Global Birkhoff coordinates for the periodic Toda lattice

Global Birkhoff co ordinates for the p erio dic T o da lattice Andreas Henrici Thomas Kapp eler ∗ No v em ber 9, 2018 Abstract In this paper w e prov e that t h e period ic T oda lattice admits globally defined Birkhoff coordin ates. 1 1 In tro duction Consider the T o da lattice with p erio d N ( N ≥ 2), ˙ q n = ∂ p n H, ˙ p n = − ∂ q n H for n ∈ Z , where the (rea l) coo rdinates ( 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 by 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 o s itive pa rameter, α > 0. F or the sta ndard T o da lattice, α = 1 . The T o da lattice was intro duced by T o da [17] and studied extens ively in the sequel. It is a n imp or ta nt mo del for an integrable system of N particles in one space dimension with near est neighbor interaction and b elongs to the family of lattices introduce and n umerically inv estigated b y F ermi, Pasta, and Ulam in their seminal pa p e r [5]. T o prove the integrability o f the T o da la ttice, Flas chk a int ro duced in [3] the (noncanonica l) co o rdinates 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 rela tive to the center of mass. Note that the total momen tum is conserved by the T o da flow, hence any tra jectory of the center of mass is a straight line. ∗ Supported in part b y the Swi ss National Science F oundation, the pr ogramm e SPECT, and the European Communit y through the FP6 Mari e Cur ie R TN ENIGMA (MR TN-CT- 2004-5652) 1 2000 Mathematics Sub ject Classification: 37J35, 70H06 1 2 1 INTR ODUCTION In these co or dinates 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 eq uations of motion are  ˙ 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 ident ify the sequences ( b n ) n ∈ Z and ( a n ) n ∈ Z with the vectors ( b n ) 1 ≤ n ≤ N ∈ R N and ( a n ) 1 ≤ n ≤ N ∈ R N > 0 . Our a im is to study the normal form of the sy s tem of equa tions (2) on the phase space M := R N × R N > 0 . This sys tem is Hamiltonia n with resp ect to the nonstandar d Poisson str ucture J ≡ J b,a , defined a t 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 brack et corr e s p o nding 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 where ∇ b and ∇ a denote the gr adients w ith resp ect to the N -vectors b = ( b 1 , . . . , b N ) and a = ( a 1 , . . . , a N ), resp ectively . Therefore , equations (2) can alter natively b e written a s ˙ b n = { b n , H } J , ˙ a n = { a n , H } J (1 ≤ n ≤ N ). F urther note 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 fo r a ny n, k with n / ∈ { k , k + 1 } . 3 Since the matrix A defined by (4) has rank N − 1, the Poisson structure J is deg enerate. It a dmits 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 gr adients ∇ b,a C i = ( ∇ b C i , ∇ a C i ) ( i = 1 , 2), given by ∇ 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 ea ch p oint ( b, a ) of M . Let M β ,α := { ( b, a ) ∈ R 2 N : ( C 1 , C 2 ) = ( β , α ) } denote the level set o f ( 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 . By (8)-(9), the sets M β ,α are real a nalytic submanifolds of M of co dimension t wo. F urthermore the Poisson structure J , restricted to M β ,α , b ecomes no ndegenerate everywhere on M β ,α and therefore induces a symplectic structure ν β ,α on M β ,α . In this way , w e obtain a symplectic foliation of M with M β ,α being its (s y mplectic) leav es. T o state the ma in re s ult of this pa p e r, we intro duce the mo del space P := R 2( N − 1) × R × R > 0 endow ed with the degener ate Poisson str uctur e J 0 whose symplectic leav es a re R 2( N − 1) × { β } × { α } endow ed with the cano nical symplectic s tructure. Theorem 1.1 . The r e exists a map Ω : ( M , J ) → ( P , J 0 ) ( b, a ) 7→ (( x n , y n ) 1 ≤ n ≤ N − 1 , C 1 , C 2 ) with t he fol lowing pr op erties: • Ω is a r e al analytic diffe omorphi sm. • Ω is c anonic al, i .e. it pr eserves t he Poisson br ackets. In p articular, t he symple ctic foliation of M by M β ,α is t rivial. • The 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. the tr ansforme d T o da Hamiltonian ˆ H = H ◦ Ω − 1 is a function of the actions ( I n ) 1 ≤ n ≤ N − 1 and C 1 , C 2 alone. 2 A smo oth function C : M → R is a Casim ir function for J if { C, ·} J ≡ 0. 4 2 PREL IMINARIES F urther prop erties of Ω are discusse d at the end o f sectio n 7. In [9], we us e the Birkho ff co ordina tes ( x n , y n ) 1 ≤ n ≤ N − 1 given by Theorem 1.1 to o btain a KAM-theorem for the p er io dic T o da la ttice. R elate d work: Theorem 1.1 improv es o n ear lier work on the no rmal form of the p erio dic T o da lattice in [1, 2 ]. In particular, we co nstruct g lobal Bir khoff co ordinates on all o f M instea d of a s ing le symplectic leaf and show that tech- niques recently developed for trea ting the K dV equatio n (cf. [10, 11]) and the defo cusing NLS equation (cf. [7 , 14]) can also b e applied for the T o da lattice. Outline of the p ap er: In section 2 we re v iew the Lax pair of the per io dic T o da lattice and collect some auxilia ry res ults on the s p e c trum of the Jacobi ma tr ix L ( b, a ) ass o ciated to an e lement ( b, a ) ∈ M . The construction of the co ordinates ( x n , y n ) 1 ≤ n ≤ N − 1 (see sec tio n 4) uses the actio n v ar iables ( I n ) 1 ≤ n ≤ N − 1 defined on M and the angle v a riables ( θ n ) 1 ≤ n ≤ N − 1 defined on a dense domain M ′ ⊂ M , bo th of which hav e been studied in detail in o ur previous pap e r [8]. W e give a brief rev iew of these a c tio n-angle v ariables in sectio n 3 . The co o rdinates ( x n , y n ) 1 ≤ n ≤ N − 1 are then defined on the dense domain M ′ by ( x n , y n ) = p 2 I n (cos θ n , sin θ n ) . In a firs t step we show that the co ordinate functions ( x n , y n ) extend to real analytic functions on all o f M . Using the canonical rela tions among the a ction- angle v ariables e s tablished in [8] w e then show in s ection 7 that Ω is a ca nonical lo cal diffeomorphism. Finally , the fact that Ω is 1-1 and onto is deduced from a prior i estimates o f the a ctions which are used to prov e that Ω is prop er (cf. section 7). 2 Preliminaries It is w ell known (cf. e.g. [17]) that the system (2) admits a Lax pair formulation ˙ L = ∂ L ∂ t = [ B , L ], where L ≡ L + ( b, a ) is the p er io dic J a cobi 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          . 5 Hence the flow of ˙ L = [ B , L ] is isos pe c tral. Prop ositi o n 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 quant ities. Let us now co llect a few results from [15] and [1 7] of the sp ectral theory o f Jacobi matrices needed in the sequel. Denote by M C the complex ific a tion 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 consider for any complex num ber λ the differenc e equa tion ( 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 erator R b,a = a k − 1 S − 1 + b k S 0 + a k S 1 (12) with S m denoting the shift op era tor of order m ∈ Z , i.e. ( S m y )( k ) = y ( k + m ) for k ∈ Z . F undamental solutions: The tw o fundamental solutio ns y 1 ( · , λ ) and y 2 ( · , λ ) of (11) are defined b y the standar d initial conditions y 1 (0 , λ ) = 1 , y 1 (1 , λ ) = 0 and y 2 (0 , λ ) = 0, y 2 (1 , λ ) = 1. They satisfy the Wro nskian 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) 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 o lynomial in λ of degree at most k − 1 and dep ends real ana lytically on ( b, a ) (see [15]). In particula r, one easily verifies that y 2 ( N + 1 , λ, b, a ) is a p o lynomial in λ of degree N with leading term α − N λ N , whereas y 1 ( N , λ ) is a poly nomial in λ of degr ee les s than N . 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 ) . (15) In the se q uel, we will often write ∆ λ for ∆( λ ). As y 2 ( N + 1 , λ ) = α − N λ N + . . . and y 1 ( N , λ ) = O ( λ N − 1 ), ∆( λ , b, a ) is a p o lynomial in λ of degree N with leading term α − N λ N , and it depends real analy tically on ( b, a ) (see e.g. [17]). According to Flo q ue t’s Theor em (see e.g. [16]), for λ ∈ C given, (11) admits a p er io dic o r antiperio dic so lution of per io d N if the discr imina nt ∆( λ ) satisfies ∆( λ ) = +2 or ∆( λ ) = − 2, r e s p e c tively . These so lutions corresp ond to eigenv ectors of L + or L − , re sp ectively , with L ± defined by (10) . It turns out to b e mor e co nv enien t to 6 2 PREL IMINARIES combine these tw o cases by co ns idering the p erio dic J acobi matr ix Q ≡ Q ( b, a ) of size 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 spec tr um of the matrix Q is the union o f the spec tr a of the matrices L + and L − and therefore the zero set of the p olyno mia l ∆ 2 λ − 4. The function ∆ 2 λ − 4 is a p olyno mia l in λ o f degre e 2 N and admits a pr o duct represe ntation ∆ 2 λ − 4 = α − 2 N 2 N Y j =1 ( λ − λ j ) . (16) The facto r α − 2 N in (16) comes fro m the ab ov e men tioned fact that the leading term o f ∆( λ ) is α − N λ N . F or a ny ( b, a ) ∈ M , the matr ix Q is symmetric a nd hence the eigenv alues ( λ j ) 1 ≤ j ≤ 2 N of Q a re rea l. When listed in increas ing o rder and with their a lge- braic mult iplicities, they sa tisfy the following relations (cf. [15 ]) λ 1 < λ 2 ≤ λ 3 < λ 4 ≤ λ 5 < . . . λ 2 N − 2 ≤ λ 2 N − 1 < λ 2 N . As explained ab ov e, the λ j are p erio dic or a ntiper io dic eigenv alues of L and th us e ig env a lues of L + or L − according to whether ∆( λ j ) = 2 o r ∆( λ j ) = − 2. One ha s (cf. [1 5]) ∆( λ 1 ) = ( − 1) N · 2 , ∆( λ 2 n ) = ∆( λ 2 n +1 ) = ( − 1) n + N · 2 , ∆( λ 2 N ) = 2 . (17) Since ∆ λ is a p olynomial of degre e N with leading term α − N λ N , ˙ ∆ λ ≡ ˙ ∆( λ ) = d dλ ∆( λ ) is a p olyno mia l o f degr ee N − 1 with leading term N α − N λ N − 1 , hence admits a pr o duct repres e nt ation o f the form ˙ ∆ λ = N α − N N − 1 Y k =1 ( λ − ˙ λ k ) . (18) The zero es ( ˙ λ n ) 1 ≤ n ≤ N − 1 of ˙ ∆ λ satisfy λ 2 n ≤ ˙ λ n ≤ λ 2 n +1 for an y 1 ≤ n ≤ N − 1. The op en interv als ( λ 2 n , λ 2 n +1 ) are r eferred to as the n -th sp e ctr al gap and γ n := λ 2 n +1 − λ 2 n as the n -t h gap length . Note that | ∆( λ ) | > 2 on the sp ectral 7 gaps. W e say that the n - th gap is op en if γ n > 0 a nd c ol lapse d otherwis e. The set of elements ( b, a ) ∈ M for which the n - th gap is collapsed is denoted by D n , D n := { ( b, a ) ∈ M : γ n = 0 } . (19) Using tha t γ 2 n (unlik e γ n ) is a rea l analytic function on M , it ca n b e shown that D n is a real analytic submanifold of M o f co dimensio n 2 (cf. [11] for a similar statement in the case o f Hill’s op er ator). Isolating n eighb orho o ds: Let ( b, a ) ∈ M b e given. The str ict inequa lities λ 2 n − 1 < λ 2 n (1 ≤ n ≤ N ) g ua rantee the existence of a fa mily of mutually disjoint o p en subsets ( U n ) 1 ≤ n ≤ N − 1 of C so that for a ny 1 ≤ n ≤ N − 1 , U n is a neighborho o d of the closed interv a l [ λ 2 n , λ 2 n +1 ]. Such a family of neighborho o ds is r eferred to as a family o f isolating neighb orho o ds for ( b, a ). In the case w her e ( b, a ) ∈ M C , we lis t the eigenv alues ( λ j ) 1 ≤ j ≤ 2 N in lexico - graphic o rdering 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 ) ∈ M C : γ n = 0 } . (21) In the seq uel, we will omit the s uper script 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 . As the lexicogr aphic ordering is not contin uous, the λ i ’s and ˙ λ i ’s no longer depend contin uously on ( b, a ) ∈ M C . How ever, if we choo s e a small enough complex neighborho o d W o f M in M C , then for any ( b, a ) ∈ W the c lo sed interv als G n ⊆ C (1 ≤ n ≤ N − 1) defined by G n := { (1 − t ) λ 2 n + tλ 2 n +1 : 0 ≤ t ≤ 1 } (22) are pairwise disjoint, and he nce , as in the re a l case, there exists a family of isolating neig hborho o ds ( U n ) 1 ≤ n ≤ N − 1 . Lemma 2.2. 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 2.3. In the se quel, we wil l have to shrink the c omplex neighb orho o d W sever al times, but c ontinue to denote it by the same letter. 3 The lexicographic or deri ng a ≺ b for complex n umbers a and b i s defined by a ≺ b : ⇐ ⇒ 8 < : Re a < Re b or Re a = Re b and Im a ≤ Im b. (20) 8 2 PREL IMINARIES Contours Γ n : F or any ( b, a ) ∈ W and any 1 ≤ n ≤ N − 1, we deno te by Γ n a c ir cuit in U n around G n with counterclockwise orientation. Isosp e ctr al set: F or ( b, a ) ∈ M , the set Iso( b, a ) of all element s ( b ′ , a ′ ) ∈ M so that Q ( b ′ , a ′ ) has the same sp ectrum as Q ( b, a ) is descr ib ed with the help o f the Dirichlet eigenv alues µ 1 < µ 2 < . . . < µ N − 1 of (11) defined by y 1 ( N + 1 , µ n ) = 0 . (23) They coincide with the eigen v a lues of the ( N − 1) × ( N − 1)-matr ix L 2 = L 2 ( b, a ) given by          b 2 a 2 0 . . . 0 a 2 . . . . . . . . . . . . 0 . . . . . . . . . 0 . . . . . . . . . . . . a N − 1 0 . . . 0 a N − 1 b N          . In the sequel, we will also r efer to µ 1 , . . . , µ N − 1 as the Dirichlet eigenv alues of L ( b, a ). Ev aluating the W ro ns kian identit y (13 ) 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 . (2 4) Hence the v a lue o f the dis criminant at µ n is given b y ∆( µ n ) = y 2 ( N + 1 , µ n ) + 1 y 2 ( N + 1 , µ n ) (25) and | ∆( µ n ) | ≥ 2. By Lemma 2.4 b elow, given the p o int ( b, a ) with b 1 = . . . = b N = β and a 1 = . . . = a N = α , one ha s λ 2 n = λ 2 n +1 and hence µ n = λ 2 n for any 1 ≤ n ≤ N − 1. It then follows from a straig ht forward deformation argument that λ 2 n ≤ µ n ≤ λ 2 n +1 everywhere in the real space M . Conv ersely , ac c ording to v an Mo er be ke [15], g iven any (real) Jaco bi matrix Q with sp ectrum λ 1 < λ 2 ≤ λ 3 < λ 4 ≤ λ 5 < . . . λ 2 N − 2 ≤ λ 2 N − 1 < λ 2 N and any se q uence ( 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 Jac o bi matr ices Q with sp ectrum ( λ n ) 1 ≤ n ≤ 2 N and Dirichlet sp ectrum ( µ n ) 1 ≤ n ≤ N − 1 , wher e r is the num ber o f n ’s with λ 2 n < µ n < λ 2 n +1 . In the case where ( b, a ) ∈ M C , we contin ue to define the Dirichlet eigenv alues ( µ n ) 1 ≤ n ≤ N − 1 by (23 ), and we list them in lexicog raphic ordering µ 1 ≺ µ 2 ≺ . . . ≺ µ N − 1 . Then the µ i ’s no longer dep end co nt inuously on ( b, a ) ∈ M C . How ev er, if we c ho ose the co mplex neighbor ho o d W of M in M C of Lemma 2.2 small enough, then for any ( b, a ) ∈ W and 1 ≤ n ≤ N − 1, ther e exist isola ting neighborho o ds ( U n ) 1 ≤ n ≤ N − 1 so that µ n is contained in the neig hborho o d U n of G n (but not necessarily in G n itself ). F or la ter use, we co mpute the sp ectra of Q ( b, a ) and L 2 ( b, a ) in the sp e- cial cas e ( b, a ) = ( β 1 N , α 1 N ) with β ∈ R and α > 0. Here 1 N denotes the 9 vector (1 , . . . , 1) ∈ R N . These points ar e the equilibrium p oints (of the re s tric- tions) o f the T o da Hamiltonian vector field (to the symplec tic leaves M β ,α ). W e compute the sp ectrum ( λ j ) 1 ≤ j ≤ 2 N of the matr ix Q ( β 1 N , α 1 N ) a nd the Dirichlet eigenv a lues ( µ l ) 1 ≤ l ≤ N − 1 of L = L ( β 1 N , α 1 N ). F urthermor e, for any 1 ≤ l ≤ N − 1 , we compute a normalize d eige nv ector corr esp onding to the eigenv alue µ l , g l =  g l ( j )  1 ≤ j ≤ N , i.e. Lg l = µ l g l , g l (1) = 0 , and a vector h l =  h l ( j )  1 ≤ j ≤ N which is the normalized so lutio n of L y = µ l y o rthogona l to g l satisfying W ( h l , g l )( N ) > 0. Lemma 2.4. The sp e ctrum ( 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 cos l π N (1 ≤ l ≤ N − 1) , λ 2 N = β + 2 α. In p articular, al l sp e ctr al gaps of Q ( β 1 N , α 1 N ) ar e c ol lapse d. F or any 1 ≤ l ≤ N − 1 , the ve ct ors 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 ) , (26) h l ( j ) = ( − 1) j r 2 N cos ( j − 1) l π N (1 ≤ j ≤ N ) (27) satisfy Ly = µ l y and the normalization c onditio ns 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 . Remark 2 . 5. R e c al l that ( g l ( j )) 1 ≤ j ≤ N is a ve ctor in R N . The normalizatio n c ondition g l (0) > 0 me ans t hat g l ( j ) = ν l y 1 ( j, µ l ) for 1 ≤ j ≤ N with ν l > 0 . 3 Action-angle v ariables In this section we summar ize the results obtained in [8] whic h we will ne e d in the sequel. First w e have to intro duce s ome mo re notation. Rie mann surfac e Σ b,a : Denote by Σ b,a the Riemann surface obtained as the compactification o f 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 sp ectral inv a riants. (Strictly sp eaking, Σ b,a is a Riemann surface only if the sp ectrum of Q ( b, a ) is simple - se e e.g. App endix A 10 3 A CTION-ANGLE V ARIABLES in [16] for details in this case. If the sp ectrum of Q ( b, a ) is not simple, Σ( b, a ) bec omes a Riemann surface a fter doubling the multiple eigenv alues - see e.g. section 2 o f [12].) Dirichlet divisors: T o the Dirichlet eigenv alue µ n (1 ≤ n ≤ N − 1) we asso ciate the p oint µ ∗ n on the s urface 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 ∆ 2 µ n − 4 = ( y 1 ( N , µ n ) − y 2 ( N + 1 , µ n )) 2 . Standar d r o ot: The standard ro o t or s -ro ot for shor t, s √ 1 − λ 2 , is defined for λ ∈ C \ [ − 1 , 1 ] by s p 1 − λ 2 := iλ + p 1 − λ − 2 . (30) More gener ally , we define for λ ∈ C \ { ta + (1 − t ) b | 0 ≤ t ≤ 1 } the s -ro ot o f a radicand o f 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 s he et and c anonic al r o ot: F or ( b , a ) ∈ M the canonical sheet of Σ b,a is given by the set of p o ints ( λ, c p ∆ 2 λ − 4) in C b,a , where the c -roo t 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 sign condition − i c q ∆ 2 λ − 4 > 0 for λ 2 N − 1 < λ < λ 2 N . (32) As a consequence o ne has for any 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 of the cano nica l sheet a nd the c - ro ot ca n b e extended to the neighborho o d W of M in M C of Lemma 2.2. Ab elian differ entials: Let ( b , a ) ∈ M and 1 ≤ n ≤ N − 1. Then there exists a unique p oly no mial ψ n ( λ ) o f degr ee at mos t N − 2 such that for an y 1 ≤ k ≤ N − 1 1 2 π Z c k ψ n ( λ ) p ∆ 2 λ − 4 dλ = δ kn . (34) Here, for a ny 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 follows fr om (34) that 1 π Z λ 2 k +1 λ 2 k ψ n ( λ ) + p ∆ 2 λ − 4 dλ = 0 . (35) Hence in e very g ap ( λ 2 k , λ 2 k +1 ) with k 6 = n the p olynomial ψ n has a zero which we denote by σ n k . If λ 2 k = λ 2 k +1 then it follo ws from (3 4) and Cauc hy’s theore m 11 that σ n k = λ 2 k = λ 2 k +1 . As ψ n ( λ ) is a p olynomia l of degree at mos t N − 2, one has ψ n ( λ ) = M n Y 1 ≤ k ≤ N − 1 k 6 = n ( λ − σ n k ) , (36) where M n ≡ M n ( b, a ) 6 = 0. In a stra ightforw ard wa y one can prov e that there exists a neig hborho o d W of M in M C , so that for any ( b, a ) ∈ W and any 1 ≤ n ≤ N − 1, there is a uniq ue po lynomial ψ n ( λ ) of degr ee at most N − 2 sa tis fying (34) for any 1 ≤ k ≤ N − 1 as well as the pro duct representation (36), a nd so that the zero es are analytic functions o n W . W e ha ve seen in the in tro duction that there are tw o Cas imir functions C 1 and C 2 for J , leading to the symplectic folia tion of M with the leav es M β ,α . In [8] w e defined glo bal action v ariables ( I n ) 1 ≤ n ≤ N − 1 on M and, for any 1 ≤ n ≤ N − 1 , the angle v ariable θ n on M \ D n where D n is given by (19) and (21). 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 (37 ) wher e ˙ ∆ λ = d dλ ∆ λ is the λ -derivative of the discriminant ∆ 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 se ction 2. Definition 3.2. F or any 1 ≤ n ≤ N − 1 , the function θ n is define d for ( b , a ) ∈ M \ D n by θ n := η n + β n ( mo d 2 π ) and β n := N − 1 X n 6 = k =1 β n k , (38) 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 ) . (39) Her e for any 1 ≤ k ≤ N − 1 , µ ∗ k is the Dirichlet divisor define d in (29), and λ 2 k is identifie d with t he r amific ation p oint ( λ 2 k , 0) on the Riemann su rfac e Σ b,a . The inte gr ation p aths on Σ b,a in (39) ar e r e quir e d t o b e admissible in the s en se that their image under the pr oje ction π : Σ b,a → C on t he first c omp onent stays inside the isolating neighb orho o ds U k . In [8] we prov ed the following results. Theorem 3.3 . (A) The r e exists a c omplex neighb orho o d W of M in M C with the fol lowing pr op erties: (i) F or any 1 ≤ n ≤ N − 1 , the funct ions I n : W → C , θ n : W \ D n → C ( mo d π ) , and β n : W → C ar e analytic. 12 4 BIRKHOFF MAP (ii) O n the r e al sp ac e M , e ach fun ction I n is r e al-value d and nonne gative. It vanishes at a p oint ( b, a ) ∈ M if and only if the n - th gap is c ol- lapse d, i.e. if γ n ( b, a ) = 0 . Mor e over β n ( b, a ) = 0 for any ( b, a ) ∈ M with γ n ( b, a ) = 0 . (iii) F or any 1 ≤ n ≤ N − 1 , the qu otient I n /γ 2 n extends analytic al ly fr om M \ D n to al l of W and has st rictly p ositive r e al p art on W . As a c onse quenc e, ξ n = + p 2 I n /γ 2 n is a wel l-define d, analytic and nonvanishing function on W , wher e + √ · is the princip al br anch of the squar e r o ot on C \ ( −∞ , 0] . (B) The variable s I n and θ n , 1 ≤ n ≤ N − 1 , ar e glob al ly define d action-angle variables for the p erio dic T o da lattic e. Mor e pr e cisely: (iv) The funct ions ( I n ) 1 ≤ n ≤ N − 1 ar e p airwise in involution and Poisson c ommu te with t he T o da H amiltonian H , i.e. for any 1 ≤ m, n ≤ N − 1 , i = 1 , 2 , { I m , I n } J = 0 , { H , I n } J = 0 and { C i , I n } J = 0 on W . (v) The funct ions θ n : W \ D n → R , 1 ≤ n ≤ N − 1 , ar e c onjugate to the variables ( I m ) 1 ≤ m ≤ N − 1 , i.e. for any 1 ≤ m ≤ N − 1 , j = 1 , 2 , { I m , θ n } J = δ mn and { C i , θ n } J = 0 on W \ D n and { θ m , θ n } J = 0 on W \ ( D m ∪ D n ) . 4 Birkhoff ma p In this section, w e construct the map Ω, Ω = ((Ω n ) 1 ≤ n ≤ N − 1 , C 1 , C 2 ), defined on W . F or an y 1 ≤ n ≤ N − 1 and ( b, a ) ∈ W \ D n the n -th comp onent of Ω, Ω n = ( x n , y n ), is defined to b e ( x n , y n ) = + p 2 I n (cos θ n , sin θ n ) . In order to extend ( x n , y n ) to all of W , we substitute the formula 2 I n = γ 2 n ξ 2 n of Theo rem 3 .3 int o the definition of ( x n , y n ). Hence, for ( b, a ) ∈ W \ D n , ( x n = ξ n γ n cos θ n = ξ n γ n e iθ n + e − iθ n 2 y n = ξ n γ n sin θ n = ξ n γ n e iθ n − e − iθ n 2 i . By Theore m 3.3, β n - a nd therefore e ± iβ n - as well as ξ n are analytic on W . Thu s it remains to analytically extend the functions z ± n := γ n e ± iη n (40) to W . Note that at this p oint, z ± n is defined o n W \ D n only . The following result is prov ed in section 5 b elow. 13 Prop ositi o n 4. 1. T he functions z ± n extend analytic al ly to W . On M ∩ D n , z ± n = 0 . Definition 4.2. F or ( b, a ) ∈ W and 1 ≤ n ≤ N − 1 ,  x n := ξ n 2 ( z + n e iβ n + z − n e − iβ n ) y n := ξ n 2 i ( z + n e iβ n − z − n e − iβ n ) (41) F or any 1 ≤ n ≤ N − 1, it follows from (41) that x n ± iy n = ξ n z ± n e iβ n . Now w e ar e ready to define the co ordina te map Ω. As the Casimir function C 2 takes only pos itive v alues, w e introduce as targe t spac e of Ω the model space P := R 2( N − 1) × R × R > 0 and define Ω : M → P ( b, a ) 7→ (( x n , y n ) 1 ≤ n ≤ N − 1 , C 1 , C 2 ) . (42) In view of Theorem 3.3 a nd Prop o sition 4.1 we then hav e proved Theorem 4.3 . The map Ω : M → P is r e al analytic. It extends analytic al ly to the c omplex neighb orho o d W of M in M C of The or em 3.3 . T o compute the differential of the ma p Ω, we first compute for a ny 1 ≤ n ≤ N − 1 the gradient ∇ b,a z + n of z + n for 1 ≤ n ≤ N − 1. Let us first recall some notation intro duce d in [8]. F o r sequences  v ( j ) j ∈ Z  ,  w ( j ) j ∈ Z  ⊆ C define the N -vectors v · w :=  v ( j ) w ( j )  1 ≤ j ≤ N , (43) v · S w :=  v ( j ) w ( j + 1 )  1 ≤ j ≤ N , (44) where S deno tes the shift op e rator of o rder 1 . F urther define the 2 N - vector v · s w := ( v · w , v · S w + w · S v ) . (45) In cas e v = w we also us e the shorter notation v 2 := v · s v . (46) W ritten co mp o nent 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 ) . The following prop osition will b e proved in section 6. 14 4 BIRKHOFF MAP Prop ositi o n 4. 4. A t any p oint ( b, a ) ∈ M ∩ D n , the gr adient ∇ b,a z + n is given by ∇ b,a z + n ≡ ( ∇ b z + n , ∇ a z + n ) = ( h n − ig n ) 2 , (47) wher e g n and h n ar e define d by (26) and (27), r esp e ctively. It is conv enient to in tro duce the complex version of Ω, Ω C :( M , J ) → ( C N − 1 × R × R > 0 , J 0 ) ( b, a ) 7→ (( x n + iy n ) 1 ≤ n ≤ N − 1 , C 1 , C 2 ) , (48) and the abbre viations s n := sin nπ N for 1 ≤ n ≤ N − 1 . Moreo ver, fo r the res t of this section we write √ · instead of + √ · for the principa l br a nch o f the squa re ro ot function, defined on C \ ( −∞ , 0]. Prop ositi o n 4.5. L et β ∈ R and α > 0 . The gr adient ∇ b,a Ω C of Ω C at ( b, a ) = ( β 1 N , α 1 N ) is given by ( 1 ≤ n ≤ N − 1 ) ( ∇ b,a x n + i ∇ b,a y n )( j ) = 1 √ 2 αN · 1 √ s n ( e (2 j − 2) iπn N (1 ≤ j ≤ N ) − 2 e (2 j − 1) iπn N ( N + 1 ≤ j ≤ 2 N ) (49) and ∇ b,a C 1 = − 1 N (1 N , 0 N ) , ∇ b,a C 2 = 1 N (0 N , 1 N ) , (50) wher e 0 N = (0 , . . . , 0) = 0 · 1 N . Pr o of. Note that the for mulas in (50) immediately follow fro m the formulas (8) and (9) for the g radients of the Casimir functions C 1 and C 2 at an arbitrar y po int ( b, a ) ∈ M . It remains to show (49). In o r der to compute the gradient of x n + iy n , we us e the fo r mula x n + iy n = ξ n z + n e iβ n stated ab ov e. By Lemma 2.4, all gaps are closed for ( b, a ) = ( β 1 N , α 1 N ). Hence, by Theorem 3.3 (ii), β n = 0 and th us e iβ n = 1 for any 1 ≤ n ≤ N − 1. In addition, by Prop o sition 4.1, z + n = 0 . The gradient of x n + i y n = ξ n z + n e iβ n with res pe c t to ( b, a ) at ( b, a ) = ( β 1 N , α 1 N ) is therefore given by ∇ b,a x n + i ∇ b,a y n = ξ n ∇ b,a z + n . (51) F urther, b y The o rem 3.3, ξ n = lim γ n → 0 √ 2 I n γ n . ( 52) The pro of in [8 ] of the results s ta ted in Theorem 3 .3 shows that ξ n = √ N 2 p χ n ( τ n ) , 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 ) , (53) 15 and τ n = 1 2 ( λ 2 n + λ 2 n +1 ) (54) As a ll g aps are colla ps ed, ˙ λ n = λ 2 n = λ 2 n +1 = τ n for any 1 ≤ n ≤ N − 1. Hence, for λ = τ n , the pro duct in (53) is equal to ( − 1) N − n − 1 and thus ξ n = √ N 2 p χ n ( τ n ) = √ N 2  ( λ 2 N − τ n )( τ n − λ 1 )  − 1 4 . By Lemma 2 .4, λ 1 = β − 2 α , λ 2 N = β + 2 α , and τ n = β − 2 α co s nπ N . Therefore ξ n = √ N 2  4 α 2  1 − cos 2 nπ N  − 1 4 =  8 α N sin nπ N  − 1 2 = r N 8 α · 1 √ s n . (55) Next, by Prop ositio n 4.4, the gradie nt ∇ b,a z + n of z + n in (51) is given b y ∇ b,a z + n = ( h n − ig n ) 2 (56) where w e use d the no tation intro duced in (46). F r o m the fo r mulas (26)-(27) for g n and h n we then obtain from (56) in the case ( b, a ) = ( β 1 N , α 1 N ) ( ∇ b,a z + n )( j ) = 2 N ( e (2 j − 2) iπn N (1 ≤ j ≤ N ) − 2 e (2 j − 1) iπn N ( N + 1 ≤ j ≤ 2 N ) . (57) Substituting (55) and (57) into (5 1) then yields the claimed fo r mula (4 9) and therefore co mpletes the pro of of Theorem 4.5. W e end this section with proving commutator relations among the v ar iables ( x n , y n ) 1 ≤ n ≤ N − 1 which will b e used in section 7 to prov e Theorem 1 .1. Prop ositi o n 4. 6. F or any ( b, a ) ∈ W and 1 ≤ k , l ≤ N − 1 , t he fol lowing r elatio ns hold: { x k , x l } J = 0 ; { y k , y l } J = 0 ; { x k , y l } J = δ kl . Pr o of. By the cont inuit y of {· , ·} J , it is sufficient to prov e the claimed rela tions for any 1 ≤ k , l ≤ N − 1 and ( b, a ) ∈ W \ ( D k ∪ D l ). In this case, x n = √ 2 I n cos θ n and y n = √ 2 I n sin θ n for n ∈ { k , l } . Let us first show { x k , y l } = δ kl . { x k , y l } J = { p 2 I k cos θ k , p 2 I l sin θ l } J = cos θ k p 2 I l { p 2 I k , sin θ l } J + sin θ l p 2 I k { cos θ k , p 2 I l } J = cos θ k cos θ l √ 2 I l √ 2 I k { I k , θ l } J + sin θ k sin θ l √ 2 I k √ 2 I l { I l , θ k } J = δ kl , where fo r the latter identit y w e used Theor em 3 .3 (B). The other tw o claimed relations are prov ed similarly . 16 5 PROOF OF PROPOSITION ?? 5 Pro of of Prop osition 4.1 T o prov e Prop osition 4.1 w e follow the ar guments used in [11] to prove a similar result for K dV. Recall from (39) that η n is the following int egra l on Σ b,a , η n = Z µ ∗ n λ 2 n ψ n ( λ ) p ∆ 2 λ − 4 dλ (mod 2 π ) , where µ ∗ n is the Dirichlet divisor introduce d in (29), and λ 2 n is iden tified with the ramification po int ( λ 2 n , 0) o n Σ b,a . Note that o n W \ D n , z ± n = γ n e ± iη n is contin uous. Indeed, p ossible disco n- tin uities of η n due to the lexico graphic o rdering o f the eigenv a lues ( λ j ) 1 ≤ j ≤ 2 N lead s imu ltaneously to a s ig n change of γ n and e ± iη n , th us leaving γ n e ± iη n unaffected. F or λ near the interv al G n , defined in (22), one has ψ n ( λ ) c p ∆ 2 λ − 4 = ζ n ( λ ) s p ( λ 2 n +1 − λ )( λ − λ 2 n ) , (58) where ζ n ( λ ) := M ′ n + p ( λ − λ 1 )( λ 2 N − λ ) Y m 6 = n λ − σ n m + p ( λ 2 m +1 − λ )( λ 2 m − λ ) , (59) with M ′ n 6 = 0. No te that ζ n is a nalytic and nonv anishing in U n . W e claim that ζ n ( µ ) = 1 + O ( | γ n | ) (60) for µ ∈ G n , lo cally uniformly o n W . Indeed, for real ( b, a ) with γ n > 0 and any µ ∈ G n we deduce from (3 4 ), using that on the interv al ( λ 2 n , λ 2 n +1 ), b oth ( − 1) N + n +1 ψ n ( λ ) and ζ n ( λ ) are p ositive, π = Z λ 2 n +1 λ 2 n ( − 1) N + n +1 ψ n ( λ ) + p ∆ 2 λ − 4 dλ = Z λ 2 n +1 λ 2 n ζ n ( µ ) +  ζ n ( λ ) − ζ n ( µ )  + p ( λ 2 n +1 − λ )( λ − λ 2 n ) dλ = π ζ n ( µ ) + O  sup λ ∈ G n | ζ n ( λ ) − ζ n ( µ ) |  , where we used that Z λ 2 n +1 λ 2 n dλ + p ( λ 2 n +1 − λ )( λ − λ 2 n ) = π . Hence for µ ∈ G n , ζ n ( µ ) = 1 + O  sup λ ∈ G n | ζ n ( λ ) − ζ n ( µ ) |  . 17 By Ca uch y’s estimate, sup λ,µ ∈ G n | ζ n ( λ ) − ζ n ( µ ) | ≤ M | γ n | , where M can b e chosen lo cally uniformly on W . This pr oves the c la imed estimate (60) for real ( b, a ). F o r complex ( b, a ) ∈ W , the preceding identities remain true at lea s t up to a sign. By the co ntin uity o f ζ n in ( b, a ) and λ , the estimate (6 0 ) r emains v alid on W . W e now inv estigate the limiting behavior of z ± n as the n -th g ap collapses . This limit exis ts and do es no t v a nish when ( b, a ) is in the o p en set X n := { ( b, a ) ∈ W : µ n ( b, a ) / ∈ G n ( b, a ) } . Note that X n do es not intersect the r e al space M , since µ n ∈ [ λ 2 n , λ 2 n +1 ] for real ( b, a ). W e now define χ n ( b, a ) := Z µ n τ n ζ n ( λ ) − ζ n ( τ n ) λ − τ n dλ, (61) with τ n = ( λ 2 n + λ 2 n +1 ) / 2. Note that τ n is analytic on W . Indeed, using the pro duct r epresentation (16) of ∆ 2 λ − 4 o ne g ets by the residue theorem τ n = 1 2 π i Z Γ n λ ∆ λ ˙ ∆ λ ∆ 2 λ − 4 dλ. (62 ) Since, lo cally on M , the co ntour Γ n can be kept fixed and ∆ λ ( b, a ) is analytic on C × W , (62) shows that τ n is a real ana lytic function o n W . As µ n and ζ n are analytic on C × W , it then follows that χ n , defined by (61), is a nalytic on W . T o facilitate the statement of the following res ult, define, for an y 1 ≤ n ≤ N − 1 , the sig n ǫ n = ± 1 for elements ( b, a ) in X n so that ψ n ( µ n ) ∗ q ∆ 2 µ n − 4 = ǫ n ζ n ( µ n ) s p ( λ 2 n +1 − µ n )( µ n − λ 2 n ) . (63) Note that the s -ro ot is well defined, since µ n / ∈ G n for ( b, a ) ∈ X n . T o pr ov e Prop os itio n 4.1 we need the fo llowing auxiliar y result: Lemma 5.1. As ( b, a ) ∈ W \ D n tends to ( b 0 , a 0 ) ∈ D n ∩ X n , γ n e ± iη n → − 2(1 ± ǫ n )( µ n − τ n ) e ± ǫ n χ n , wher e ǫ n is define d by (63). Pr o of. Since X n is op en and ( b 0 , a 0 ) ∈ X n ∩ D n , it follows that ( b, a ) ∈ X n for all ( b , a ) sufficiently close to ( b 0 , a 0 ). Also, ( b, a ) / ∈ D n by assumption. F or ( b, a ) ∈ X n \ D n one ha s, mo dulo 2 π , η n = Z µ ∗ n λ 2 n ψ n ( λ ) p ∆ 2 λ − 4 dλ 18 5 PROOF OF PROPOSITION ?? = ǫ n Z µ n λ 2 n ζ n ( λ ) s p ( λ 2 n +1 − λ )( λ − λ 2 n ) dλ = ǫ n Z µ n λ 2 n ζ n ( λ 2 n ) s p ( λ 2 n +1 − λ )( λ − λ 2 n ) dλ + ǫ n Z µ n λ 2 n ζ n ( λ ) − ζ n ( λ 2 n ) s p ( λ 2 n +1 − λ )( λ − λ 2 n ) dλ = η (1) n + η (2) n mo d 2 π . The limiting b ehavior of the s e c ond term η (2) n is s traightforw ard. If ( b, a ) → ( b 0 , a 0 ), then γ n → 0 and so for λ 6 = τ n ( b 0 , a 0 ), s p ( λ 2 n +1 − λ )( λ − λ 2 n ) → i ( λ − τ n ) by the definitio n of the s -ro ot. Hence, by the definition of χ n , iη (2) n → ǫ n Z µ n τ n ζ n ( λ ) − ζ n ( τ n ) λ − τ n dλ = ǫ n χ n . Consequently , a s ( b , a ) → ( b 0 , a 0 ), e iη (2) n → e ǫ n χ n . T urning to η (1) n , make the substitution λ = τ n + z γ n / 2. Then, by the definition (31) o f the s -ro o t, Z µ n λ 2 n dλ s p ( λ 2 n +1 − λ )( λ − λ 2 n ) = Z ρ n − 1 dz s √ 1 − z 2 = φ ( ρ n ) (64) with ρ n = µ n − τ n γ n / 2 , φ ( w ) := Z w − 1 dz s √ 1 − z 2 . (65) It follows that e iφ ( w ) = − w + i s p 1 − w 2 , (66) as both sides o f the latter identit y are analytic, univ alent functions on C \ [ − 1 , 1], which hav e the same limit at − 1 a nd satisfy the same differential equation f ′ ( w ) f ( w ) = i s √ 1 − w 2 . Hence, writing exp( iη (1) n ) = exp  iǫ n φ ( ρ n ) ζ n ( λ 2 n )  = exp  iφ ( ρ n ) ǫ n  exp  iǫ n φ ( ρ n ) ˆ ζ n  with ˆ ζ n = ζ n ( λ 2 n ) − 1 , w e o btain fo r ( b, a ) ∈ X n \ D n γ n e iη (1) n = γ n ( − ρ n + i s p 1 − ρ 2 n ) ǫ n · e iǫ n φ ( ρ n ) ˆ ζ n . (67) Passing to the limit ( b, a ) → ( b 0 , a 0 ), w e hav e γ n → 0, while v n := µ n − τ n tends to a limit different from zero, and hence | ρ n | → ∞ . By the definition (30) of 19 the s -r o ot, the limit of the fir s t tw o fa ctors o n the rig ht hand side o f the ab ove equation can then be computed as follows. γ n ( − ρ n + i s p 1 − ρ 2 n ) ǫ n = γ n ( − ρ n + iǫ n s p 1 − ρ 2 n ) = γ n ( − ρ n − ǫ n ρ n + q 1 − ρ − 2 n ) = − 2 v n − 2 v n ǫ n + q 1 − ρ − 2 n ) →− 2 v n (1 + ǫ n ) . As to the thir d factor in (6 7), o bserve that for | ρ n | large, | φ ( ρ n ) | =    Z ρ n − 1 dz s √ 1 − z 2    ≤    Z 1 − 1 dz s √ 1 − z 2    + Z | ρ n | 1 dt √ t − 1 √ t + 1 = O ( p | ρ n | ) . Since ˆ ζ n = O ( | γ n | ) by (60), we th us conclude that φ ( ρ n ) ˆ ζ n → 0 and so e iǫ n φ ( ρ n ) ˆ ζ n → 1 as ( b, a ) → ( b 0 , a 0 ) . T og ether with the res ult for e iη (2) n we conclude that for ( b, a ) → ( b 0 , a 0 ) γ n e iη n → − 2 v n (1 + ǫ n ) e ǫ n χ n ( b 0 ,a 0 ) as claimed. The limit of γ n e − iη n is a simple v ariation of this arg ument. Pr o of of Pr op osition 4.1. W e extend the functions z ± n to D n ∩ W as follows z ± n =  − 2(1 ± ǫ n )( µ n − τ n ) e ± ǫ n χ n on D n ∩ X n , 0 on D n \ X n . (68) W e hav e alre ady seen that the functions z ± n are analytic on W \ D n . It is straightforward to v erify that z ± n are contin uous at every p oint of D n ∩ X n and of D n \ X n . Thus z ± n are co nt inuous on all o f W . In view of Theorem A.6 in [11] it remains to show that they a re weakly analy tic, when res tr icted to D n ∩ W , i.e. that the restrictio n of z ± n to any one-dimensio nal complex disc D c o ntained in D n ∩ W is a na lytic. If the center of D is in X n , the entire disc D is in X n , if chosen sufficiently small. The analyticity of z ± n = γ n e ± iη n on D is then evide nt from fo rmula (6 8), the definition of χ n , and the lo cal constancy of ǫ n on X n . If the center of D do es no t b elong to X n we argue as follows. The function µ n − τ n is analytic on D . It either v anishes identically on D in which case z ± n v anishe s ident ically , too . Or it v anishes in o nly finitely man y p oints. Outside these p oints, D is in X n , hence z ± n is analytic there. By contin uity and analytic contin uation, these functions a r e a nalytic on all of D . W e th us ha ve shown that z ± n are analytic on D . As χ n is analytic and ǫ n is lo cally constant, it follows that z ± n is weakly analytic o n D n ∩ W . This proves the analy ticit y of z ± n on W . 20 6 PROOF OF PROPOSITION ?? 6 Pro of of Prop osition 4.4 T o prove Pr op osition 4.4 we follow the a rguments used in [11] to show s imila r results for KdV. W e b egin with some pr eparations for the pro of o f Prop ositio n 4.4. T o compute the g radient of z + n at a p oint ( b, a ) in M ∩ D n , w e a pproximate ( b, a ) by elements ( b ′ , a ′ ) in B n := { ( b, a ) ∈ M \ D n : µ n = τ n and sign ∗ q ∆ 2 µ n − 4 = ( − 1) N + n +1 } . (69) It follows from the results of the sp ectra l theo ry o f Jacobi matrices reviewed in section 2 that B n 6 = ∅ . As a preliminary step tow ards the computation of ∇ b,a z + n , we need the following t wo lemmas. Lemma 6.1. F or any ( b, a ) ∈ M ∩ D n , ∇ b,a z + n = 2( ∇ b,a τ n − ∇ b,a µ n ) + i lim B n ∋ ( b ′ ,a ′ ) → ( b,a ) ( f 2 2 n +1 − f 2 2 n ) , (70) wher e for i ∈ { 2 n, 2 n + 1 } and ( b ′ , a ′ ) ∈ B n we denote by f i the eigenve ctor of L ( b ′ , a ′ ) asso ciate d t o λ i , n ormalize d by N X j =1 f i ( j ) 2 = 1 and  f i (1) , f i (2)  ∈ ( R > 0 × R ) ∪ ( { 0 } × R > 0 ) . Pr o of of L emma 6.1. Rec all tha t w e have introduced ψ n ( λ ) and ζ n ( λ ) in (36) and (59 ), resp ectively . F or ( b ′ , a ′ ) ∈ B n , sign ψ n ( µ n ) = ( − 1) N + n +1 and sign ζ n ( µ n ) = 1 (see disc us sions after (36) and (58)), and hence the identit y (58 ) reads ψ n ( µ n ) ∗ q ∆ 2 µ n − 4 = ζ n ( µ n ) + p ( λ 2 n +1 − µ n )( µ n − λ 2 n ) . Going through the calculations in the pro of of Lemma 5.1 with (63) replaced by the latter identit y , a ll s -r o ots replaced b y the principal branch + √ · , and with ǫ n = 1, we can write z + n = γ n e iη n = γ n e iη (1) n · e iη (2) n where, mo d 2 π , η (1) n = ζ n ( λ 2 n ) Z µ n λ 2 n dλ + p ( λ 2 n +1 − λ )( λ − λ 2 n ) and η (2) n = Z µ n λ 2 n ζ n ( λ ) − ζ n ( λ 2 n ) + p ( λ 2 n +1 − λ )( λ − λ 2 n ) dλ. 21 Note that with λ = τ n + z γ n / 2, + p ( λ 2 n +1 − λ )( λ − λ 2 n ) = γ n 2 √ 1 − z 2 and hence on B n η (1) n = ζ n ( λ 2 n ) Z 0 − 1 dz √ 1 − z 2 = π 4 ζ n ( λ 2 n ) In view of (60) we then get in the limit ( b ′ , a ′ ) → ( b, a ), with ( b ′ , a ′ ) ∈ B n , η (1) n → π 4 (mo d 2 π ) Using a gain λ ≡ λ ( z ) = τ n + z γ n / 2 one computes fo r ( b ′ , a ′ ) ∈ B n η (2) n = Z 0 − 1  Z 1 0 ζ ′ n ( λ 2 n + s ( λ − λ 2 n )) ds  γ n (1 + z ) 2 √ 1 − z 2 dz and thus η (2) n → 0 as ( b ′ , a ′ ) → ( b, a ), or e iη ′′ n → 1. Since for ( b, a ) ∈ M ∩ D n , one has γ n e iη ′ n = 0 by Pro po sition 4.1, it then follows that ∇ b,a z + n = lim B n ∋ ( b ′ ,a ′ ) → ( b,a ) ∇ b,a  γ n e iη ′ n  , Moreov er, let v n := µ n − τ n and as in (6 5), in tro duce ρ n = µ n − τ n γ n / 2 . Then − 1 ≤ ρ n ≤ 1 and by (64) we get for ( b ′ , a ′ ) ∈ B n , e iη (1) n = e iφ ( ρ n ) e iφ ( ρ n ) ˆ ζ n , wher e ˆ ζ n = ζ n ( λ 2 n ) − 1. By (66) it fo llows that γ n e iη (1) n =  − 2 v n + iγ n + p 1 − ρ 2 n  ·  − ρ n + i + p 1 − ρ 2 n  ˆ ζ n . The gradients o f b oth factors hav e a limit as ( b ′ , a ′ ) → ( b, a ), and the pro duct rule can b e applied. F o r ( b ′ , a ′ ) ∈ B n , we hav e µ n = τ n and hence v n = 0 as well as ρ n = 0. Thus, the fir s t factor e quals iγ n and hence , in the limit, v anishes, while the second factor equals ( i ) ˆ ζ n and thus, by (60), conv erges to 1. As a consequence, by the pro duct rule, ∇ b,a z + n = lim ( b ′ ,a ′ ) → ( b,a ) ∇ b,a  γ n e iη ′ n  = lim ( b ′ ,a ′ ) → ( b,a ) ∇ b,a  − 2 v n + iγ n + p 1 − ρ 2 n  = 2( ∇ b,a τ n − ∇ b,a µ n ) + i lim ( b ′ ,a ′ ) → ( b,a ) ∇ b,a γ n = 2( ∇ b,a τ n − ∇ b,a µ n ) + i lim ( b ′ ,a ′ ) → ( b,a )  f 2 2 n +1 − f 2 2 n  where for the la tter ident ity we used that on M \ D n , ∇ b,a λ i = f 2 i for i ∈ { 2 n, 2 n + 1 } (cf. [8], Pr o p osition 5.3) a nd that lim ( b ′ ,a ′ ) → ( b,a )  f 2 2 n +1 − f 2 2 n  exists, as z + n , µ n , and τ n are analytic. Lemma 6.2. As ( b ′ , a ′ ) ∈ B n tends to ( b, a ) ∈ M ∩ D n , the p erio dic eigenve ctors f 2 n and f 2 n +1 of L ( b ′ , a ′ ) , n ormalize d as in Le mma 6.1 , c onver ge t o normalize d eigenve ctors of L ( b, a ) , denote d by the same symb ols, such that in the limit, h f 2 n +1 , g n i < 0 < h f 2 n , g n i 22 6 PROOF OF PROPOSITION ?? and f 2 n (1) h f 2 n +1 , g n i = − f 2 n +1 (1) h f 2 n , g n i . (71) Pr o of of L emma 6.2. As ( b ′ , a ′ ) tends to ( b, a ), the initial data  f i (1) , f i (2)  of the norma lized eigenv ectors f i ( i ∈ { 2 n, 2 n + 1 } ) is a vector in the unit disc of R 2 . Cho ose a convergen t subsequence of initial data. Then, for i ∈ { 2 n, 2 n + 1 } , f i can b e expresse d as a linear combination of the fundamental solutions y 1 and y 2 , f i = f i (0) y 1 + f i (1) y 2 . As the fundamental s olutions depend a nalytically on λ and ( b ′ , a ′ ), the eigenv ectors f 2 n +1 and f 2 n then con verge to some eigenv ectors of L ( b, a ), which we denote by ¯ f 2 n +1 and ¯ f 2 n . Note that, by the norma lization of f i , one has ¯ f i (1) ≥ 0 for i ∈ { 2 n, 2 n + 1 } . By (70), lim ( b ′ ,a ′ ) → ( b,a )  f 2 2 n +1 − f 2 2 n  exists. As τ n is ana lytic and, on M \ D n , ∇ b,a τ n =  f 2 2 n +1 + f 2 2 n  / 2, it follows tha t lim ( b ′ ,a ′ ) → ( b,a )  f 2 2 n +1 + f 2 2 n  ex- ists as well. Hence the limits o f f 2 i , i ∈ { 2 n , 2 n + 1 } exist, and ¯ f 2 n +1 and ¯ f 2 n are uniquely determined up to a sign. As ¯ f i (1) ≥ 0 for i ∈ { 2 n, 2 n + 1 } , this sign is uniquely determined once we show that ¯ f i (1) 6 = 0 . (72) T o simplify notation, write tempora rily f and g instead of f 2 n +1 and g n . T o prov e (72), observe that ( λ 2 n +1 − µ n ) h f , g i = h λ 2 n +1 f , g i − h f , µ n g i = h Lf , g i − h f , Lg i = N X j =1  [ b j f ( j ) + a j f ( j + 1) + a j − 1 f ( j − 1 )] g ( j ) − f ( j ) [ b j g ( j ) + a j g ( j + 1 ) + a j − 1 g ( j − 1)]  = N X j =1  a j f ( j + 1 ) g ( j ) − a j − 1 f ( j ) g ( j − 1)  + N X j =1  a j − 1 f ( j − 1) g ( j ) − a j f ( j ) g ( j + 1)  . Note that the la tter tw o sums are telesco ping, hence ( λ 2 n +1 − µ n ) h f , g i = a N  f ( N + 1) g ( N ) − f (1) g (0) + f (0) g (1) − f ( N ) g ( N + 1)  = a N f (1)  ( − 1) n + N g ( N ) − g (0)  , where for the latter equality we used that g (1 ) = 0 = g ( N + 1) and f ( N + 1) = ( − 1) n + N f (1) acco rding to whether λ 2 n +1 is a perio dic or antiperio dic eigen v alue - s e e (17 ) in section 2. Hence we hav e ( λ 2 n +1 − µ n ) h f 2 n +1 , g n i = a N f 2 n +1 (1)  ( − 1) n + N g n ( N ) − g n (0)  . (73) A similar computation shows that ( λ 2 n − µ n ) h f 2 n , g n i = a N f 2 n (1)  ( − 1) n + N g n ( N ) − g n (0)  . (74) 23 F or ( b ′ , a ′ ) ∈ B n , one has λ 2 n +1 − µ n = µ n − λ 2 n as well as f i (1) > 0 ( i ∈ { 2 n, 2 n + 1 } ) and ( − 1) n + N g n ( N ) 6 = g n (0). Hence the quotients of the left and rig ht hand sides of (73) and (74) a re w ell defined, and we obtain f 2 n (1) h f 2 n +1 , g n i = − f 2 n +1 (1) h f 2 n , g n i . Passing to the limit a s ( b ′ , a ′ ) → ( b, a ) we obtain ¯ f 2 n (1) h ¯ f 2 n +1 , g n i = − ¯ f 2 n +1 (1) h ¯ f 2 n , g n i . (75 ) W e claim that κ n := lim ( b ′ ,a ′ ) → ( b,a ) ( − 1) n + N g n ( N ) − g n (0) λ 2 n +1 − µ n (76) exists and tha t κ n < 0 . T o see it, divide (73) by ( λ 2 n +1 − µ n ). Then the existence of the limit in (76) implies that one can take limits of bo th sides o f the resulting equation as ( b ′ , a ′ ) → ( b, a ) to get h ¯ f 2 n +1 , g n i = a N κ n ¯ f 2 n +1 (1) . (77) If ¯ f 2 n +1 (1) = 0, then, as ¯ f 2 n +1 is p er io dic o r antiperio dic, ¯ f 2 n +1 ( N + 1) = 0 as well. Hence ¯ f 2 n +1 is a (non trivial) scalar multiple o f g n and thus h ¯ f 2 n +1 , g n i 6 = 0, contradicting (77). Hence the claim that κ n < 0 implies that ¯ f 2 n +1 (1) > 0, i.e. ¯ f 2 n +1 satisfies all the normaliza tio n conditions listed in Lemma 6 .2. It rema ins to pro ve that th e limit (76) exis ts a nd that κ n < 0, or equiv alen tly , lim ( b ′ ,a ′ ) → ( b,a ) ( − 1) n + N y 1 ( N , µ n ) − 1 λ 2 n +1 − µ n < 0 . (78) Recall that ∗ p ∆ 2 ( µ n ) − 4 = y 1 ( N , µ n ) − y 2 ( N + 1 , µ n ). Hence, for ( b ′ , a ′ ) ∈ B n , 2 y 1 ( N , µ n ) =  y 1 ( N , µ n ) + y 2 ( N + 1 , µ n )  +  y 1 ( N , µ n ) − y 2 ( N + 1 , µ n )  = ∆( µ n ) + ∗ p ∆ 2 ( µ n ) − 4 = ∆( µ n ) + ( − 1) N − n − 1 + p ∆ 2 ( µ n ) − 4 , the last equality b eing a conseq uenc e of the definition (69) of B n . Reca ll that, according to (17), 2 = ( − 1) n + N ∆( λ 2 n +1 ). Substituting 4 = ∆ 2 ( λ 2 n +1 ) into the formula above, the inequa lit y (78) can then b e equiv alen tly written as lim ( b ′ ,a ′ ) → ( b,a ) ( − 1) n + N ∆( µ n ) − ∆( λ 2 n +1 ) λ 2 n +1 − µ n − + p ∆ 2 ( µ n ) − ∆ 2 ( λ 2 n +1 ) λ 2 n +1 − µ n ! < 0 . (79 ) Concerning the first term in the ab ove express io n, w e get in the limit, as ( b ′ , a ′ ) → ( b, a ) ∆( µ n ) − ∆( λ 2 n +1 ) µ n − λ 2 n +1 → ˙ ∆( λ 2 n +1 ) = 0 , as λ 2 n +1 is a double eigenv alue of Q ( b, a ). Co ncerning the second ter m in (79), write ∆ 2 ( µ n ) − ∆ 2 ( λ 2 n +1 ) = − 2 Z λ 2 n +1 µ n ∆( λ ) ˙ ∆( λ ) dλ 24 6 PROOF OF PROPOSITION ?? = − 2 Z λ 2 n +1 µ n ∆( λ ) Z λ ˙ λ n ¨ ∆( µ ) dµ ! dλ, (80) where ˙ λ n is the unique ro ot o f ˙ ∆ in the n -th ga p. Note that lo ca lly uniformly on M , Z λ ˙ λ n ¨ ∆( µ ) dµ = ( λ − ˙ λ n ) ¨ ∆( ˙ λ n ) + O (( λ − ˙ λ n ) 2 ) and − 2 Z λ 2 n +1 µ n ∆( λ )( λ − ˙ λ n ) ¨ ∆( ˙ λ n ) dλ = − 2 ∆( µ n ) ¨ ∆( ˙ λ n ) Z λ 2 n +1 µ n ( λ − ˙ λ n ) dλ + O ( γ 3 n ) The first term o n the r ight hand side of the latter ex pression can b e co mputed to b e, using that µ n = τ n on B n and ˙ λ n = τ n + O ( γ 2 n ) lo cally unifor mly − 2 ∆( µ n ) ¨ ∆( ˙ λ n ) 1 2  ( λ 2 n +1 − ˙ λ n ) 2 − ( µ n − ˙ λ n ) 2  = − ∆( µ n ) ¨ ∆( ˙ λ n )  γ n 2  2 + O ( γ 3 n ) . Dividing (80) by ( µ n − λ 2 n +1 ) 2 and taking the limit thus leads to lim ( b ′ ,a ′ ) → ( b,a ) ∆ 2 ( µ n ) − ∆ 2 ( λ 2 n +1 ) ( µ n − λ 2 n +1 ) 2 = − ∆( λ 2 n +1 ) ¨ ∆( λ 2 n +1 ) . As ∆( λ 2 n +1 ) = ( − 1) n + N · 2 (see (17)) and ( − 1) n + N ¨ ∆( λ 2 n +1 ) < 0 (again by (17) a nd the fa ct that ∆( λ ) is a p olynomial o f degree N ), one concludes that − ∆( λ 2 n +1 ) ¨ ∆( λ 2 n +1 ) > 0. This pr ov es the estimate (79), hence by (73), h ¯ f 2 n +1 , g n i < 0. By a similar arg ument, one shows that ¯ f 2 n (1) > 0, and there- fore, (75) implies h f 2 n , g n i > 0. Pr o of of Pr op osition 4.4. According to Le mma 6.2 , lim ( b ′ ,a ′ ) → ( b,a )  f 2 2 n +1 − f 2 2 n  = f 2 2 n +1 − f 2 2 n (81) where the limiting eigenv ectors f 2 n and f 2 n +1 are or thonormal a nd sa tis fy the inequalities h f 2 n +1 , g n i < 0 < h f 2 n , g n i . By definition, g n and h n are orthonor - mal and spa n the same subspac e a s f 2 n and f 2 n +1 . Hence there exist s, t ≥ 0 with s 2 + t 2 = 1 such that f 2 n +1 = s h n − tg n , f 2 n = t h n + sg n . Substituting these for mulas into equation (71), we obtain t 2 = s 2 , and hence s = t = 1 √ 2 . Thus we hav e f 2 2 n +1 − f 2 2 n = − 2 h n · s g n , f 2 2 n +1 + f 2 2 n = h 2 n + g 2 n . (82) By [8 ], ∇ b,a µ n = g 2 n and ∇ b,a τ n = ( f 2 2 n +1 + f 2 2 n ) / 2, hence 2( ∇ b,a τ n − ∇ b,a µ n ) = h 2 n − g 2 n . ( 83) In view of (81)-(83), formula (47) then follows from (70). 25 7 Pro of of Theorem 1.1 In this s ection we show Theorem 1.1. Its three statements ar e co ntained in Theorem 7.1, Theo rem 7.8, and Co rollar y 7.11, resp ectively . Theorem 7.1 . The map Ω : M → P ( b, a ) 7→ (( x n , y n ) 1 ≤ n ≤ N − 1 , C 1 , C 2 ) is a glob al, r e al analytic diffe omorphism. L o c al Pr op erties In a first step we establish that Ω is a lo cal diffeomorphism everywhere in pha se s pace. Prop ositi o n 7.2. At every p oint ( b, a ) ∈ M , the differ ential d ( b,a ) Ω : T ( b,a ) M → T Ω( b,a ) P is a line ar isomorphism. Let us first introduce some additional no ta tion. F or ( b, a ) ∈ M and 1 ≤ n ≤ N − 1 , define d n := ∇ b,a x n and d − n := ∇ b,a y n . F urther w e r ecall Lemma 7.2 in [8], neede d later . Lemma 7.3. A t every p oint ( b, a ) in M , the ve ctors  ( ∇ b,a I n ) n ∈ K , ∇ b,a C 1 , ∇ b,a C 2  ar e line arly indep endent. Her e K = K ( b, a ) denotes the index set of op en gaps, K := { 1 ≤ n ≤ N − 1 : γ n ( b, a ) > 0 } . Prop os itio n 7.2 follows from the following lemma. Lemma 7.4. A t every p oint ( b, a ) ∈ M , the 2 N ve ctors  ( d n ) 1 ≤ n ≤ N − 1 , ( d − n ) 1 ≤ n ≤ N − 1 , ∇ b,a C 1 , ∇ b,a C 2  (84) ar e line arly indep endent. Pr o of of L emma 7.4. T o verify t he c la imed sta tement, consider an arbitrar y linear combination f = P 1 ≤| n |≤ N − 1 r n d n + s 1 ∇ b,a C 1 + s 2 ∇ b,a C 2 with rea l co ef- ficient s ( r n ) 1 ≤| n |≤ N , s 1 , s 2 such that f = 0. F or m ∈ K take the scala r pro duct of b oth sides of f = 0 with J ∇ b,a I m and us e Lemma 4.6 together with the ident ity I m = ( x 2 m + y 2 m ) / 2 to g et 0 = h f , J ∇ b,a I m i = r m { x m , I m } J + r − m { y m , I m } J = r m y m − r − m x m . Hence the 2- vectors ( r m , r − m ) and ( x m , y m ) are par allel, i.e. ( r m , r − m ) = c m (cos θ m , sin θ m ) with c m ∈ R satisfying c 2 m = r 2 m + r 2 − m . Thus, if c m v an- ishes, r m and r − m bo th v anis h. F ur thermore, by the definition of d ± m , r m d m + r − m d − m = r m ∇ b,a x m + r − m ∇ b,a y m = c m (cos θ m ∇ b,a x m + sin θ m ∇ b,a y m ) = c m √ 2 I m ∇ b,a I m , 26 7 PROOF OF THEOREM ?? where for the last e quality we used the identit y ∇ b,a I m = p 2 I m (cos θ m ∇ b,a x m + sin θ m ∇ b,a y m ) , obtained from 2 I m = x 2 m + y 2 m by differen tiation. Hence the equation f = 0 reads X n ∈ K c n √ 2 I n ∇ b,a I n + X n / ∈ K ( r n ∇ b,a x n + r − n ∇ b,a y n ) + s 1 ∇ b,a C 1 + s 2 ∇ b,a C 2 = 0 . (85) Next, for m / ∈ K , take the s c alar pr o duct o f b oth sides of (85) with J ∇ b,a y m and J ∇ b,a x m . By the commut ator relations of Lemma 4 .6 one then o bta ins the ident ities 0 = h f , J ∇ b,a y m i = r m { x m , y m } J = r m and 0 = h f , J ∇ b,a x m i = − r − m { x m , y m } J = − r − m . Hence (85) be c omes X n ∈ K c n √ 2 I n ∇ b,a I n + s 1 ∇ b,a C 1 + s 2 ∇ b,a C 2 = 0 . By Lemma 7.3, c n = 0 - hence r n = r − n = 0 by the r e ma rk ab ove - for any n ∈ K and s 1 = s 2 = 0. Glob al Pr op erties In a second step, we show that Ω is bijective a nd ca nonical. First we show Prop ositi o n 7. 5. T he map Ω is pr op er, i.e. the pr eimage of any c omp act set is c omp act. T o prov e Prop os itio n 7.5 we need tw o auxiliary results. Lemma 7.6. F or any ( b, a ) ∈ M and any 1 ≤ n ≤ N − 1 , γ 2 n ≤ 3 π ( λ 2 N − λ 1 ) I n (86) and 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 . (87) A pro of of Lemma 7 .6 can b e found in Appendix A of [8]; in the cas e α = 1 (86) ha s b een prov ed in [1] (p.601-60 2). Lemma 7.7. F or 1 ≤ n ≤ N and any ( b, a ) ∈ M , | b n | ≤ | C 1 ( b, a ) | + N − 1 X k =1 γ k ( b, a ) , ( 88) 27 0 < a n ≤ N 2 π C 2 ( b, a ) + | C 1 ( b, a ) | + N − 1 X k =1 γ k ( b, a ) ! , (89) λ 2 N ( b, a ) − λ 1 ( b, a ) ≤ 2 π C 2 ( b, a ) + N − 1 X k =1 γ k ( b, a ) . (90) Lemma 7.7 is s hown in App endix A. In a weak er for m, it has b een prov ed in [2] (p.564-565 ). Pr o of of Pr op osition 7.5. Let ( b ( m ) , a ( m ) ) m ≥ 1 ⊆ M b e a sequence in M so that  Ω( b ( m ) , a ( m ) )  m ≥ 1 conv erges in P . Then, for any 1 ≤ n ≤ N − 1, the seq uence  I n ( b ( m ) , a ( m ) )  m ≥ 1 of action v a riables is a Cauch y sequence, as well as the se- quence ( C ( m ) 1 , C ( m ) 2 ) m ≥ 1 of the v alues of the Casimir functions C 1 and C 2 . By Lemma 7.6 and Lemma 7.7 it then follows that ( b ( m ) , a ( m ) ) m ≥ 1 admits a sub- sequence which co nv erges to an element ( b, a ) in R N × R N ≥ 0 . As by assumption, the sequence  C 2 ( a ( m ) )  m ≥ 1 conv erges to α > 0 and α = lim m →∞ C 2 ( a ( m ) ) = N Y n =1 a n ! 1 / N it follows that a n > 0 for all 1 ≤ n ≤ N , i.e. ( b, a ) ∈ M . Pr o of of The or em 7.1. By Prop ositio n 7.2, Ω is op en, and by Prop osition 7.5, it is closed. As P is connected, Ω( M ) = P , hence Ω is onto. It is also 1 -1, since, by the same reasoning, the set B of all po ints in P with more than one preimag e is op en and close d. W e claim that (0 N − 1 , 0 N − 1 , 0 , 1) / ∈ B and hence B = ∅ . Here 0 N − 1 denotes the vector (0 , . . . , 0) ∈ R N − 1 . T o see that (0 N − 1 , 0 N − 1 , 0 , 1) / ∈ B note that for any ( b, a ) ∈ M with Ω( b, a ) = (0 N − 1 , 0 N − 1 , 0 , 1), all action v ar iables v anish, and he nce , by Theorem 3.3 (ii), a ll g aps must be co llapsed. By the res ults reviewed in section 2, it then follows that µ n = λ 2 n for any 1 ≤ n ≤ N − 1 and hence there is exac tly one ma tr ix Q with γ n = 0 for any 1 ≤ n ≤ N − 1 and ( β , α ) = (0 , 1). Since we hav e already shown the r eal analyticity of Ω in Theo rem 4 .3, this completes the pro of of Theor e m 7.1. Next we show that Ω : M → P is canonical. Recall that the phase space M is endow ed with the Poisson bra ck et {· , ·} J defined by (5). It is degenera te and has C 1 , C 2 as Ca simir functions. The mo del space P is endowed with the standard Poisson str uctur e o n R 2( N − 1) , i.e. among the co ordina te functions ( x n , y n ) 1 ≤ n ≤ N − 1 , β , α , all Poisson brack ets v a nish, except for 1 ≤ n ≤ N − 1, { x n , y n } 0 = −{ y n , x n } 0 = 1 . Note that o n the mo del space P , the co ordinate functions β and α are tw o independent Casimir s defining a triv ial foliation with symplectic leaves P β ,α := R 2( N − 1) × { β } × { α } . 28 7 PROOF OF THEOREM ?? Theorem 7.8. The map Ω is c anonic al, i.e. it pr eserves the Poisson br ackets. Pr o of. W e have to verify that { F , G } 0 ◦ Ω = { F ◦ Ω , G ◦ Ω } J for arbitrar y functions F , G in C 1 ( P ). Clearly , the pullbac ks of the functions β , α on P ar e the Casimir functions C 1 , C 2 of {· , ·} J . Mo r eov er, by L e mma 4.6, { x k , y l } J = δ kl and { x k , x l } J = { y k , y l } J = 0 e verywhere on M , hence Ω is canonical. Let π : P → R × R > 0 denote the pro jection of P = R 2( N − 1) × R × R > 0 onto the las t tw o factor s. Then π defines a symplectic foliatio n with leaves P β ,α = R 2( N − 1) × { β } × { α } . The definitio n o f Ω to gether with Theorem 7 .8 then leads to the following result. Corollary 7.9. F or every β ∈ R and α > 0 , Ω( M β ,α ) = P β ,α , and Ω | M β ,α : M β ,α → P β ,α is a symple ctomorph ism. In p articular, the map ( I n ) 1 ≤ n ≤ N − 1 : M β ,α →  R ≥ 0  N − 1 is ont o. T o formulate the last result of this s ection, recall that in section 2, for any ( b, a ) ∈ M , we hav e introduce d the isosp ectra l set Iso ( b, a ) = { ( b ′ , a ′ ) ∈ M : sp ec Q ( b ′ , a ′ ) = sp ec Q ( b, a ) } . F or ( x, y , β , α ) ∈ P , let T ( x, y , β , α ) deno te the torus T ( x, y , β , α ) := { ( u n , v n ) 1 ≤ n ≤ N − 1 | u 2 n + v 2 n = x 2 n + y 2 n ∀ n } × { β } × { α } . Prop ositi o n 7.10. F or any ( b, a ) ∈ M , Ω( Iso ( b , a )) = T (Ω( b, a )) . (91) Pr o of. Note tha t the action v ar iables I n are defined in terms of the discriminant ∆ λ , and ∆ λ is a sp ectral inv aria nt . Hence Ω(Iso( b, a )) ⊂ T (Ω( b, a )). As Iso( b, a ) and T (Ω( b, a )) are b oth tor i of the same dimensio n, (91) then follows. Corollary 7.11. The pul lb ack ˆ H = H ◦ Ω − 1 of the Hamiltonian of the p erio dic T o da lattic e H by Ω − 1 is a function of the action variables ( I n ) 1 ≤ n ≤ N − 1 and the Casimir functions C 1 , C 2 alone. In other wor ds, ˆ H is in Birkhoff normal form. Ther efor e, the 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 of the p erio dic T o da lattic e. Pr o of. Note that H ca n b e wr itten 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 eigenv alues of L ( b, a ). Hence, by Prop osition 7.10, ˆ H = H ◦ Ω − 1 is co ns tant on T ( x, y , β , α ), i.e. ˆ H is a function of ( I n ) 1 ≤ n ≤ N − 1 , C 1 , C 2 alone. 29 A Pro of of Lemma 7.7 W e b egin b y proving the estimate (88). In a first step we use a tr ace formula observed by v an Mo er b eke [15] whic h express es b 1 as a linear combination of the traces o f L + ≡ L + ( b, a ), L − ≡ L − ( b, a ) and L 2 ≡ L 2 ( b, a ), the matrix obtained from L ( b, a ) by r e moving the fir st column and the first row: b 1 = 1 2 (tr L + + tr L − ) − tr L 2 . (92) The Dirichlet eigenv alues ( µ n ) 1 ≤ n ≤ N − 1 can b e characterized as the sp e c trum o f L 2 . Hence we ca n rewr ite (92 ) as b 1 = 1 2 ( λ 1 + λ 2 N ) + 1 2 N − 1 X k =1 ( λ 2 k + λ 2 k +1 − 2 µ k ) . (93) F or b n +1 with 1 ≤ n ≤ N − 1, a s imilar formula ca n b e der ived, b n +1 = 1 2 ( λ 1 + λ 2 N ) + 1 2 N − 1 X k =1 ( λ 2 k + λ 2 k +1 − 2 µ ( n ) k ) . (94) Here ( µ ( n ) i ) 1 ≤ i ≤ N − 1 denotes the Diric hlet sp ectrum of L ( S n b, S n a ). Note that ( S n b ) 1 = b n +1 and sp ec Q ( S n b, S n a ) = sp ec Q ( b, a ) , since the discrimina nt s (15) for ( b, a ) and ( S n b, S n a ) coincide. Hence (94) can be obtained b y a pplying (9 3) to the elemen t ( S n b, S n a ) instead of ( b, a ). Note that for a ny 1 ≤ k ≤ N − 1 , the eig env a lue µ ( n ) k lies in the clos ed interv al [ λ 2 k , λ 2 k +1 ]. It follows from (94) that the difference b i − b j can then b e estimated by | b i − b j | ≤ N − 1 X k =1 γ k ( b, a ) . (95) As for any 1 ≤ n ≤ N we hav e N b n − N C 1 = N b n − N X j =1 b j = N X j =1 ( b n − b j ) , it follows that N | b n | ≤ N | C 1 | + N P N − 1 k =1 γ k , and fo rmula (88) of Lemma 7.7 is established. T o obtain (89), choose an ar bitr ary L 2 -orthonor mal basis ( f j ) 1 ≤ j ≤ N ⊆ R N of eige n vectors asso c ia ted to the eigenv a lues ( λ + j ) 1 ≤ j ≤ N of L + ( b, a ). W e claim that a k = N X j =1 λ + j f j ( k ) f j ( k + 1) . (96) 30 REFERENCES T o verify (96), multiply a k − 1 f j ( k − 1) + b k f j ( k ) + a k f j ( k + 1) = λ + j f j ( k ) by f j ( k + 1) and s um over j . As ( f j ) 1 ≤ j ≤ N is a n or thonormal basis , the N × N - matrix ( f j ( k )) j,k is orthog onal, hence P N j =1 f j ( k ) f j ( l ) = δ kl for 1 ≤ k , l ≤ N , and (96 ) follows. Since | f j ( k ) | ≤ 1 ∀ j, k , the identit y (96) implies that a k ≤ N X j =1 | λ + j | . (97) T o estimate | λ + j | , note that λ + j − C 1 = λ + j − 1 N N X k =1 λ + k = 1 N N X k =1 ( λ + j − λ + k ) . Hence for any 1 ≤ j ≤ N , − ( λ 2 N − λ 1 ) ≤ λ + j + C 1 ≤ λ 2 N − λ 1 or | λ + j | ≤ | C 1 | + ( λ 2 N − λ 1 ) leading to | λ + j | ≤ | C 1 | + 2 N − 1 X n =1 ( λ n +1 − λ n ) = | C 1 | + N − 1 X n =1 γ n + N X n =1  λ 2 n − λ 2 n − 1  . (98) W e now rec all from ([8], App endix B) that fo r a ny 1 ≤ n ≤ N , λ 2 n − λ 2 n − 1 ≤ 2 π C 2 N . 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