ λ } J = y 2 ( N + 1 , µ n ) a N ˙ y 1 ( N + 1 , µ n ) h y 2 1 ( µ n ) , J ∇ b,a ∆ λ i . (109) Substituting the formula (7 6) for J ∇ b,a ∆ λ we obtain h y 2 1 ( µ n ) , J ∇ b,a ∆ λ i = − 1 a N y 2 ( N , λ ) h y 2 1 ( µ n ) , J y 2 1 ( λ ) i − 1 a N ( y 2 ( N + 1 , λ ) − y 1 ( N , λ )) h y 2 1 ( µ n ) , J y 1 ( λ ) · s y 2 ( λ ) i + 1 a N y 1 ( N + 1 , λ ) h y 2 1 ( µ n ) , J y 2 2 ( λ ) i . (110) T o ev aluate the right side o f (1 10), w e apply Corolla ry 6.4 and get λ − µ n a 1 a N h y 2 1 ( µ n ) , J ∇ b,a ∆ λ i = 1 a 1 y 2 ( N + 1 , µ n ) 2  − y 2 ( N , λ ) y 1 ( N + 1 , λ ) 2 − ( y 2 ( N + 1 , λ ) − y 1 ( N , λ )) y 1 ( N + 1 , λ ) y 2 ( N + 1 , λ ) + y 1 ( N + 1 , λ ) y 2 ( N + 1 , λ ) 2  − y 1 ( N + 1 , λ ) a 1 . Using the W ro nskian identit y (14), the sum of the terms in the square brack et of the latter expr ession simpliﬁes, and one obtains λ − µ n a 1 a N h y 2 1 ( µ n ) , J ∇ b,a ∆ λ i = y 1 ( N + 1 , λ ) a 1 y 2 ( N + 1 , µ n ) 2 − y 1 ( N + 1 , λ ) a 1 = y 1 ( N + 1 , λ ) a 1 ( y 1 ( N , µ n ) 2 − 1) , (111) where for the latter equa lity we again used (14). Substituting (111) into (1 09), we get λ − µ n a 1 a N { µ n , ∆ λ } J = y 2 ( N + 1 , µ n ) y 1 ( N + 1 , λ ) a 1 a N ˙ y 1 ( N + 1 , µ n ) ( y 1 ( N , µ n ) 2 − 1) = y 1 ( N + 1 , λ ) a 1 a N ˙ y 1 ( N + 1 , µ n ) ∗ q ∆ 2 µ n − 4 , where we used that, b y the deﬁnition of the starred square ro ot (29), ∗ q ∆ 2 µ n − 4 = y 1 ( N , µ n ) − y 2 ( N + 1 , µ n ) = y 2 ( N + 1 , µ n )( y 1 ( N , µ n ) 2 − 1) . This prov es (10 8). Prop osition 6 . 6. F or any λ ∈ R , 1 ≤ n ≤ N − 1 , and ( b, a ) ∈ M \ D n , { θ n , ∆ λ } J = ψ n ( λ ) . (112) 31 Pr o of. Recall that θ n = P N − 1 k =1 β n k (mo d 2 π ) with β n k given b y (91)-(92). T o compute { β n k , ∆ λ } J , w e ﬁrst consider the ca se where ( b, a ) / ∈ S N − 1 k =1 D k and λ 2 k < µ k < λ 2 k +1 for any 1 ≤ k ≤ N − 1 . Then λ 2 k and µ ∗ k are smo oth near ( b, a ) and, by Leibniz’s r ule, we g et { β n k , ∆ λ } J = Z µ ∗ k λ 2 k { ψ n ( µ ) q ∆ 2 µ − 4 , ∆ λ } J dµ + ψ n ( µ k ) ∗ q ∆ 2 µ k − 4 { µ k , ∆ λ } J − ψ n ( λ 2 k ) ∗ q ∆ 2 λ 2 k − 4 { λ 2 k , ∆ λ } J ! . By Corollar y 6.3, { λ 2 k , ∆ λ } J = 0. Moreover, as the gra dien t ∇ b,a ψ n ( µ ) √ ∆ 2 µ − 4 is orthogo nal to T b,a Iso ( b, a ) and J ∇ b,a ∆ λ ∈ T b,a Iso ( b, a ) it follows that the Poisson br a c ket { ψ n ( µ ) √ ∆ 2 µ − 4 , ∆ λ } J v anishes for any µ in the is olating neig h bo rho od U n of G n . Hence { β n k , ∆ λ } J = ψ n ( µ k ) ∗ q ∆ 2 µ k − 4 { µ k , ∆ λ } J . By (108), we then obtain { θ n , ∆ λ } J = N − 1 X k =1 ψ n ( µ k ) ˙ y 1 ( N + 1 , µ k ) y 1 ( N + 1 , λ ) λ − µ k = ψ n ( λ ) , where for the latter equality we used that P N − 1 k =1 ψ n ( µ k ) ˙ y 1 ( N +1 ,µ k ) y 1 ( N +1 ,λ ) λ − µ k and ψ n ( λ ) are both poly nomials in λ of deg ree at most N − 2 whic h agree at the N − 1 po in ts ( µ k ) 1 ≤ k ≤ N − 1 . In the general case, where ( b, a ) ∈ M \ D n and the Dirichlet eig en v alues are arbitrar y , λ 2 k ≤ µ k ≤ λ 2 k +1 for any 1 ≤ k ≤ N − 1, the claimed result follows from the case treated ab ov e by c on tinuit y . Prop osition 6.7 . L et 1 ≤ n, m, k , l ≤ N − 1 and let ( b, a ) ∈ M with λ 2 i ( b, a ) = µ i ( b, a ) for i = k , l . Then { β n k , β m l } J = 0 . Pr o of. In view of Pro p osition 5.4, this amounts to showing that the sca lar pr o d- uct h ( g k · s h k ) , J ( g l · s h l ) i v a nishes. F or k = l , this follows from the skew-symmetry of the Poisson bracket, hence we can ass ume k 6 = l . W e a pply Lemma 6.1 with v 1 := g k , w 1 := h k , v 2 := h l and w 2 := g l , which implies that W 1 = W ( g k , g l ) and W 2 = W ( h k , h l ). Since g k (1), g l (1), g k ( N + 1) and g l ( N + 1) all v anish, we conclude that W 1 ( N ) = W 1 (0) = 0 and ( S W 1 )( N ) = ( S W 1 )(0) = 0, hence the expressions V a nd E , deﬁned in (10 2) and (103), v anish. This prov es the claim. 32 7 CANONICAL RELA TIONS 7 Canonical relations In this section we complete the pro of of Theore m 1.1 a nd Corollary 1.2. In particular we sho w that the v ariables ( I n ) 1 ≤ n ≤ N − 1 , ( θ n ) 1 ≤ n ≤ N − 1 satisfy the canonical relations stated in Theo rem 1 .1. Using the res ults of the pr eceding sections, we can now co mpute the Poisson brack ets amo ng the actio n and angle v aria bles introduced in s e c tion 3. Theorem 7.1. The action-angle variables ( I n ) 1 ≤ n ≤ N − 1 and ( θ n ) 1 ≤ n ≤ N − 1 sat- isfy the fol lowing c anonic al r elations for 1 ≤ n, m ≤ N − 1 : (i) on M , { I n , I m } J = 0; (113) (ii) on M \ D n , { θ n , I m } J = −{ I m , θ n } J = − δ nm . (114) Pr o of. Recall that d dt arcosh ( t ) = ( t 2 − 1) − 1 2 . Hence for any ( b, a ) ∈ M I n = 1 2 π Z Γ n λ d dλ arcosh     ∆ λ 2     dλ and therefore ∇ b,a I n = 1 2 π Z Γ n λ d dλ ∇ b,a ∆ λ c p ∆ 2 λ − 4 dλ. Int egra ting by parts we g et ∇ b,a I n = − 1 2 π Z Γ n ∇ b,a ∆ λ c p ∆ 2 λ − 4 dλ. (115 ) As { ∆ λ , ∆ µ } J = 0 fo r all λ, µ ∈ C b y Cor ollary 6.2, it follows that { I n , I m } J = 0 on M for any 1 ≤ n, m ≤ N − 1. T o pr o ve (114), use (115) and then Pr opo sition 6.6 to get { θ n , I m } J = − 1 2 π Z Γ m { θ n , ∆ λ } J c p ∆ 2 λ − 4 dλ = − 1 2 π Z Γ m ψ n ( λ ) c p ∆ 2 λ − 4 dλ = − δ nm , by the normaliz ing condition (36) of ψ n . T o prove that the angles ( θ n ) 1 ≤ n ≤ N − 1 pairwise Poisson commute we need the following lemma. W e denote by K = K ( b, a ) the index set of the op en g aps, i.e. K ( b, a ) = { 1 ≤ n ≤ N − 1 : γ n ( b, a ) > 0 } . Lemma 7. 2. At every p oint ( b, a ) in M , t he set of ve ctors (i)  ( ∇ b,a I n ) n ∈ K , ∇ b,a C 1 , ∇ b,a C 2  and 33 (ii) ( J ∇ b,a I n ) n ∈ K ar e b oth line arly indep endent. Pr o of. The cla imed statements follow from the ortho gonality r elations stated in Theor em 7.1: Let ( b, a ) ∈ M and supp ose that for some real co eﬃcients ( r n ) n ∈ K ⊆ R and s 1 , s 2 ∈ R we have X n ∈ K r n ∇ b,a I n + s 1 ∇ b,a C 1 + s 2 ∇ C 2 = 0 . F or a n y m ∈ K , take the scalar pro duct of this identit y with J ∇ b,a θ m . Using that { I n , θ m } J = δ nm and that C 1 and C 2 are Casimir functions of {· , ·} J one obtains 0 = X n ∈ K r n { I n , θ m } J = X n ∈ K r n δ nm = r m . Thu s r m = 0 for all m ∈ K , and it follows that s 1 ∇ b,a C 1 + s 2 ∇ b,a C 2 = 0. By (8) and (9), ∇ b,a C 1 and ∇ b,a C 2 are linearly independent, hence s 1 = s 2 = 0 . This shows (i). The pr oo f of (i) also shows that (ii) holds . Theorem 7.3. In addition to the c anonic al r elations s t ate d in The or em 7.1, the angle variables ( θ n ) 1 ≤ n ≤ N − 1 satisfy for any 1 ≤ n , m ≤ N − 1 on M \ ( D n ∪ D m ) { θ n , θ m } J = 0 . (11 6) Pr o of. Let 1 ≤ n, m ≤ N − 1. By contin uity , it suﬃces to prove the iden tit y (116) for ( b, a ) ∈ M \  S N − 1 l =1 D l  . Let ( b, a ) be an arbitr a ry element in M \  S N − 1 l =1 D l  . Recall that Iso( b, a ) denotes the set of all elements ( b ′ , a ′ ) in M with sp ec( Q b ′ ,a ′ ) = sp ec( Q b,a ), Iso ( b, a ) = { ( b ′ , a ′ ) ∈ M : ∆( · , b ′ , a ′ ) = ∆( · , b, a ) } . Then Iso( b, a ) is a torus cont ained in M \  S N − 1 l =1 D l  , and a s a ll eig en v alues of Q ( b, a ) ar e simple, its dimension is N − 1. By Lemma 7.2, at any p oint ( b ′ , a ′ ) ∈ Iso( b, a ), the vectors ( J ∇ b ′ ,a ′ I k ) 1 ≤ k ≤ N − 1 are linearly independent. Using the formula (115) for the gra dien t of I k , one se e s that, by Corolla r y 6 .2, for a n y µ ∈ R , 1 ≤ k ≤ N − 1, h∇ b ′ ,a ′ ∆ µ , J ∇ b ′ ,a ′ I k i = − 1 2 π Z Γ n { ∆ µ , ∆ λ } J c p ∆ 2 λ − 4 dλ = 0 . Hence for any ( b ′ , a ′ ) ∈ Iso( b, a ), ( J ∇ b ′ ,a ′ I k ) 1 ≤ k ≤ N − 1 ∈ T b ′ ,a ′ Iso ( b, a ) , and therefore these vectors for m a basis of T b ′ ,a ′ Iso ( b, a ). T o prove the ident ity (11 6), we apply the Jacobi identit y { F , { G, H } J } J + { G, { H , F } J } J + { H , { F , G } J } J = 0 34 7 CANONICAL RELA TIONS to the functions I k , θ n and θ m . Since by Theo rem 7 .1, { I k , θ n } J = δ kn , w e obtain { I k , { θ n , θ m } J } J = 0 on M \ N − 1 [ l =1 D l ! for any 1 ≤ k ≤ N − 1 . It then follows by the ab ov e consideratio ns that ∇ b ′ ,a ′ { θ n , θ m } J is or thogonal to T b ′ ,a ′ Iso ( b, a ) for all ( b ′ , a ′ ) ∈ Iso( b, a ), i.e. { θ n , θ m } J is c o nstan t on Iso( b, a ), { θ n , θ m } J ( b ′ , a ′ ) = { θ n , θ m } J ( b, a ) ∀ ( b ′ , a ′ ) ∈ Iso ( b, a ) . By [17], Theorem 2.1, there ex ists a unique elemen t ( b ′ , a ′ ) ∈ Iso ( b , a ) satisfying µ k ( b ′ , a ′ ) = λ 2 k ( b, a ) for all 1 ≤ k ≤ N − 1. The claimed identit y (11 6 ) then follows from Pr opo sition 6.7. Pr o of of The or em 1.1. By Theor em 3.5 and Theore m 4.2, the action and angle v ariable s introduced in Deﬁnitions 3.1 and 4.1, respec tiv ely , hav e the claimed analyticity prop erties. The canonica l relations among these v ariables hav e b een veriﬁed in Theorem 7.1 a nd Theo r em 7.3, a nd the r elations { C i , I n } J = 0 (on M ) and { C i , θ n } J = 0 (on M \ D n ) follow from the fact that C 1 and C 2 are Casimir functions. It remains to sho w that the actions Poisson commute with the T o da Hamiltonian. T o this e nd note that that the Hamiltonian H can b e written as H = 1 2 N X n =1 b 2 n + N X n =1 a 2 n = 1 2 tr ( L ( b, a ) 2 ) = 1 2 N X j =1 ( λ + j ) 2 where ( λ + j ) 1 ≤ j ≤ N are the N eig en v alues of L ( b, a ). Recall that on the dense op en subset M \ ∪ N − 1 k =1 D k of M , the λ + i ’s (1 ≤ i ≤ N ) are simple eig en v alues and hence real ana lytic. It then follows by (11 5) tha t for any 1 ≤ n ≤ N − 1, { H , I n } J = N X i =1 λ + i { λ + i , I n } J = − N X i =1 λ + i 2 π Z Γ n { λ + i , ∆ λ } J c p ∆ 2 λ − 4 dλ = 0 , where for the latter identit y we used Corolla ry 6.3. Hence for any 1 ≤ n ≤ N − 1, { H , I n } J = 0 on M \ ∪ N − 1 k =1 D k . By contin uity it then follows that { H, I n } J = 0 everywhere on M . Pr o of of Cor ol lary 1.2. Since for any β ∈ R and α > 0 the sy mplectic leaf M β ,α is a subma nifo ld of M of dimensio n 2( N − 1), there ar e at most N − 1 indep endent int egra ls in inv olution on M β ,α . F or any given ( b , a ) ∈ M β ,α let π β ,α denote the orthogo nal pr o jection T b,a M → T b,a M β ,α . Then the gra dien t of the restr iction I n | M β ,α of I n to M β ,α (1 ≤ n ≤ N − 1 ) is given by π β ,α ∇ b,a I n . By Lemma 7.2 the vectors ( π β ,α ∇ b,a I n ) n ∈ K are linearly independent. As M β ,α \ ∪ N − 1 k =1 D k is dense in M β ,α , it then fo llows that ( I n | M β ,α ) 1 ≤ n ≤ N − 1 are functionally indepe nden t. 35 Finally , as C 1 and C 2 are Casimir functions of {· , ·} J , it follows that for an y ( b, a ) ∈ M β ,α { I n | M β ,α , I m | M β ,α } J ( b, a ) = h π β ,α ∇ b,a I n , π β ,α J ∇ b,a I m i = h∇ b,a I n , J ∇ b,a I m i = { I n , I m } J = 0 , i.e. the restr ictions I n | M β ,α of I n (1 ≤ n ≤ N − 1 ) are in inv olutio n. A Pro of of Lemma 3.7 In this App endix, we pr o ve Lemma 3.7. It tur ns out that the pro of in ([1], p. 601- 602) of the sp ecial case w he r e the par ameter α in (1) equa ls 1 can b e adapted for arbitra ry v alues. Pr o of of L emma 3.7. Let ( b, a ) b e a n a rbitrary ele ment in M a nd 1 ≤ n ≤ N − 1. First note that I n = 1 π R λ 2 n +1 λ 2 n arcosh | 1 2 ∆( λ ) | dλ and use d dt arcosh ( t ) = 1 √ t 2 − 1 to obtain I n = 1 π Z λ 2 n +1 λ 2 n Z | ∆( λ ) | / 2 1 1 √ t 2 − 1 dt dλ. Since the in tegra nd of the inner integral is nonincreasing, we estimate it from below by its v a lue at | ∆( λ ) | 2 . This leads to I n ≥ 1 π Z λ 2 n +1 λ 2 n p | ∆( λ ) | − 2 p | ∆( λ ) | + 2 dλ. (117) W e will show b elow that fo r λ 2 n ≤ λ ≤ λ 2 n +1 p | ∆( λ ) | − 2 p | ∆( λ ) | + 2 ≥ √ λ − λ 2 n p λ 2 n +1 − λ λ 2 N − λ 1 . (118) W e then s ubs titute (118) into the integral (117) a nd s plit the integration interv al int o tw o eq ual pa r ts, I n ≥ 2 π 1 λ 2 N − λ 1 Z τ n λ 2 n √ λ − λ 2 n p λ 2 n +1 − λ λ 2 N − λ 1 dλ, where τ n = ( λ 2 n + λ 2 n +1 ) / 2. F o r λ 2 n ≤ λ ≤ τ n we estimate the qua n tit y λ 2 n +1 − λ from b elow by γ n / 2, yielding I n ≥ 2 π 1 λ 2 N − λ 1 Z τ n λ 2 n r γ n 2 p λ − λ 2 n dλ = 1 3 π ( λ 2 N − λ 1 ) γ 2 n . It remains to verify (1 18). Recall that λ 2 n and λ 2 n +1 are either both p erio dic or b oth antiperio dic eig e n v alues of L . If λ 2 n and λ 2 n +1 are p erio dic eigenv alues, we hav e ∆( λ ) ≥ 2 for λ 2 n ≤ λ ≤ λ 2 n +1 , i.e. | ∆( λ ) | = ∆( λ ). In order to ma ke writing easier, let us assume that N is even - the cas e where N is o dd is tr eated 36 B PROOF OF THEORE M ?? in the same w ay . Then by (19), λ 1 and λ 2 N are p erio dic e igen v alues of L and th us for any λ 2 n ≤ λ ≤ λ 2 n +1 , the left side of (118) can b e estimated from below by p ∆( λ ) − 2 p ∆( λ ) + 2 = s ( λ − λ 2 n )( λ − λ 2 n +1 ) ( λ − λ 2 )( λ − λ 2 N − 1 ) · R ≥ √ λ − λ 2 n p λ 2 n +1 − λ λ 2 N − λ 1 · R , where R ≡ R ( λ ) = s λ − λ 1 λ − λ 3 · · · λ − λ 2 n − 3 λ − λ 2 n − 1 λ 2 n +4 − λ λ 2 n +2 − λ · · · λ 2 N − λ λ 2 N − 2 − λ . (119) As each of the the fra ctions under the squar e ro ot in (11 9) can b e estimated fro m below by 1, for any λ 2 n ≤ λ ≤ λ 2 n +1 it follows that R ( λ ) ≥ 1 o n [ λ 2 n , λ 2 n +1 ], leading to the claimed es timate (11 8). B Pro of of Theorem 3.9 In this App endix we prov e Theor em 3.9 using estimates derived in [1]. Let ( b, a ) be in M β ,α with β ∈ R and α > 0 ar bitrary . T o s how Theo rem 3 .9 we need the following Prop osition B. 1. F or any ( b, a ) ∈ M β ,α with β ∈ R , α > 0 arbitr ary and any 1 ≤ n ≤ N , λ 2 n ( b, a ) − λ 2 n − 1 ( b, a ) ≤ 2 π α N . (120) Before pr o ving Pr o pos itio n B.1 we show how to use it to prove Theorem 3.9. Pr o of of The or em 3.9. W e be g in b y adding up the inequa lities (60) and get N − 1 X n =1 γ 2 n ≤ 3 π ( λ 2 N − λ 1 ) N − 1 X n =1 I n ! . (121) Note that λ 2 N − λ 1 = N − 1 X n =1 γ n + N X n =1 ( λ 2 n − λ 2 n − 1 ) . By the estimate of Pr opo s ition B.1 we get for any ( b, a ) ∈ M β ,α λ 2 N − λ 1 ≤ 2 π α + N − 1 X n =1 γ n which we substitute into (121) to yield N − 1 X n =1 γ 2 n ≤ 6 απ 2 N − 1 X n =1 I n ! + 3 π N − 1 X n =1 γ n ! N − 1 X n =1 I n ! . 37 Using the inequality 2 ab ≤ ǫ 2 a 2 + 1 ǫ 2 b 2 ( a, b ∈ R , ǫ > 0) with a = P N − 1 n =1 γ n , b = P N − 1 n =1 I n , and ǫ 2 = 1 3 π ( N − 1) , one gets N − 1 X n =1 γ 2 n ≤ 6 π 2 α N − 1 X n =1 I n ! + 3 π 2   1 3 π ( N − 1 ) N − 1 X n =1 γ n ! 2 + 3 π ( N − 1) N − 1 X n =1 I n ! 2   . As  P N − 1 n =1 γ n  2 ≤ ( N − 1 )  P N − 1 n =1 γ 2 n  , one then concludes that 1 2 N − 1 X n =1 γ 2 n ≤ 6 π 2 α N − 1 X n =1 I n ! + 9 π 2 2 ( N − 1 ) N − 1 X n =1 I n ! 2 , (122) which is the cla imed estimate (61). T o prov e Prop osition B.1 w e ﬁrs t need to make some preparatio ns . Note that for an element o f the for m ( b, a ) = ( β 1 N , α 1 N ) one has, by L e mma 2.6, λ 2 n ( β 1 N , α 1 N ) − λ 2 n − 1 ( β 1 N , α 1 N ) = 2 α  cos ( n − 1) π N − cos nπ N  = 4 α sin (2 n − 1) π 2 N sin π 2 N < 2 π α N . Hence to prov e Prop o sition B.1 it s uﬃces to show that for any ( b, a ) ∈ M β ,α and any 1 ≤ n ≤ N λ 2 n ( b, a ) − λ 2 n − 1 ( b, a ) ≤ λ 2 n ( − β 1 N , α 1 N ) − λ 2 n − 1 ( − β 1 N , α 1 N ) . (123) T o this end, fo llowing [15] (cf. also [7]), we in tro duce the confor mal map δ ( λ ) := ( − 1) N Z λ λ 1 ˙ ∆( µ ) p 4 − ∆ 2 ( µ ) dµ, (124) where the sign o f the s quare ro ot is chosen such that for µ < λ 1 , p 4 − ∆ 2 ( µ ) has po sitiv e imagina ry part. It is deﬁned on the upp er half plane U := { Im z > 0 } and its image is the spike do main Ω( b, a ) := { x + iy : 0 < x < N π , y > 0 } \ N − 1 [ n =1 T n where for 1 ≤ n ≤ N − 1, T n denotes the spike T n := ( nπ + it : 0 < t ≤ ar cosh ( − 1) N + n ∆( ˙ λ n ) 2 !) . 38 B PROOF OF THEORE M ?? T o see that δ ( U ) = Ω( b, a ), note that for a n y ( b , a ) ∈ M and λ ∈ U the discriminant ∆( λ ) a nd the function δ ( λ ) are re la ted by the formula ∆( λ ) = 2( − 1 ) N cos δ ( λ ) . (125) T o prov e (125), recall tha t for − 1 < t < 1, one has d dt arccos t = 1 + √ 1 − t 2 . This formula remains v alid for any t in C \ (( −∞ , − 1 ] ∪ [1 , ∞ )). Thus δ ( λ ) = ( − 1) N Z λ λ 1 ˙ ∆( µ ) p 4 − ∆ 2 ( µ ) dµ = Z λ λ 1 d dµ arccos  ( − 1) N ∆( µ ) 2  . Since by (19), ∆( λ 1 ) = 2 ( − 1) N , w e then ge t δ ( λ ) = arccos  ( − 1) N ∆( µ ) 2     λ λ 1 = ar ccos  ( − 1) N ∆( λ ) 2  , (126) leading to formula (12 5) a nd the cla imed statement that δ ( U ) = Ω( b, a ). The map δ can be extended con tin uously to the closure { Im z ≥ 0 } of the upper half plane. This extensio n, aga in deno ted by δ , is 2-1 ov er e ac h nontrivial spike T n and 1 -1 otherwise. Since the n -th spike T n is the image under δ of the n -th gap ( λ 2 n , λ 2 n +1 ), all spikes are empt y iﬀ all gaps are co llapsed. By Lemma 2.6, all gaps ar e collapsed for ( b, a ) = ( − β 1 N , α 1 N ), hence Ω( b, a ) ⊂ Ω( − β 1 N , α 1 N ) for any ( b, a ) ∈ M β ,α . Note that λ 2 n ( b, a ) − λ 2 n − 1 ( b, a ) = δ − 1 ( nπ − ) − δ − 1 (( n − 1) π +) = Z ∞ −∞ u ( n ) ( δ ( λ )) d λ, (127) where u ( n ) : Ω( b, a ) → R , z 7→ u ( n ) ( z ; b, a ) is the har monic mea sure of the op en subset (( n − 1) π , n π ) of ∂ Ω( b, a ) (see e.g. [6] for the notion of the harmonic measure). W e need tw o lemmas from complex and harmonic analys is , resp ectively . Lemma B.2. F or ( b, a ) = ( − β 1 N , α 1 N ) with β ∈ R , α > 0 arbitr ary, the map δ ( λ ) deﬁne d by (124) is given by δ ( λ ) = N arccos  − λ + β 2 α  . (128) F or arbitr ary ( b, a ) in M β ,α and ξ ∈ R , the fol lowing asymptotic estimate holds as η → ∞ δ ( ξ + iη ) = N a rccos  − ξ + iη + β 2 α  + O ( η − 2 ) , (129) lo c al ly uniformly in ξ . Pr o of of L emma B.2 . In view of the formulas (51) and (52) for the fundamen- tal solutions y 1 and y 2 for ( b, a ) = ( − β 1 N , α 1 N ), the discrimina n t ∆( λ ) = 39 ∆( λ, − β 1 N , α 1 N ) is given by ∆( λ ) = y 1 ( N , λ ) + y 2 ( N + 1 , λ ) = − sin( ρ ( N − 1)) sin ρ + sin( ρ ( N + 1 )) sin ρ = 2 cos( ρN ) , where π < ρ < 2 π is determined by cos ρ = λ + β 2 α . Hence ∆( λ ) = 2 T N  λ + β 2 α  (130) where for any z ∈ U , T N ( z ) = cos( N arc cos z ) . (131) Actually , T N ( z ) is a p olynomial in z of degree N , r e ferred to as Cheb ychev po lynomial of the ﬁrst kind. Substituting (1 30) in to (126), we obtain δ ( λ ) = arccos  ( − 1) N ∆( λ ) 2  = arccos  ( − 1) N T N  λ + β 2 α  . The claimed identit y (12 8) now follows from the elementary symmetry T N ( z ) = ( − 1 ) N T N ( − z ) ∀ z ∈ C . Now let ( b, a ) ∈ M β ,α . The a symptotic estimate (1 29) follows by compar- ing the polyno mials ∆( λ ) cor rep o nding to ( b, a ) and the one co rresp onding to ( − β 1 N , α 1 N ). By (17) (and the discussion following it), in b oth ca s es, ∆( λ ) = α − N λ N  1 + N β λ − 1 + O ( λ − 2 )  as | λ | → ∞ . This implies that ∆ b,a ( λ ) = ∆ − β 1 N ,α 1 N ( λ ) · (1 + O ( λ − 2 )) , hence b y (12 8) and (13 0) δ b,a ( λ ) = arccos  ( − 1) N ∆ b,a ( λ ) 2  = arccos  ( − 1) N 2 ∆ − β 1 N ,α 1 N ( λ ) · (1 + O ( λ − 2 ))  . = arccos  ( − 1) N T N  λ + β 2 α  · (1 + O ( λ − 2 ))  = arccos  T N  − λ + β 2 α  · (1 + O ( λ − 2 ))  . Substituting formula (13 1) for T N , one then concludes that δ b,a ( λ ) = arccos  cos  N arccos  − λ + β 2 α  · (1 + O ( λ − 2 ))  = N arccos  − λ + β 2 α  + O ( λ − 2 ) , (132) 40 B PROOF OF THEORE M ?? where in the last step we us ed tha t a rccos z = − i log( z + i + √ 1 − z 2 ) for any z in C \ (( −∞ , − 1] ∪ [1 , ∞ )). Lemma B.3. L et u : Ω( b, a ) → R b e a b ounde d harmonic function su ch that the nontangential limit of u ( z ) on ∂ Ω( b, a ) has c omp act supp ort, and let U ( λ ) := u ( δ ( λ )) , wher e δ ( λ ) is t he function deﬁne d by (124). Then for almost every t ∈ R , the limit U ( t ) := lim η → 0 U ( t + iη ) exists and is inte gr able, and Z ∞ −∞ U ( t ) dt = lim x →∞ 2 π α sinh  x N  u  N π 2 + ix  . (133 ) Pr o of of L emma B.3 . Aga in b y F a tou’s theorem, for a.e. t the (non ta ngen- tial) limit lim η → 0 U ( t + iη ) exists, since U is a b o unded harmonic function on { Im ( z ) > 0 } . Since u is bounded on Ω( b, a ) and its nontangen tial limit to ∂ Ω has compa ct suppo rt, U ( t ) is bounded and compa ctly supp o rted and th us in particular integrable. F or λ = ξ + iη , one then has the Poisson represe ntation U ( ξ + iη ) = η π Z ∞ −∞ U ( t ) ( t − ξ ) 2 + η 2 , (134) and b y domina ted convergence we conclude that Z ∞ −∞ U ( t ) dt = lim η →∞ π η U ( ξ + iη ) . (135) (In particular , the limit in the latter expres sion exists.) In order to compute the right hand side o f (135), let ξ + i η b e given by ξ + iη = δ − 1  N π 2 + ix  , for x suﬃciently large. Then U ( ξ + iη ) = u  N π 2 + ix  , and by (132), it follows that π 2 + i N x = ar ccos  − ( ξ + β ) + iη 2 α  + O  ( ξ + iη ) − 2  ( x → ∞ ) . (136) T aking the co sine of both sides o f (136), m ultiplying by − 2 α and using that cos( π 2 + it ) = − i sinh t for t ∈ R , we obtain 2 iα sinh x N = (( ξ + β ) + iη )  1 + O  ( ξ + iη ) − 2  ( x → ∞ ) . Hence, as x → ∞ , ξ = O (1) , η = 2 α s inh x N + O (1) . (137) Substituting (137) into (135) lea ds to the claimed formula (1 33). REFERENCES 41 Pr o of of Pr op osition B.1 . Let ( b, a ) ∈ M β ,α . Bes ides the har monic measur e u ( n ) of the set E := (( n − 1) π , nπ ) ⊂ ∂ Ω( b, a ) we a lso consider the harmo nic mea sure u ( n ) β ,α of E ⊂ ∂ Ω( − β 1 N , α 1 N ); note that Ω( − β 1 N , α 1 N ) = { x + iy | 0 < x < N π , y > 0 } and hence Ω( b, a ) ⊆ Ω( − β 1 N , α 1 N ). Accor ding to [6], b o th u ( n ) and u ( n ) β ,α satisfy the hypotheses of Lemma B.3. Le t us r ecall the monotonicit y pro p erty of the harmonic mea sures u ( z , E , Ω) with resp ect to Ω (see e.g. [6]): If Ω 1 ⊆ Ω 2 , E ⊂ ∂ Ω 1 ∩ ∂ Ω 2 , and u ( z , E , Ω i ) ( i = 1 , 2) denotes the har monic measure of E ⊂ ∂ Ω i , then for any z ∈ Ω 1 , u ( z , E , Ω 1 ) ≤ u ( z , E , Ω 2 ). Apply this ge ne r al principle to Ω 1 := Ω( b, a ) and Ω 2 := Ω( − β 1 N , α 1 N ) to get u ( n ) ( x ) ≤ u ( n ) β ,α ( x ) . (138) W riting U ( n ) ( λ ) := u ( n ) ( δ ( λ )) as well as U ( n ) β ,α ( λ ) := u ( n ) β ,α ( δ ( λ )) and combin- ing (127), (133), and (138), we c onclude that λ 2 n ( b, a ) − λ 2 n − 1 ( b, a ) = Z ∞ −∞ U ( n ) ( λ ) d λ = lim x →∞ 2 π α sinh  x N  u ( n )  N π 2 + ix  ≤ lim x →∞ 2 π α sinh  x N  u ( n ) β ,α  N π 2 + ix  = Z ∞ −∞ U ( n ) β ,α ( λ ) d λ = λ 2 n ( − β 1 N , α 1 N ) − λ 2 n − 1 ( − β 1 N , α 1 N ) . This co mpletes the pro of of the estimate (1 20) and therefore of Prop osition B.1. References [1] D. B ¨ attig, A. M. Bloch, J. C. Guillot & T. Kappeler , O n the sym- plectic structure of the phase space for p erio dic KdV, T o da, and defocus ing NLS. Duke Math. J. 79 (1995 ), 54 9-604. [2] D. B ¨ attig, B. Gr ´ eber t, J. C. Guillot & T. Ka ppeler , Fibration of the phase space of the perio dic T o da lattice. J. Math. Pur es Appl. 7 2 (1993), 553 -565. [3] H . Flaschka , The T o da lattice. I. Existence of integrals. Phys. R ev. , Sect. 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V aninsky , Actio n-angle v aria ble s for the cubic Schroedinger equation. Comm. Pur e Appl. Math. 50 (1997 ), 489-5 62. [17] P. v an Moerbeke , The spe c trum of Jacobi matrices. Invent. Math. 37 (1976), 45-81 . [18] G. Teschl , Jac obi Op er ators and Completely Inte gr able Nonline ar L at- tic es . Math. Surv eys and Monographs 72 , Amer. Ma th. So c., Pr ovidence, 2000. [19] M. Toda , The ory of Nonline ar L attic es , 2nd enl. ed., Springer Series in Solid-State Sciences 20 , Springer , Ber lin, 19 89. [20] M. Tsuji , Potent ial The ory in Mo dern F u nction The ory , Mruzen, T okyo, 1959. REFERENCES 43 Institut f ¨ ur Ma thema tik, Universit ¨ at Z ¨ urich, Winter th u rerstrass e 190, CH-8057 Z ¨ urich, S witzerl a n d E-mail addr ess: a ndreas.h enrici@math.unizh.ch Institut f ¨ ur Ma thema tik, Universit ¨ at Z ¨ urich, Winter th u rerstrass e 190, CH-8057 Z ¨ urich, S witzerl a n d E-mail addr ess: t homas.ka ppeler@math.unizh.ch

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