Global action-angle variables for the periodic Toda lattice
aa r X i v : . [ n li n . S I] F e b Global action-angle variablesfor the periodic Toda lattice
Andreas Henrici Thomas Kappeler ∗ October 24, 2018
Abstract
In this paper we construct global action-angle variables for the periodicToda lattice. Consider the Toda lattice with period N ( N ≥ q n = ∂ p n H, ˙ p n = − ∂ q n H for n ∈ Z , where the (real) coordinates ( q n , p n ) n ∈ Z satisfy ( q n + N , p n + N ) =( q n , p n ) for any n ∈ Z and the Hamiltonian H T oda is given by H T oda = 12 N X n =1 p n + α N X n =1 e q n − q n +1 (1)where α is a positive parameter, α >
0. For the standard Toda lattice, α = 1.The Toda lattice was introduced by Toda [19] and studied extensively in thesequel. It is an important model for an integrable system of N particles in onespace dimension with nearest neighbor interaction and belongs to the family oflattices introduce and numerically investigated by Fermi, Pasta, and Ulam intheir seminal paper [5]. To prove the integrability of the Toda lattice, Flaschkaintroduced in [3] the (noncanonical) coordinates b n := − p n ∈ R , a n := αe ( q n − q n +1 ) ∈ R > ( n ∈ Z ) . These coordinates describe the motion of the Toda lattice relative to the centerof mass. Note that the total momentum is conserved by the Toda flow, henceany trajectory of the center of mass is a straight line. ∗ Supported in part by the Swiss National Science Foundation, the programme SPECT,and the European Community through the FP6 Marie Curie RTN ENIGMA (MRTN-CT-2004-5652) In these coordinates the Hamiltonian H T oda takes the simple form H = 12 N X n =1 b n + N X n =1 a n , and the equations of motion are (cid:26) ˙ b n = a n − a n − ˙ a n = 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 Nn =1 a n = α N . Hence wecan identify the sequences ( b n ) n ∈ Z and ( a n ) n ∈ Z with the vectors ( b n ) ≤ n ≤ N ∈ R N and ( a n ) ≤ n ≤ N ∈ R N> . Our aim is to study the normal form of the systemof equations (2) on the phase space M := R N × R N> . This system is Hamiltonian with respect to the nonstandard Poisson structure J ≡ J b,a , defined at a point ( b, a ) = (( b n , a n ) ≤ n ≤ N by J = (cid:18) A − t A (cid:19) , (3)where A is the b -independent N × N -matrix A = 12 a . . . − a N − a a − a a . . . ...... . . . . . . . . . 00 . . . − a N − a N . (4)The Poisson bracket corresponding to (3) is then given by { F, G } J ( b, a ) = h ( ∇ b F, ∇ a F ) , J ( ∇ b G, ∇ a G ) i R 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 ( M ) and where ∇ b and ∇ a denote the gradients with respectto the N -vectors b = ( b , . . . , b N ) and a = ( a , . . . , a N ), respectively. Therefore,equations (2) can alternatively be written as ˙ b n = { b n , H } J , ˙ a n = { a n , H } J (1 ≤ n ≤ N ). Further note that { b n , a n } J = a n { b n +1 , a n } J = − a n , (6)while { b n , a k } J = 0 for any n, k with n / ∈ { k, k + 1 } .Since the matrix A defined by (4) has rank N −
1, the Poisson structure J is degenerate. It admits the two Casimir functions C := − N N X n =1 b n and C := N Y n =1 a n ! N (7)whose gradients ∇ b,a C i = ( ∇ b C i , ∇ a C i ) ( i = 1 , ∇ b C = − N (1 , . . . , , ∇ a C = 0 , (8) ∇ b C = 0 , ∇ a C = C N (cid:18) a , . . . , a N (cid:19) , (9)are linearly independent at each point ( b, a ) of M .The main result of this paper ist the following one: Theorem 1.1.
The periodic Toda lattice admits globally defined action-anglevariables. More precisely:(i) There exist real analytic functions ( I n ) ≤ n ≤ N − on M which are pairwisein involution and which Poisson commute with the Toda Hamiltonian H and the two Casimir functions C , C , i.e. for any ≤ m, n ≤ N − , i = 1 , , { I m , I n } J = 0 on M and { H, I n } J = 0 and { C i , I n } J = 0 on M . (ii) For any ≤ n ≤ N − there exist a real analytic submanifold D n ofcodimension and a function θ n : M \ D n → R , defined mod π and realanalytic when considered mod π , so that on M \ S N − n =1 D n , ( θ n ) ≤ n ≤ N − and ( I n ) ≤ n ≤ N − are conjugate variables. More precisely, for any ≤ m, n ≤ N − , i = 1 , { 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 N : ( C , C ) = ( β, α ) } denote the level set of( C , C ) for ( β, α ) ∈ R × R > . Note that ( − β N , α N ) ∈ M β,α where 1 N =(1 , . . . , ∈ R N . As the gradients ∇ b,a C and ∇ b,a C are linearly independenteverywhere on M , the sets M β,α are (real analytic) submanifolds of M of codi-mension two. Furthermore the Poisson structure J , restricted to M β,α , becomesnondegenerate everywhere on M β,α and therefore induces a symplectic structure ν β,α on M β,α . In this way, we obtain a symplectic foliation of M with M β,α being the symplectic leaves. A smooth function C : M → R is a Casimir function for J if { C, ·} J ≡ Corollary 1.2.
On each symplectic leaf M β,α , the action variables ( I n ) ≤ n ≤ N − are a maximal set of functionally independent integrals in involution of the pe-riodic Toda lattice. In subsequent work [9], we will use Theorem 1.1 to construct global
Birkhoffcoordinates for the periodic Toda lattice. More precisely, we introduce the modelspace P := R N − × R × R > endowed with the degenerate Poisson structure J whose symplectic leaves are R N − × { β } × { α } endowed with the standardPoisson structure, and prove the following theorem: Theorem 1.3.
There exists a real analytic, canonical diffeomorphism
Ω : ( M , J ) → ( P , J )( b, a ) (( x n , y n ) ≤ n ≤ N − , C , C ) such that the coordinates ( x n , y n ) ≤ n ≤ N − , C , C are global Birkhoff coordinatesfor the periodic Toda lattice, i.e. ( x n , y n ) ≤ n ≤ N − are canonical coordinates, C , C are the Casimirs and the transformed Toda Hamiltonian ˆ H = H ◦ Ω − is a function of the actions ( I n ) ≤ n ≤ N − and C , C alone. In [10] we used Theorem 1.3 to obtain a KAM theorem for Hamiltonianperturbations of the periodic Toda lattice.
Related work:
Theorem 1.1 and Theorem 1.3 improve on earlier work on thenormal form of the periodic Toda lattice in [1, 2]. In particular, we constructglobal Birkhoff coordinates on all of M instead of a single symplectic leaf andshow that techniques recently developed for treating the KdV equation (cf.[11, 12]) and the defocusing NLS equation (cf. [8, 16]) can also be applied forthe Toda lattice. Outline of the paper:
In section 2 we review the Lax pair of the periodicToda lattice and collect some auxiliary results on the spectrum of the Jacobimatrix L ( b, a ) associated to an element ( b, a ) ∈ M . In section 3 we study theaction variables ( I n ) ≤ n ≤ N − , and in section 4 we define the angle variables( θ n ) ≤ n ≤ N − on M \ ∪ nn =1 D n using holomorphic differentials defined on thehyperelliptic Riemann surface associated to the spectrum of L ( b, a ). In sections5 and 6 we establish formulas of the gradients of the actions and angles in termsof products of fundamental solutions and prove orthogonality relations betweensuch products which are then used in section 7 to show that ( I n ) ≤ n ≤ N − and( θ n ) ≤ n ≤ N − are canonical variables and to prove Theorem 1.1 and Corollary1.2. 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 periodic Jacobi matrix defined by L ± ( b, a ) := b a . . . ± a N a b a . . . ...0 a b . . . 0... . . . . . . . . . a N − ± a N . . . a N − b N , (10)and B the skew-symmetric matrix B = a . . . − a N − a a . . . ...0 − a . . . . . . 0... . . . . . . . . . a N − a N . . . − a N − . Hence the flow of ˙ L = [ B, L ] is isospectral.
Proposition 2.1.
For a solution (cid:0) b ( t ) , a ( t ) (cid:1) of the periodic Toda lattice (2),the eigenvalues ( λ + j ) ≤ j ≤ N of L (cid:0) b ( t ) , a ( t ) (cid:1) are conserved quantities. Let us now collect a few results from [17] and [19] of the spectral theory ofJacobi matrices needed in the sequel. Denote by M C the complexification of thephase space M , M C = { ( b, a ) ∈ C N : Re a j > ∀ ≤ j ≤ N } . For ( b, a ) ∈ M C we consider for any complex number λ the difference equation( 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 operator R b,a = a k − S − + b k S + a k S (12)with S m denoting the shift operator of order m ∈ Z , i.e.( S m y )( k ) = y ( k + m ) for k ∈ Z . Fundamental solutions:
The two fundamental solutions y ( · , λ ) and y ( · , λ )of (11) are defined by the standard initial conditions y (0 , λ ) = 1, y (1 , λ ) = 0and y (0 , λ ) = 0, y (1 , λ ) = 1. They satisfy the Wronskian identity W ( n ) := y ( n, λ ) y ( n + 1 , λ ) − y ( n + 1 , λ ) y ( n, λ ) = a N a n . (13) Note that for n = N one gets W ( N ) = 1 . (14)For each k ∈ N , y i ( k, λ, b, a ), i = 1 ,
2, is a polynomial in λ of degree at most k − b, a ) (see [17]). In particular, one easilyverifies that y ( N + 1 , λ, b, a ) is a polynomial in λ of degree N with leadingcoefficient α − N . Wronskian:
More generally, one defines for any two sequences ( v ( n )) n ∈ Z and( w ( n )) n ∈ Z the Wronskian 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 properties of the Wronskian, which can be easilyverified.
Lemma 2.2. (i) If y and z are solutions of (11) for λ = λ and λ = λ ,respectively, then W = W ( y, z ) satisfies for any k ∈ Z a k W ( k ) = a k − W ( k −
1) + ( λ − λ ) y ( k ) z ( k ) . (15) (ii) If y ( · , λ ) is a -parameter-family of solutions of (11) which is continuouslydifferentiable with respect to the parameter 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 − W ( k −
1) + y ( k, λ ) . (16) Discriminant:
We denote by ∆( λ ) ≡ ∆( λ, b, a ) the discriminant of (11),defined by ∆( λ ) := y ( N, λ ) + y ( N + 1 , λ ) . (17)In the sequel, we will often write ∆ λ for ∆( λ ). Note that y ( N + 1 , λ ) isa polynomial in λ of degree N with leading term α − N λ N , whereas y ( N, λ )is a polynomial in λ of degree less than N , hence ∆( λ, b, a ) is a polynomialin λ of degree N with leading term α − N λ N , and it depends real analyticallyon ( b, a ) (see e.g. [19]). According to Floquet’s Theorem (see e.g. [18]), for λ ∈ C given, (11) admits a periodic or antiperiodic solution of period N if thediscriminant ∆( λ ) satisfies ∆( λ ) = +2 or ∆( λ ) = −
2, respectively. (Thesesolutions correspond to eigenvectors of L + or L − , respectively, with L ± definedby (10).) It turns out to be more convenient to combine these two cases by l l l l l l l l (a) N = 4 l l l l l l l l l l (b) N = 5 Figure 1: Examples of the discriminant ∆( λ )considering the periodic Jacobi matrix Q ≡ Q ( b, a ) of size 2 N defined by Q = b a . . . . . . a N a b . . . ... 0 . . . a N − ... ...0 . . . a N − b N a N . . . . . . a N b a . . . . . . a b . . . ...... ... ... . . . . . . a N − a N . . . a N − b N . Then the spectrum of the matrix Q is the union of the spectra of the matrices L + and L − and therefore the zero set of the polynomial ∆ λ −
4. The function∆ λ − λ of degree 2 N and admits a product representation∆ λ − α − N N Y j =1 ( λ − λ j ) . (18)The factor α − N in (18) comes from the above mentioned fact that the leadingterm of ∆( λ ) is α − N λ N .For any ( b, a ) ∈ M , the matrix Q is symmetric and hence the eigenvalues( λ j ) ≤ j ≤ N of Q are real. When listed in increasing order and with their alge-braic multiplicities, they satisfy the following relations (cf. [17]) λ < λ ≤ λ < λ ≤ λ < . . . λ N − ≤ λ N − < λ N . As explained above, the λ j are periodic or antiperiodic eigenvalues of L andthus eigenvalues of L + or L − according to whether ∆( λ ) = 2 or ∆( λ ) = − λ ) = ( − N · , ∆( λ n ) = ∆( λ n +1 ) = ( − n + N · , ∆( λ N ) = 2 . (19) Since ∆ λ is a polynomial of degree N , ˙∆ λ ≡ ˙∆( λ ) = ddλ ∆( λ ) is a polynomialof degree N −
1, and it admits a product representation of the form˙∆ λ = N α − N N − Y k =1 ( λ − ˙ λ k ) . (20)The zeroes ( ˙ λ n ) ≤ n ≤ N − of ˙∆ λ satisfy λ n ≤ ˙ λ n ≤ λ n +1 for any 1 ≤ n ≤ N −
1. The intervals ( λ n , λ n +1 ) are referred to as the n -th spectral gap and γ n := λ n +1 − λ n as the n -th gap length . Note that | ∆( λ ) | > n -th gap is open if γ n > collapsed otherwise. Theset 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 } . (21)By writing the condition γ n = 0 as γ n = 0 and exploiting the fact that γ n (unlike γ n ) is a real analytic function on M , it can be shown as in [12] that D n is a real analytic submanifold of M of codimension 2. Isolating neighborhoods:
Let ( b, a ) ∈ M be given. The strict inequality λ n − < λ n guarantees the existence of a family of mutually disjoint opensubsets ( U n ) ≤ n ≤ N − of C so that for any 1 ≤ n ≤ N − U n is a neighborhoodof the closed interval [ λ n , λ n +1 ]. Such a family of neighborhoods is referred toas a family of isolating neighborhoods (for ( b, a )).In the case where ( b, a ) ∈ M C , we list the eigenvalues ( λ j ) ≤ j ≤ N in lexico-graphic ordering λ ≺ λ ≺ λ ≺ . . . ≺ λ N . We then extend the gap lenghts γ n to all of M C by γ n := λ n +1 − λ n (1 ≤ n ≤ N − D C n := { ( b, a ) ∈ W : γ n = 0 } . (23)In the sequel, we will omit the superscript and always write D n for D C n .Similarly, we do this for the zeroes ( ˙ λ n ) ≤ n ≤ N − of ˙∆ λ . The λ i ’s and ˙ λ i ’sno longer depend continuously on ( b, a ) ∈ M C . However, if we choose a smallenough complex neighborhood W of M in M C , then for any ( b, a ) ∈ W the closedintervals G n ⊆ C (1 ≤ n ≤ N −
1) defined by G n := { (1 − t ) λ n + tλ n +1 : 0 ≤ t ≤ } (24)are pairwise disjoint, and hence, as in the real case, there exists a family ofisolating neighborhoods ( U n ) ≤ n ≤ N − . The lexicographic ordering a ≺ b for complex numbers a and b is defined by a ≺ b : ⇐⇒ Re a < Re b orRe a = Re b and Im a ≤ Im b. (22) Lemma 2.3.
There exists a neighborhood W of M in M C such that for any ( b, a ) ∈ W , there are neighborhoods U n of G n in C ( ≤ n ≤ N − ) which arepairwise disjoint. Remark 2.4.
In the sequel, we will have to shrink the complex neighborhood W several times, but continue to denote it by the same letter.Contours Γ n : For any ( b, a ) ∈ W and any 1 ≤ n ≤ N −
1, we denote by Γ n a circuit in U n around G n with counterclockwise orientation. Isospectral set:
For ( b, a ) ∈ M , the set Iso( b, a ) of all elements ( b ′ , a ′ ) ∈ M so that Q ( b ′ , a ′ ) has the same spectrum as Q ( b, a ) is described with the help ofthe Dirichlet eigenvalues µ < µ < . . . < µ N − of (11) defined by y ( N + 1 , µ n ) = 0 . (25)They coincide with the eigenvalues of the ( N − × ( N − L = L ( b, a )given by b a . . . a . . . . . . . . . ...0 . . . . . . . . . 0... . . . . . . . . . a N − . . . a N − b N . In the sequel, we will also refer to µ , . . . , µ N − as the Dirichlet eigenvaluesof L ( b, a ). Evaluating the Wronskian identity (13) at λ = µ n one sees that µ n lies in the closure of the n -th spectral gap. More precisely, substituting y ( N + 1 , µ n ) = 0 in the identity (13) with λ = µ n yields y ( N, µ n ) y ( N + 1 , µ n ) = 1 . (26)Hence the value of the discriminant at µ n is given by∆( µ n ) = y ( N + 1 , µ n ) + 1 y ( N + 1 , µ n ) (27)and | ∆( µ n ) | ≥
2. By Lemma 2.6 below, given the point ( b, a ) with b = . . . = b N = β and a = . . . = a N = α , one has λ n = λ n +1 and hence µ n = λ n for any 1 ≤ n ≤ N −
1. It then follows from a straightforward deformationargument that λ n ≤ µ n ≤ λ n +1 everywhere in the real space M .Conversely, according to van Moerbeke [17], given any (real) Jacobi matrix Q with spectrum λ < λ ≤ λ < λ ≤ λ < . . . λ N − ≤ λ N − < λ N and any sequence ( µ n ) ≤ n ≤ N − with λ n ≤ µ n ≤ λ n +1 for n = 1 , . . . , N − r N -periodic Jacobi matrices Q with spectrum ( λ n ) ≤ n ≤ N and Dirichlet spectrum ( µ n ) ≤ n ≤ N − , where r is the number of n ’s with λ n <µ n < λ n +1 .In the case where ( b, a ) ∈ M C , we continue to define the Dirichlet eigenvalues( µ n ) ≤ n ≤ N − by (25), and we list them in lexicographic ordering µ ≺ µ ≺ . . . ≺ µ N − . Then the µ i ’s no longer depend continuously on ( b, a ) ∈ M C .However, if we choose the complex neighborhood W of M in M C of Lemma 2.3small enough, then for any ( b, a ) ∈ W and 1 ≤ n ≤ N − µ n is contained inthe neighborhood U n of G n (but not necessarily in G n itself). Riemann surface Σ b,a : Denote by Σ b,a the Riemann surface obtained as thecompactification of the affine curve C b,a defined by { ( λ, z ) ∈ C : z = ∆ ( λ, b, a ) − } . (28)Note that C b,a and Σ b,a are spectral invariants. (Strictly speaking, Σ b,a is aRiemann surface only if the spectrum of Q ( b, a ) is simple - see e.g. Appendix Ain [18] for details in this case. If the spectrum of Q ( b, a ) is not simple, Σ( b, a )becomes a Riemann surface after doubling the multiple eigenvalues - see e.g.section 2 of [13].) Dirichlet divisors:
To the Dirichlet eigenvalue µ n (1 ≤ n ≤ N −
1) weassociate the point µ ∗ n on the surface Σ b,a , µ ∗ n := (cid:16) µ n , ∗ q ∆ µ n − (cid:17) with ∗ q ∆ µ n − y ( N, µ n ) − y ( N + 1 , µ n ) , (29)where we used that, in view of (26) and the Wronskian identiy (14),∆ µ n − (cid:0) y ( N, µ n ) − y ( N + 1 , µ n ) (cid:1) . Standard root:
The standard root or s -root for short, s √ − λ , is defined for λ ∈ C \ [ − ,
1] by s p − λ := iλ + p − λ − . (30)More generally, we define for λ ∈ C \ { ta + (1 − t ) b | ≤ t ≤ } the s -root of aradicand of the form ( b − λ )( λ − a ) with a ≺ b, a = b by s p ( b − λ )( λ − a ) := γ s p − w , (31)where γ := b − a , τ := b + a and w := λ − τγ/ . Canonical sheet and canonical root:
For ( b, a ) ∈ M the canonical sheet ofΣ b,a is given by the set of points ( λ, c p ∆ λ −
4) in C b,a , where the c -root c p ∆ λ − C \ S Nn =0 ( λ n , λ n +1 ) (where λ := −∞ and λ N +1 := ∞ ) anddetermined by the sign condition − i c q ∆ λ − > λ N − < λ < λ N . (32)As a consequence one has for any 1 ≤ n ≤ N sign c q ∆ λ − i − − N + n − for λ n < λ < λ n +1 . (33)The definition of the canonical sheet and the c -root can be extended to theneighborhood W of M in M C of Lemma 2.3.1The s -root and the c -root will be used together in the following way: By theproduct representations (20) and (18) of ˙∆ λ and ∆ λ −
4, respectively, one seesthat for any ( b, a ) in W \ D n with 1 ≤ n ≤ N − λ c p ∆ λ − N ( λ − ˙ λ n ) s p ( λ n +1 − λ )( λ − λ n ) χ n ( λ ) ∀ λ ∈ Γ n (34)where χ n ( λ ) = ( − N + n − + p ( λ − λ )( λ N − λ ) Y m = n λ − ˙ λ m + p ( λ − λ m +1 )( λ − λ m ) . (35)Note that the principal branches of the square roots in (35) are well defined for λ near G n and that the function χ n is analytic and nonvanishing on U n . Inaddition, for ( b, a ) real, χ n is nonnegative on the interval ( λ n , λ n +1 ). Abelian differentials:
Let ( b, a ) ∈ M and 1 ≤ n ≤ N −
1. Then there exists aunique polynomial ψ n ( λ ) of degree at most N − ≤ k ≤ N − π Z c k ψ n ( λ ) p ∆ λ − dλ = δ kn . (36)Here, for any 1 ≤ k ≤ N − c k denotes the lift of the contour Γ k to thecanonical sheet of Σ b,a . For any k = n with λ k = λ k +1 , it follows from (36)that 1 π Z λ k +1 λ k ψ n ( λ ) + p ∆ λ − dλ = 0 . (37)Hence in every gap ( λ k , λ k +1 ) with k = n the polynomial ψ n has a zero whichwe denote by σ nk . If λ k = λ k +1 then it follows from (36) and Cauchy’s theoremthat σ nk = λ k = λ k +1 . As ψ n ( λ ) is a polynomial of degree at most N −
2, onehas ψ n ( λ ) = M n Y ≤ k ≤ N − k = n ( λ − σ nk ) , (38)where M n ≡ M n ( b, a ) = 0.In a straightforward way one can prove that there exists a neighborhood W of M in M C , so that for any ( b, a ) ∈ W and any 1 ≤ n ≤ N −
1, there is a uniquepolynomial ψ n ( λ ) of degree at most N − ≤ k ≤ N − W . Lemma 2.5.
Let ≤ n ≤ N − be fixed. Then the zeroes ( ˙ λ k ) ≤ k ≤ N − of ˙∆( λ ) and ( σ nk ) ≤ k ≤ N − ,k = n of ψ n ( λ ) satisfy the estimates ˙ λ k − τ k = O ( γ k ) , (39) σ nk − τ k = O ( γ k ) . (40) near any given point ( b, a ) ∈ W , where τ k = ( λ k +1 + λ k ) . Proof.
To verify (39), write ∆ λ − λ − λ − λ n )( λ n +1 − λ ) p n ( λ ) (41)where p n is a polynomial which does not vanish for λ ∈ U n . Then (39) followsby differentiating (41) with respect to λ at ˙ λ n .Fix 1 ≤ k, n ≤ N − k = n . In a first step we prove that σ nk − τ k = O ( γ k )near any given point ( b, a ) ∈ W . If γ k = 0, then σ nk = τ k , and (40) is clearlyfulfilled. Hence we assume in the sequel that γ k = 0. By the product formulas(38) and (18) for ψ n ( λ ) and ∆ λ −
4, respectively, we obtain, for λ near G k , ψ n ( λ ) c p ∆ λ − λ − σ nk s p ( λ k +1 − λ )( λ − λ k ) ζ nk ( λ ) (42)where ζ nk ( λ ) = M ′ n ( b, a )( λ − σ nn ) + p ( λ − λ )( λ N − λ ) Y m = k λ − σ nm + p ( λ m +1 − λ )( λ m − λ ) , (43)with σ nn := τ n and M ′ n ( b, a ) = 0. The function ζ nk is analytic and nonvanishingin U k . Substituting (42) into (36) one gets12 π Z Γ k λ − σ nk s p ( λ k +1 − λ )( λ − λ k ) ζ nk ( λ ) dλ = 0 . (44)We now drop the superscript n for the remainder of this proof and write ζ k as ζ k ( λ ) = ξ k + ( ζ k ( λ ) − ξ k ) with ξ k := ζ k ( τ k ) = 0. Note that12 π Z Γ k λ − σ k s p ( λ k +1 − λ )( λ − λ k ) dλ = τ k − σ k and hence (44) becomes( σ k − τ k ) ξ k = 12 π Z Γ k ( λ − σ k )( ζ k ( λ ) − ξ k ) s p ( λ k +1 − λ )( λ − λ k ) dλ. (45)To estimate the integral on the right hand side of (45), note that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π Z Γ k f ( λ ) s p ( λ k +1 − λ )( λ − λ k ) dλ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max λ ∈ G k | f ( λ ) | (46)for an arbitrary function f analytic on U k . We want to apply (46) for 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 ≤ k ≤ N − {| 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 ) . | ξ k | 6 = 0, we get | 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 improve the estimate (47). Note thatsup 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).For later use, we compute the spectra of Q ( b, a ) and L ( b, a ) in the spe-cial case ( b, a ) = ( β N , α N ) with β ∈ R and α >
0. Here 1 N denotes thevector (1 , . . . , ∈ R N . These points are the equilibrium points (of the restric-tions) of the Toda Hamiltonian vector field (to the symplectic leaves M β,α ). Wecompute the spectrum ( λ j ) ≤ j ≤ N of the matrix Q ( β N , α N ) and the Dirich-let eigenvalues ( µ k ) ≤ k ≤ N − of L = L ( β N , α N ) together with a normalizedeigenvector g l = (cid:0) g l ( j ) (cid:1) ≤ j ≤ N of µ l , i.e. Lg l = µ l g l , g l (1) = 0, and a vector h l = (cid:0) h l ( j ) (cid:1) ≤ j ≤ N which is the normalized solution of Ly = µ l y orthogonal to g l satisfying W ( h l , g l )( N ) > Lemma 2.6.
The spectrum ( λ j ) ≤ j ≤ N of Q ( β N , α N ) and the Dirichlet eigen-values ( µ l ) ≤ l ≤ N − of L ( β N , α N ) are given by λ = β − α,λ l = λ l +1 = µ l = β − α cos lπN (1 ≤ l ≤ N − ,λ N = β + 2 α. In particular, all spectral gaps of Q ( β N , α N ) are collapsed. For any ≤ l ≤ N − , the vectors g l and h l defined by g l ( j ) = ( − j +1 r N sin ( j − lπN (1 ≤ j ≤ N ) , (49) h l ( j ) = ( − j r N cos ( j − lπN (1 ≤ j ≤ N ) (50) satisfy Ly = µ l y and the normalization conditions N X j =1 g l ( j ) = N X j =1 h l ( j ) = 1 , g l (0) > , g l (1) = 0; W ( h l , g l )( N ) > , h h l , g l i R N = 0 . Remark 2.7.
For ( b, a ) = ( β N , α N ) the fundamental solutions y and y aregiven by y ( j, λ ) = − sin( ρ ( j − ρ ( j ∈ Z ) (51) y ( j, λ ) = sin( ρj )sin ρ ( j ∈ Z ) (52) where π < ρ < π is determined by cos ρ = λ − β α .Proof. For any λ ∈ R , the difference equation (11) for ( β N , α N ) reads( R β,α y )( k ) := βy ( k ) + αy ( k −
1) + αy ( k + 1) = λy ( k ) (53)and can be written as y ( k −
1) + y ( k + 1) = λ − βα y ( k ) . (54)Since we are looking for periodic solutions of (54), we make the ansatz y ( k ) = e ± iρk . This leads to the characteristic equation2 cos ρ ≡ e iρ + e − iρ = λ − βα . For the solution to be 2 N -periodic, it is required that ρ ∈ πN Z . To put theeigenvalues in ascending order, introduce ρ l = (1 + lN ) π with 0 ≤ l ≤ N . Thenfor any 1 ≤ j ≤ N , there exists 0 ≤ l ≤ N such that λ j = β + 2 α cos ρ l = β − α cos lπN . Note that for l = 0, λ = β − α is an eigenvalue of Q ( β N , α N ) with eigenvector y ( k ) = e iπk = ( − k . Similarly, for l = N , λ N = β + 2 α is an eigenvalue witheigenvector y ( k ) ≡
1. For the eigenvalue λ l = β − α cos lπN (1 ≤ l ≤ N − y ± ( k ) = e ± iρ l k are two linearly independent eigenvectors. As there are 2 N eigenvalues allto-gether, λ l is double for any 1 ≤ l ≤ N −
1, and λ and λ N are both simple. Itfollows that all N − µ l = λ l for all 1 ≤ l ≤ N − g k and h k , one easily verifies that for any realnumber λ = ± α + β , the fundamental solution y ( · , λ ) of (54) with y (0 , λ ) = 1and y (1 , λ ) = 0 is given by y ( j, λ ) = − sin (cid:0) ρ ( j − (cid:1) sin ρ ( j ∈ Z )where π < ρ < π is determined by cos ρ = λ − β α , thus proving (51). In the sameway, one verifies (52). For λ = µ l = β − α cos lπN we then getsin (cid:0) ρ l ( j − (cid:1) = sin (cid:0) (1 + lN ) π ( j − (cid:1) = ( − j +1 sin ( j − lπN . (cid:0) ρ l ( j − (cid:1) = 0 for j = 1 and j = N + 1. As N X j =1 sin ( j − lπN = N X j =1 cos ( j − lπN and these two sums add up to N , one sees that N X j =1 sin ( j − lπN = N , (55)yielding the claimed formula (49) for g l .By the same argument one shows that ˜ h l given by ( − j q N cos ( j − lπN (i.e.the right side of (50)) satisfies R β,α ˜ h l = µ l ˜ h l and the normalization condition P Nj =1 ˜ h l ( j ) = 1. Using standard trigonometric identities one verifies that h g l , ˜ h l i = N X j =1 g l ( j )˜ h l ( j ) = 0and W (˜ h l , g l )( N ) can be computed to be˜ 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) > . Hence ˜ h l is indeed the eigenvector with the required normalization, i.e. h l = ˜ h l ,thus proving (50). In the next two sections, we define the candidates for action-angle variableson the phase space M and investigate some of their properties. In this sec-tion we introduce globally defined action variables ( I n ) ≤ n ≤ N − as proposed byFlaschka-McLaughlin [4]. Definition 3.1.
Let ( b, a ) ∈ M . For ≤ n ≤ N − , I n := 12 π Z Γ n λ ˙∆ λ c p ∆ λ − dλ (56) where ˙∆ λ = ddλ ∆ λ is the λ -derivative of the discriminant ∆ λ = ∆( λ, b, a ) andthe contour Γ n and the canonical root c √· are given as in section 2. Remark 3.2.
The contours Γ n can be chosen locally independently of ( b, a ) .In view of the fact that ∆ λ is a spectral invariant of L ( b, a ) the actions I n areentirely determined by the spectrum of L ( b, a ) . In particular, ( I n ) ≤ n ≤ N − areconstants of motion, since by Proposition 2.1, the Toda flow is isospectral. Remark 3.3.
The variables ( I n ) ≤ n ≤ N − can also be represented as integralson the Riemann surface Σ b,a . If c n denotes the lift of Γ n to the canonical sheetof Σ b,a , (56) becomes I n = 12 π Z c n λ ˙∆ λ p ∆ λ − dλ (1 ≤ n ≤ N − . (57)From the definition (56), the following result can be deduced: Proposition 3.4.
On the real space M , each function I n is real, nonnegative,and it vanishes if γ n = 0 .Proof. Since Z Γ n ˙∆ λ c p ∆ λ − dλ = 0 , it follows that I n = 12 π Z Γ n ( λ − ˙ λ n ) ˙∆ λ c p ∆ λ − dλ. (58)By shrinking the contour of integration to the real interval, we get I n = 1 π Z λ n +1 λ n ( − N + n − ( λ − ˙ λ n ) ˙∆ λ + p ∆ λ − dλ by taking into account the definition (32) of the c -root. Since sign( λ − ˙ λ n ) ˙∆ λ =( − N + n − on [ λ n , λ n +1 ] \ { ˙ λ n } , the integrand is real and nonnegative, hence I n is real and nonnegative on M , as claimed.If γ n = 0, then λ n = λ n +1 . Hence ˙ λ n = λ n = λ n +1 = τ n and λ − ˙ λ n = i s p ( λ n +1 − λ )( λ − λ n ) . Therefore the integrand in (56) is holomorphic in the interior of the contour Γ n ,and by Cauchy’s theorem the integral in (56) vanishes.The action variables ( I n ) ≤ n ≤ N − can be extended in a straightforward wayto a complex neighborhood W of M in M C . Theorem 3.5.
There exists a complex neighborhood W of M in M C such thatfor all ≤ n ≤ N − , the functions I n defined by (56) extend analytically to W , I n : W → C .Proof. Let W denote a neighborhood of M in M C of Lemma 2.3 and define forany 1 ≤ n ≤ N − I n on W by the formula (56). Let ( b, a ) ∈ W be given. Then there exists a neighborhood W b,a of ( b, a ) in W so that theintegration contours Γ n in (56) can be chosen to be the same for any elementin W b,a and ˙∆ λ / c p ∆ λ − B ε (Γ n ) × W b,a , where B ε (Γ n ) := { λ ∈ C | dist ( λ, Γ n ) < ε } is the ε -neighborhood of Γ n with ε sufficiently small. Thisshows that I n is analytic on W .7 Proposition 3.6.
There exists a complex neighborhood W of M in M C suchthat for any ≤ n ≤ N − , the quotient I n /γ n extends analytically from M \ D n to all of W and has strictly positive real part on W . As a consequence, ξ n = + p I n /γ n is an analytic and nonvanishing function on W , where + √· isthe principal branch of the square root on C \ ( −∞ , .Proof. Let W be the complex neighborhood of Theorem 3.5. Substituting (34)into (58) leads to the following identity on W \ D n I n = N π Z Γ n ( λ − ˙ λ n ) s p ( λ n +1 − λ )( λ − λ n ) χ n ( λ ) dλ, where χ n is given by (35). Upon the substitution λ ( ζ ) = τ n + γ n ζ , with τ n = ( λ n + λ n +1 ) and δ n =
2( ˙ λ n − τ n ) γ n , one then obtains2 I n γ n = N π Z Γ ′ n ( ζ − δ n ) s p − ζ χ n ( τ n + γ n ζ ) dζ, (59)where Γ ′ n is the pullback of Γ n under the substitution λ = λ ( ζ ), i.e. a circuit in C around [ − , λ n − τ n = O ( γ n ), and hence δ n → γ n →
0. Weconclude thatlim γ n → I n γ n = N π Z Γ ′ n χ n ( τ n ) ζ dζ s p − ζ = χ n ( τ n ) N π Z − t dt + √ − t = N χ n ( τ n ) . By defining I n γ n by N χ n ( τ n ) on W ∩ D n , it follows that I n γ n is a continuousfunction on all of W . This extended function is analytic on W \ D n as is itsrestriction to W ∩ D n . By Theorem A.6 in [12] it then follows that I n γ n is analyticon all of W .By Lemma 3.7 below, the quotient I n /γ n can be bounded away from zeroon M , I n γ n ≥ π ( λ N − λ ) . By shrinking W , if necessary, it then follows that forany 1 ≤ n ≤ N −
1, the real part of I n /γ n is positive and never vanishes on W .Hence the principal branch of the square root of 2 I n /γ n is well defined on W and ξ n has the claimed properties.To show that + q I n γ n is well defined on W , we used in the proof of Proposition3.6 the following auxiliary result, which we prove in Appendix A: Lemma 3.7.
For any ( b, a ) ∈ M and any ≤ n ≤ N − , γ n ≤ π ( λ N − λ ) I n . (60)From the definition (56), Proposition 3.4, and the estimate (60) one obtains Corollary 3.8.
For any ( b, a ) ∈ M and any ≤ n ≤ N − , I n = 0 if and only if γ n = 0 . Actually, Lemma 3.7 can be improved. We finish this section with an a prioriestimate of the gap lengths γ n in terms of the action variables and the value ofthe Casimir C alone, which will be shown in Appendix B. Theorem 3.9.
For any ( b, a ) ∈ M β,α with β ∈ R , α > arbitrary, N − X n =1 γ n ≤ π α N − X n =1 I n ! + 9 π ( N − N − X n =1 I n ! . (61) In this section, we define and study the angle coordinates ( θ n ) ≤ n ≤ N − . Each θ n is defined mod 2 π on W \ D n , where W is a complex neighborhood of M in M C as in Lemma 2.3 and D n is given by (23). Definition 4.1.
For any ≤ n ≤ N − , the function θ n is defined for ( b, a ) ∈ M \ D n by θ n := η n + N − X n = k =1 β nk mod π, (62) where for k = n , β nk = Z µ ∗ k λ k ψ n ( λ ) p ∆ λ − dλ, η n = Z µ ∗ n λ n ψ n ( λ ) p ∆ λ − dλ ( mod π ) , (63) and where for ≤ k ≤ N − , µ ∗ k is the Dirichlet divisor defined in (29), and λ k is identified with the ramification point ( λ k , on the Riemann surface Σ b,a .The integration paths on Σ b,a in (63) are required to be admissible in the sensethat their image under the projection π : Σ b,a → C on the first component staysinside the isolating neighborhoods U k . Note that, in view of the normalization conditions (36) of ψ n , the aboverestriction of the paths of integration in (63) implies that η n and hence θ n arewell-defined mod 2 π . Theorem 4.2.
Let W be the complex neighborhood of M in M C introduced inLemma 2.3. Then for any ≤ n ≤ N − , the function θ n : W \ D n → C ( mod π ) is analytic. Remark 4.3.
As the lexicographic ordering of the eigenvalues of Q ( b, a ) is notcontinuous on W , it follows that η n and hence θ n are only continuous mod π on W .Proof of Theorem 4.2. To see that θ n : W \ D n → C (mod π ) is analytic, definefor any 1 ≤ k ≤ N − E k := { ( b, a ) ∈ M C : µ k ( b, a ) ∈ { λ k ( b, a ) , λ k +1 ( b, a ) }} . ≤ k ≤ N − k = n , β nk is analytic on W \ ( D k ∪ E k ), that its restrictions to D k ∩ W and E k ∩ W are weakly analytic ,and that it is continuous on W . Together with the fact that E k ∩ W and D k ∩ W are analytic subvarieties of W it then follows that β nk is analytic on W - seeTheorem A.6 in [12]. Similar results can be shown for β nn = η n (mod π ) on W \ D n , and one concludes that θ n (mod π ) is analytic on W \ D n .To prove that β nk , k = n , is analytic on W \ ( D k ∪ E k ), note that since λ k is a simple eigenvalue on W \ D k , it is analytic there. Furthermore, µ ∗ k is ananalytic function on the (sufficiently small) neighborhood W of M in M C . On W \ ( D k ∪ E k ) we can use the substitution λ = λ k + z to get β nk = Z µ ∗ k λ k ψ n ( λ ) p ∆ λ − dλ = Z µ ∗ k − λ k ψ n ( λ k + z ) √ z p D ( z ) dz, where D ( z ) = ∆ ( λ k + z ) − z is analytic near z = 0 and D (0) = 0. Note that D ( z ) does not vanish for z on an admissible integration path not going through λ k +1 . Such a path exists since ( b, a ) is in the complement of E k . Furthermore ψ n ( λ k + z ) and D ( z ) are analytic in z near such a path and depend analyticallyon ( b, a ) ∈ W \ ( D k ∪ E k ). Combining these arguments shows that β nk is analyticon W \ ( D k ∪ E k ).For k = n with λ k = λ k +1 one has Z λ k +1 λ k ψ n ( λ ) p ∆ λ − dλ = 0 . (64)As σ nk = λ k if λ k = λ k +1 one sees that (64) continues to hold for ( b, a ) ∈ E k ∩ W with λ k = λ k +1 and we have β nk | E k ∩ W ≡
0. To prove the analyticityof β nk | D k ∩ W consider the representation (42) of ψ n ( λ ) √ ∆ λ − . For ( b, a ) ∈ D k ∩ W ,one has λ k = λ k +1 = τ k = σ nk , which implies that the factor λ − σ nks √ ( λ k +1 − λ )( λ − λ k ) in (42) equals ± i . Hence wecan write β nk = Z µ ∗ k λ k ψ n ( λ ) p ∆ λ − dλ = ± i Z µ k τ k ζ nk ( λ ) dλ. As µ k is analytic on W , it then follows that β nk | D k ∩ W is analytic. To see that β nk is continuous on W , one separately shows that β nk is continuous at points in W \ ( D k ∪ E k ), E k ∩ W \ D k , D k ∩ W \ E k and D k ∩ E k ∩ W , where for the proofof the continuity of β nk at points in D k ∩ E k ∩ W we use (42) and the estimate σ nk − τ k = O ( γ k ) of Lemma 2.5.By (64), η n vanishes mod π on E n ∩ W \ D n . Arguing in a similar way asfor β nk one then concludes that η n (mod π ) is analytic on W \ D n . Let E and F be complex Banach spaces, and let U ⊂ E be open. The map f : U → F is weakly analytic on U , if for each u ∈ U , h ∈ E and L ∈ F ∗ , the function z Lf ( u + zh ) isanalytic in some neighborhood of the origin in C . In this section we establish formulas of the gradients of I n , θ n (1 ≤ n ≤ N − M in terms of products of the fundamental solutions y and y .Consider the discriminant for a fixed value of λ as a function on M ,∆ λ ( b, a ) = y ( N ) + y ( N + 1) . Then ∆ λ is a real analytic function on M . To obtain a formula for the gradientsof y ( N ) and y ( N + 1) with respect to b , differentiate R b,a y i = λy i with respectto b in the direction v ∈ R N to get( R b,a − λ ) h∇ b y i , v i ( k ) = − v k y i ( k ) . (65)Differentiating R b,a y i = λy i with respect to a in the direction u ∈ R N leads to( R b,a − λ ) h∇ a y i , u i ( k ) = − u k − y i ( k − − u k y i ( k + 1) . (66)Taking the sum of (65) and (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 can rewrite 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 standard scalar product in R N , whereas in(65), (66), and (67) it is the one in R N . The inhomogeneous Jacobi differenceequation (68) for the sequence h∇ b,a y i , ( v, u ) i ( k ) can be integrated using thediscrete analogue of the method of the variation of constants used for inhomo-geneous differential equations. 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 ≥ h∇ b,a y i , ( v, u ) i ( m ) = − y ( m ) a N m X k =1 y ( k )( R v,u y i )( k ) − y ( m ) a N m X k =1 y ( k )( R v,u y i )( k ) ! . (69)In the sequel, we will use (69) to derive various formulas for the gradi-ents. The common feature among these formulas is that they involve productsbetween the fundamental solutions y and y of (11). Whereas the gradientswith respect to b = ( b , . . . , b N ) involve products computed by componentwisemultiplication, the gradients with respect to a = ( a , . . . , a N ) involve productsobtained by multiplying shifted components, reflecting the fact that the b j arethe diagonal elements of the symmetric matrix L ( b, a ), whereas the a j are theoff-diagonal elements of L ( b, a ).1To simplify notation for the formulas in this section, we define for sequences (cid:0) v ( j ) j ∈ Z (cid:1) , (cid:0) w ( j ) j ∈ Z (cid:1) ⊆ C the N -vectors v · w := ( v ( j ) w ( j )) ≤ j ≤ N , (70) v · Sw := ( v ( j ) w ( j + 1)) ≤ j ≤ N , (71)where S denotes the shift operator of order 1. Combining (70) and (71), wedefine the 2 N -vector v · s w := ( v · w, v · Sw + w · Sv ) . (72)In case v = w we also use the shorter notation v := v · s v. (73)Written componentwise, v · s w is the 2 N -vector( v · s w )( j ) = (cid:26) v ( j ) w ( j ) (1 ≤ j ≤ N ) v ( j − N ) w ( j − N +1) + v ( j − N +1) w ( j − N ) ( N < j ≤ N ) Proposition 5.1.
For any ( b, a ) ∈ M , the gradient ∇ b,a ∆ λ = ( ∇ b ∆ λ , ∇ a ∆ λ ) is given by − a N ∇ b ∆ λ = y ( N ) y · y − y ( N +1) y · y + (cid:0) y ( N +1) − y ( N ) (cid:1) y · y (74) − a N ∇ a ∆ λ = 2 y ( N ) y · Sy − y ( N + 1) y · Sy + (cid:0) y ( N + 1) − y ( N ) (cid:1)(cid:0) y · Sy + y · Sy (cid:1) (75) or in the notation introduced above ∇ b,a ∆ λ = − a N (cid:0) y ( N ) y − y ( N + 1) y + ( y ( N + 1) − y ( N )) y · s y (cid:1) . (76) The gradients ∇ b ∆ λ and ∇ a ∆ λ admit the representations (1 ≤ m ≤ N ) ∂ ∆ λ ∂b m = − a m y ( N, λ, S m b, S m a ) , (77) ∂ ∆ λ ∂a m = − (cid:18) a m y ( N +1 , λ, S m b, S m a )+ 1 a m +1 y ( N − , λ, S m +1 b, S m +1 a ) (cid:19) . (78) Proof.
The claimed formula (76) follows from the defintion of ∆ λ and formula(69). Indeed, evaluate (69) for i = 1 and m = N to get h∇ b,a y , ( v,u ) i ( N )= − y ( N ) a N N X k =1 y ( k ) ( u k − y ( k −
1) + v k y ( k ) + u k y ( k + 1))+ y ( N ) a N N X k =1 y ( k ) ( u k − y ( k −
1) + v k y ( k ) + u k y ( k + 1)) . (79)2 In order to identify these two sums with h y , ( v, u ) i and h y · s y , ( v, u ) i , respec-tively, note that N X k =1 u k − y ( k ) y ( k −
1) = N X k =1 u k y ( k ) y ( k + 1) + u N T where T := y (0) y (1) − y ( N ) y ( N + 1) . For the second sum in (79), we get an expression of the same type with a similarcorrection term T := y (0) y (1) − y ( N ) y ( N + 1) . Taking into account the initial conditions of the fundamental solutions and theWronskian identity (13), one sees that y ( N ) T − y ( N ) T vanishes. Hence wehave the formula h∇ b,a y , ( v, u ) i ( N ) = − a N (cid:0) y ( N ) h y , ( v, u ) i − y ( N ) h y · s y , ( v, u ) i (cid:1) . (80)Similarly, evaluating formula (69) for i = 2 and m = N + 1 leads to h∇ b,a y , ( v, u ) i ( N +1) = − a N (cid:0) y ( N +1) h y · s y , ( v, u ) i− y ( N +1) h y , ( v, u ) i (cid:1) . (81)Here we used that the value of the right side of (69) does not change when weomit the term for k = m = N + 1 in both sums.It remains to prove the two formulas (77) and (78). We first note that y ( n, λ, S m b, S m a ) = a m a N (cid:16) y ( n + m, λ, b, a ) y ( m, λ, b, a ) − y ( n + m, λ, b, a ) y ( m, λ, b, a ) (cid:17) , (82)since both sides of (82) are solutions of R S m b,S m a y = λy (for fixed m ∈ Z )with the same initial conditions at n = 0 ,
1. For n = 1 this follows from theWronskian identity (13). Similarly, one shows that y ( N + m, λ ) = y ( N, λ ) y ( m, λ ) + y ( N + 1 , λ ) y ( m, λ ) , (83) y ( N + m, λ ) = y ( N, λ ) y ( m, λ ) + y ( N + 1 , λ ) y ( m, λ ) . (84)for any ( b, a ) ∈ M . Hence, suppressing the variable λ , we get y ( N, S m b, S m a ) = a m a N (cid:16)(cid:0) y ( N ) y ( m ) + y ( N + 1) y ( m ) (cid:1) y ( m ) − (cid:0) y ( N ) y ( m ) + y ( N + 1) y ( m ) (cid:1) y ( m ) (cid:17) = a m a N (cid:16) y ( N ) y ( m ) + (cid:0) y ( N + 1) − y ( N ) (cid:1) y ( m ) y ( m ) − y ( N + 1) y ( m ) (cid:17) . y ( N, S m b, S m a ) = − a m ∂ ∆ λ ∂b m and formula (77) is established. To prove (78), we first conclude from (82) that a N a m +1 y ( N − , S m +1 b, S m +1 a ) = y ( N + m, b, a ) y ( m + 1 , b, a ) − y ( N + m, b, a ) y ( m + 1 , b, a ) (85)and a N a m y ( N + 1 , S m b, S m a ) = y ( N + m + 1 , b, a ) y ( m, b, a ) − y ( N + m + 1 , b, a ) y ( 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) and (86) is − a N ∂ ∆ λ ∂a m , thus proving (78).As a next step, we compute the gradients of the Dirichlet and periodic eigen-values. In the following lemma, we consider the fundamental solution y ( · , µ )as an N -vector y ( j, µ ) ≤ j ≤ N . Let k y ( µ ) k = P Nj =1 y ( j, µ ) , and denote by ˙the derivative with respect to λ . Lemma 5.2. If µ is a Dirichlet eigenvalue of L ( b, a ) , then a N y ( N, µ ) ˙ y ( N + 1 , µ ) = k y ( µ ) k > . (87) In particular, ˙ y ( N + 1 , µ ) = 0 , which implies that all Dirichlet eigenvalues aresimple.Proof. This follows from adding up the relations (16).As the Dirichlet eigenvalues ( µ n ) ≤ n ≤ N − of L ( b, a ) coincide with the rootsof y ( N +1 , µ ) and these roots are simple, they are real analytic on M . Similarly,the eigenvalues λ and λ N are real analytic on M , whereas for any 1 ≤ n ≤ N − λ n and λ n +1 are real analytic on M \ D n . Note that for ( b, a ) ∈ M \ D n and i ∈ { n, n + 1 } , we have ˙∆ λ i = 0 as λ i is a simple eigenvalue. Proposition 5.3.
For any ≤ n ≤ N − , the gradients of the periodic eigen-values λ i ( i = 2 n, n + 1 ) on M \ D n and of the Dirichlet eigenvalues µ n on M are given by ∇ b,a λ i = − ∇ b,a ∆ λ | λ = λ i ˙∆ λ i = f i and ∇ b,a µ n = g n , (88) where we denote by f i the eigenvector of L ( b, a ) associated to λ i , normalized by N X j =1 f i ( j ) = 1 and (cid:0) f i (1) , f i (2) (cid:1) ∈ ( R > × R ) ∪ ( { } × R > ) , and where g n = ( g n ( j )) ≤ j ≤ N is the fundamental solution y ( · , µ n ) normalizedso that P Nj =1 g n ( j ) = 1 . Proof.
We first show the second formula in (88). Differentiating y ( N +1 , µ n ) =0 with respect to ( b, a ), one obtains ∇ b,a µ n = − ∇ b,a y ( N + 1 , λ ) | λ = µ n ˙ y ( N + 1 , µ n ) . (89)Here we used that ˙ y ( N + 1 , µ n ) = 0 by Lemma 5.2. To compute the gradient ∇ b,a y ( N + 1 , λ ) | λ = µ n , we evaluate (69) for i = 1 and m = N + 1. In view of y ( N + 1 , µ n ) = 0 and taking into account (26), one then gets ∇ b,a µ n = y ( µ n ) a N y ( N, µ n ) ˙ y ( N + 1 , µ n ) . (90)The claimed formula ∇ b,a µ n = g n now follows from Lemma 5.2. By differenti-ating ∆ λ i = ± b, a ), one obtains ∇ b,a λ i = −∇ b,a ∆ λ | λ = λ i / ˙∆ λ i in a similar fashion. To see that ∇ b,a λ i = f i , differentiate R b,a f i = λ i f i withrespect to ( b, a ) in the direction ( v, u ) ∈ R 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 product in R N . Take the scalar product(in R N ) of the above equation 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 i , ( v, u ) i R N , to conclude that ∇ b,a λ i = f i holds.To compute the Poisson brackets involving angle variables we need to estab-lish some additional auxiliary results. Recall from section 3 that for 1 ≤ k, n ≤ N − k = n and ( b, a ) ∈ M , β nk is given by β nk = Z µ ∗ k λ k ψ n ( λ ) p ∆ λ − dλ, (91)whereas β nn := η n = Z µ ∗ n λ n ψ n ( λ ) p ∆ λ − dλ (mod 2 π ) . (92)By Theorem 4.2, the functions β nk with k = n are real analytic on M , whereas β nn , when considered mod π , is real analytic on M \ D n . Proposition 5.4.
Let ≤ k ≤ N − and ( b, a ) ∈ M . If γ k > and λ k = µ k ,then for any ≤ n ≤ N − , ∇ b,a β nk = − ψ n ( µ k ) a N ˙∆ µ k g k · s h k , where h k denotes the solution of R b,a y = µ k y orthogonal to g k , i.e. N X j =1 g k ( j ) h k ( j ) = 0 , satisfying the normalization condition W ( h k , g k )( N ) = 1 .Proof. We use a limiting procedure first introduced in [16] for the nonlinearSchr¨odinger equation and subsequently used for the KdV equation in [11], [12].We approximate ( b, a ) ∈ M with λ k ( b, a ) = µ k ( b, a ) < λ k +1 ( b, a ) by ( b ′ , a ′ ) ∈ Iso( b, a ) satisfying λ k ( b, a ) < µ k ( b ′ , a ′ ) < λ k +1 ( b, a ). For such ( b ′ , a ′ ), usingthe substitution λ = λ k + z in the integral of (91), we obtain β nk ( b ′ , a ′ ) = Z µ ∗ k λ k ψ n ( λ ) p ∆ λ − dλ = Z µ k − λ k ψ n ( λ k + z ) √ z p D ( z ) dz, (93)where D ( z ) ≡ D ( λ k , z ) := (∆ ( λ k + z ) − /z . Taking the gradient, undoingthe substitution, and recalling the definition (29) of the starred square root thenleads to ∇ b,a β nk = ψ n ( µ k ) ∗ q ∆ µ k − ∇ b,a µ k − ∇ b,a λ k ) + E ( b ′ , a ′ ) , (94)with the remainder term E ( b ′ , a ′ ) given by E ( b ′ , a ′ ) = Z µ k − λ k ∇ b ′ ,a ′ ψ n ( λ k + z ) p D ( λ k , z ) ! dz √ z . As the gradient in the latter integral is a bounded function in z near z = 0,locally uniformly in ( b ′ , a ′ ), it follows by the dominated convergence theoremthat lim ( b ′ ,a ′ ) → ( b,a ) E ( b ′ , a ′ ) = 0.The gradient ∇ b,a β nk depends continuously on ( b, a ) ∈ M , hence we canconclude by (94) that it can be written as ∇ b,a β nk = lim ( b ′ ,a ′ ) → ( b,a ) ψ n ( µ k ) ∗ q ∆ µ k − ∇ b ′ ,a ′ µ k − ∇ b ′ ,a ′ λ k ) . (95)The gradient of both sides of the Wronskian identity (13), y ( N, λ ) y ( N + 1 , λ ) − y ( N + 1 , λ ) y ( N, λ ) = 1 , leads to y ( N + 1) ∇ b,a y ( N ) + y ( N ) ∇ b,a y ( N + 1) (96)= y ( N +1) ∇ b,a ∆ + ( y ( N ) − y ( N + 1)) ∇ b,a y ( N +1) , The λ -derivative can then be computed to be y ( N + 1) ˙ y ( N ) + ˙ y ( N + 1) y ( N )= y ( N + 1) ˙∆ + ( y ( N ) − y ( N + 1)) ˙ y ( N + 1) . (97)6 Using (96), (97), and y ( N + 1 , µ k ) = 0, formula (89) for ∇ b,a µ k leads to ∇ b,a µ k = − y ( N + 1) ∇ b,a ∆ + ( y ( N ) − y ( N + 1)) ∇ b,a y ( N + 1) y ( N + 1) ˙∆ + ( y ( N ) − y ( N + 1)) ˙ y ( N + 1) (cid:12)(cid:12)(cid:12) µ k . (98)Further by (88), ∇ b,a λ k = − ∇ b,a ∆˙∆ (cid:12)(cid:12)(cid:12) λ = λ k . (99)Now substitute (98) and (99) into ( ∇ b ′ ,a ′ µ k − ∇ b ′ ,a ′ λ k ) and use that by (29), ∗ q ∆ µ k − y ( N ) − y ( N + 1)) | µ k . We claim thatlim ( b ′ ,a ′ ) → ( b,a ) ∇ b,a µ k −∇ b,a λ k ∗ q ∆ µ k − y ( N +1) ∇ b,a y ( N ) − ˙ y ( N ) ∇ b,a y ( N +1)˙∆ ˙ y ( N + 1) y ( N ) (cid:12)(cid:12)(cid:12) λ k . (100)Indeed, to obtain (100) after the above mentioned substitutions, we split thefraction ∗ √ ∆ µk − ( ∇ b ′ ,a ′ µ k − ∇ b ′ ,a ′ λ k ) into two parts which are treated sepa-rately. In the first part we collect all terms in ∗ √ ∆ µk − ( ∇ b ′ ,a ′ µ k − ∇ b ′ ,a ′ λ k )which contain ( y ( N ) − y ( N + 1)) | µ k in the nominator, I ( a ′ , b ′ ) := − ˙∆ | λ k · ∇ b ′ ,a ′ y ( N + 1) | µ k + ∇ b ′ ,a ′ ∆ | λ k · ˙ y ( N + 1) | µ k ˙∆ | λ k · ( y ( N + 1) ˙∆ + ( y ( N ) − y ( N + 1)) ˙ y ( N + 1)) | µ k . Using again (97) we then getlim ( b ′ ,a ′ ) → ( b,a ) I ( b ′ , a ′ ) = ˙ y ( N + 1) ∇ b,a y ( N ) − ˙ y ( N ) ∇ b,a y ( N + 1)˙∆ ˙ y ( N + 1) y ( N ) (cid:12)(cid:12)(cid:12) λ k . The second term is then given by II ( b ′ , a ′ ) = y ( N + 1) | µ k · ( ∇ b ′ ,a ′ ∆ | λ k · ˙∆ | µ k − ˙∆ | λ k · ∇ b ′ ,a ′ ∆ | µ k )˙∆ | λ k · ( y ( N ) − y ( N + 1)) | µ k · ( ˙ y ( N + 1) y ( N )) | µ k . Note that the nominator of II ( b ′ , a ′ ) is of the order O ( µ k − λ k ). In view of(29), we have ( y ( N ) − y ( N + 1)) | µ k = O ( p µ k − λ k )whereas the other terms in the denominator of II ( b ′ , a ′ ) are bounded away fromzero. Indeed, λ k being a simple eigenvalue for ( b, a ) means ˙∆ | λ k = 0 for ( b ′ , a ′ )near ( b, a ). Further, use a version of (97) in the case λ k = µ k to conclude that˙ y ( N + 1) y ( N ) = y ( N + 1) ˙∆ λ k . Hence ˙ y ( N + 1) y ( N ) | µ k = 0 for ( b ′ , a ′ ) near ( b, a ) and II ( b ′ , a ′ ) vanishes inthe limit of µ k → λ k .7Substituting (80) and (81) into (100), we obtain˙ y ( N + 1) ∇ b,a y ( N ) − ˙ y ( N ) ∇ b,a y ( N + 1)˙∆ ˙ y ( N + 1) y ( N ) (cid:12)(cid:12)(cid:12) µ k = − a N ˙∆ y · s y with y = ˙ y ( N +1)˙ y ( N +1) y − y . Hence ∇ b,a β nk = − ψ n ( µ k ) a N ˙∆( µ k ) y · s y . Since β nk is invariant under the translation b b + t (1 , . . . , h∇ b,a β nk , ( , ) i R N vanishes. Hence0 = N X j =1 ∂β nk ∂b j = − ψ n ( µ k ) a N ˙∆( µ k ) N X j =1 y ( j ) y ( j ) . It means that y and y are orthogonal to each other. Finally we introduce h k := k y k y and verify that W ( h k , g k ) = W (cid:18) k y k y , y k y k (cid:19) = W ( y , y ) = W (cid:18) ˙ y ( N + 1)˙ y ( N + 1) y − y , y (cid:19) . By (13), it then follows that W ( h k , g k ) = − W ( y , y ) = W ( y , y ) . Hence by (14), W ( h k , g k )( N ) = W ( y , y )( N ) = 1 . This completes the proof of Proposition 5.4.
In Propositions 5.1, 5.3, and 5.4, we have expressed the gradients of ∆ λ , µ n ,and, on a subset of M , of β nk in terms of products of fundamental solutions ofthe difference equation (11). In this section we establish orthogonality relationsbetween such products - see [2] for similar computations. Recall that in (72) wehave introduced for arbitrary sequences ( v j ) j ∈ Z , ( w j ) j ∈ Z the 2 N -vector v · s w . Lemma 6.1.
For any ( b, a ) ∈ M , let v , w and v , w be pairs of solutions of(11) for arbitrarily given real numbers µ and λ , respectively. Then λ − µ ) a a N h v · s w , J ( v · s w ) i = V + B, (101) where V := (cid:0) W · S W (cid:1)(cid:12)(cid:12) N + (cid:0) S W · W (cid:1)(cid:12)(cid:12) N (102)8 with W and W denoting the Wronskians W := W ( v , w ) , W := W ( w , v ) ,and where B is given by B :=( λ − µ ) a (cid:16) ( v · w ) | N +11 ( v · s w )(2 N ) − ( v · w ) | N +11 ( v · s w )(2 N ) (cid:17) . (103) Proof.
We prove (101) by a straightforward calculation, using the recurrenceproperty (15) of the Wronskian sequences W and W . By the definition (3) of J we can write 2 h v · s w , J ( v · s w ) i = E + B , where E := N X k =1 a k (cid:2) ( v · s w )( k )( v · s w )( N + k ) − ( v · s w )( k + 1)( v · s w )( N + k ) − ( v · s w )( N + k )( v · s w )( k ) + ( v · s w )( N + k )( v · s w )( k + 1) (cid:3) and B := a N (cid:0) ( v · s w )( N + 1) − ( v · s w )(1) (cid:1) ( v · s w )(2 N )+ a N (cid:0) ( v · s w )(1) − ( v · s w )( N + 1) (cid:1) ( v · s w )(2 N ) . Let us first consider E . Calculating the products v j · s w j according to (72), weobtain, after regrouping, E = N X k =1 a k (cid:2) ( v ( k ) w ( k ) + v ( k + 1) w ( k + 1)) W ( k )+( v ( k ) w ( k ) + v ( k + 1) w ( k + 1)) W ( k ) (cid:3) + B with B := a N (cid:0) v ( N + 1) w ( N + 1) − ( v · s w )( N + 1) (cid:1) ( v · s w )(2 N )+ a N (cid:0) ( v · s w )( N +1) − v ( N +1) w ( N +1) (cid:1) ( v · s w )(2 N ) . Multiply E by ( λ − µ ) and use the recurrence relation (15) to express ( λ − µ ) v ( k ) w ( k ), ( λ − µ ) v ( k + 1) w ( k + 1), ( λ − µ ) v ( k ) w ( k ), and ( λ − µ ) v ( k +1) w ( k + 1) in terms of the Wronskians W and W to get( λ − µ ) E = N X k =1 [ a k a k +1 ( W ( k ) W ( k + 1) + W ( k + 1) W ( k )) − a k − a k ( W ( k − W ( k ) + W ( k ) W ( k − λ − µ ) B . The sum on the right hand side of the latter identity is a telescoping sumand equals the term a a N V with V defined in (102). In a straightforward wayone sees that ( λ − µ ) a a N ( B + B ) equals the expression B defined by (103), henceformula (101) is established.9 Corollary 6.2.
For any λ, µ ∈ C , { ∆ λ , ∆ µ } J = 0 . (104) Proof.
By the formula (76) for the gradient of ∆ λ , { ∆ λ , ∆ µ } J = h∇ b,a ∆ λ , J ∇ b,a ∆ µ i is a linear combination of terms of the form h v · s w , J ( v · s w ) i for pairs offundamental solutions v , w and v , w of (11) for µ and λ , respectively. Inview of (77) and (78), ∇ b ∆ λ and ∇ a ∆ λ are both N -periodic. In the case λ = µ we use Lemma 6.1 and note that the boundary terms (102) and (103) in Lemma6.1 vanish, hence { ∆ λ , ∆ µ } J = 0. In the case λ = µ the identity (104) followsfrom the skew-symmetry of {· , ·} J . Corollary 6.3.
Let ≤ k ≤ N and λ ∈ C . On the open subset of M where λ k is a simple eigenvalue of Q ( b, a ) one has { λ k , ∆ λ } J = 0 . Proof.
Using formula (88) for ∇ b,a λ k , we conclude from Corollary 6.2 that { λ k , ∆ λ } J = − λ k { ∆ µ , ∆ λ } J | µ = λ k = 0 . Corollary 6.4.
Let µ n be the n -th Dirichlet eigenvalue of L ( b, a ) and λ = µ n a real number. Then ( λ − µ n ) h y ( µ n ) , Jy ( λ ) i = (cid:18) a N y ( N + 1 , λ ) y ( N + 1 , µ n ) (cid:19) (105)( λ − µ n ) h y ( µ n ) , Jy ( λ ) · s y ( λ ) i = a N y ( N + 1 , λ ) y ( N + 1 , λ ) y ( N + 1 , µ n ) (106)( λ − µ n ) h y ( µ n ) , Jy ( λ ) i = a N (cid:18) y ( N + 1 , λ ) y ( N + 1 , µ n ) (cid:19) − ! (107) Proof.
The three stated identities follow from Lemma 6.1, using that y ( N +1 , µ n ) = 0, y (2 , µ n ) = − a N /a , and, by the Wronskian identity (26), y ( N, µ n ) · y ( N + 1 , µ n ) = 1. Corollary 6.5.
Let µ n be the n -th Dirichlet eigenvalue of L ( b, a ) and λ = µ n a real number. Then { µ n , ∆ λ } J = y ( N + 1 , λ )˙ y ( N + 1 , µ n ) ∗ q ∆ µ n − λ − µ n . (108)0 Proof.
By (90), combined with (26), we get { µ n , ∆ λ } J = y ( N + 1 , µ n ) a N ˙ y ( N + 1 , µ n ) h y ( µ n ) , J ∇ b,a ∆ λ i . (109)Substituting the formula (76) for J ∇ b,a ∆ λ we obtain h y ( µ n ) , J ∇ b,a ∆ λ i = − a N y ( N, λ ) h y ( µ n ) , Jy ( λ ) i− a N ( y ( N + 1 , λ ) − y ( N, λ )) h y ( µ n ) , Jy ( λ ) · s y ( λ ) i + 1 a N y ( N + 1 , λ ) h y ( µ n ) , Jy ( λ ) i . (110)To evaluate the right side of (110), we apply Corollary 6.4 and get λ − µ n a a N h y ( µ n ) , J ∇ b,a ∆ λ i = 1 a y ( N + 1 , µ n ) (cid:16) − y ( N, λ ) y ( N + 1 , λ ) − ( y ( N + 1 , λ ) − y ( N, λ )) y ( N + 1 , λ ) y ( N + 1 , λ )+ y ( N + 1 , λ ) y ( N + 1 , λ ) (cid:17) − y ( N + 1 , λ ) a . Using the Wronskian identity (14), the sum of the terms in the square bracketof the latter expression simplifies, and one obtains λ − µ n a a N h y ( µ n ) , J ∇ b,a ∆ λ i = y ( N + 1 , λ ) a y ( N + 1 , µ n ) − y ( N + 1 , λ ) a = y ( N + 1 , λ ) a ( y ( N, µ n ) − , (111)where for the latter equality we again used (14). Substituting (111) into (109),we get λ − µ n a a N { µ n , ∆ λ } J = y ( N + 1 , µ n ) y ( N + 1 , λ ) a a N ˙ y ( N + 1 , µ n ) ( y ( N, µ n ) − y ( N + 1 , λ ) a a N ˙ y ( N + 1 , µ n ) ∗ q ∆ µ n − , where we used that, by the definition of the starred square root (29), ∗ q ∆ µ n − y ( N, µ n ) − y ( N + 1 , µ n ) = y ( N + 1 , µ n )( y ( N, µ n ) − . This proves (108).
Proposition 6.6.
For any λ ∈ R , ≤ n ≤ N − , and ( b, a ) ∈ M \ D n , { θ n , ∆ λ } J = ψ n ( λ ) . (112)1 Proof.
Recall that θ n = P N − k =1 β nk (mod 2 π ) with β nk given by (91)-(92). Tocompute { β nk , ∆ λ } J , we first consider the case where ( b, a ) / ∈ S N − k =1 D k and λ k < µ k < λ k +1 for any 1 ≤ k ≤ N −
1. Then λ k and µ ∗ k are smooth near( b, a ) and, by Leibniz’s rule, we get { β nk , ∆ λ } J = Z µ ∗ k λ k { ψ n ( µ ) q ∆ µ − , ∆ λ } J dµ + ψ n ( µ k ) ∗ q ∆ µ k − { µ k , ∆ λ } J − ψ n ( λ k ) ∗ q ∆ λ k − { λ k , ∆ λ } J ! . By Corollary 6.3, { λ k , ∆ λ } J = 0. Moreover, as the gradient ∇ b,a ψ n ( µ ) √ ∆ µ − is orthogonal to T b,a Iso ( b, a ) and J ∇ b,a ∆ λ ∈ T b,a Iso ( b, a ) it follows that thePoisson bracket { ψ n ( µ ) √ ∆ µ − , ∆ λ } J vanishes for any µ in the isolating neighborhood U n of G n . Hence { β nk , ∆ λ } J = ψ n ( µ k ) ∗ q ∆ µ k − { µ k , ∆ λ } J . By (108), we then obtain { θ n , ∆ λ } J = N − X k =1 ψ n ( µ k )˙ y ( N + 1 , µ k ) y ( N + 1 , λ ) λ − µ k = ψ n ( λ ) , where for the latter equality we used that P N − k =1 ψ n ( µ k )˙ y ( N +1 ,µ k ) y ( N +1 ,λ ) λ − µ k and ψ n ( λ )are both polynomials in λ of degree at most N − N − µ k ) ≤ k ≤ N − .In the general case, where ( b, a ) ∈ M \ D n and the Dirichlet eigenvalues arearbitrary, λ k ≤ µ k ≤ λ k +1 for any 1 ≤ k ≤ N −
1, the claimed result followsfrom the case treated above by continuity.
Proposition 6.7.
Let ≤ n, m, k, l ≤ N − and let ( b, a ) ∈ M with λ i ( b, a ) = µ i ( b, a ) for i = k, l . Then { β nk , β ml } J = 0 . Proof.
In view of Proposition 5.4, this amounts to showing that the scalar prod-uct h ( g k · s h k ) , J ( g l · s h l ) i vanishes. For k = l , this follows from the skew-symmetryof the Poisson bracket, hence we can assume k = l . We apply Lemma 6.1 with v := g k , w := h k , v := h l and w := g l , which implies that W = W ( g k , g l )and W = W ( h k , h l ). Since g k (1), g l (1), g k ( N + 1) and g l ( N + 1) all vanish,we conclude that W ( N ) = W (0) = 0 and ( SW )( N ) = ( SW )(0) = 0, hencethe expressions V and E , defined in (102) and (103), vanish. This proves theclaim.2 In this section we complete the proof of Theorem 1.1 and Corollary 1.2. Inparticular we show that the variables ( I n ) ≤ n ≤ N − , ( θ n ) ≤ n ≤ N − satisfy thecanonical relations stated in Theorem 1.1.Using the results of the preceding sections, we can now compute the Poissonbrackets among the action and angle variables introduced in section 3. Theorem 7.1.
The action-angle variables ( I n ) ≤ n ≤ N − and ( θ n ) ≤ n ≤ N − sat-isfy the following canonical relations for ≤ n, m ≤ N − :(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) Proof.
Recall that ddt arcosh ( t ) = ( t − − . Hence for any ( b, a ) ∈ M I n = 12 π Z Γ n λ ddλ arcosh (cid:12)(cid:12)(cid:12)(cid:12) ∆ λ (cid:12)(cid:12)(cid:12)(cid:12) dλ and therefore ∇ b,a I n = 12 π Z Γ n λ ddλ ∇ b,a ∆ λ c p ∆ λ − dλ. Integrating by parts we get ∇ b,a I n = − π Z Γ n ∇ b,a ∆ λ c p ∆ λ − dλ. (115)As { ∆ λ , ∆ µ } J = 0 for all λ, µ ∈ C by Corollary 6.2, it follows that { I n , I m } J = 0on M for any 1 ≤ n, m ≤ N − { θ n , I m } J = − π Z Γ m { θ n , ∆ λ } J c p ∆ λ − dλ = − π Z Γ m ψ n ( λ ) c p ∆ λ − dλ = − δ nm , by the normalizing condition (36) of ψ n .To prove that the angles ( θ n ) ≤ n ≤ N − pairwise Poisson commute we needthe following lemma. We denote by K = K ( b, a ) the index set of the open gaps,i.e. K ( b, a ) = { ≤ n ≤ N − γ n ( b, a ) > } . Lemma 7.2.
At every point ( b, a ) in M , the set of vectors(i) (cid:0) ( ∇ b,a I n ) n ∈ K , ∇ b,a C , ∇ b,a C (cid:1) and (ii) ( J ∇ b,a I n ) n ∈ K are both linearly independent.Proof. The claimed statements follow from the orthogonality relations statedin Theorem 7.1: Let ( b, a ) ∈ M and suppose that for some real coefficients( r n ) n ∈ K ⊆ R and s , s ∈ R we have X n ∈ K r n ∇ b,a I n + s ∇ b,a C + s ∇ C = 0 . For any m ∈ K , take the scalar product of this identity with J ∇ b,a θ m . Usingthat { I n , θ m } J = δ nm and that C and C are Casimir functions of {· , ·} J oneobtains 0 = X n ∈ K r n { I n , θ m } J = X n ∈ K r n δ nm = r m . Thus r m = 0 for all m ∈ K , and it follows that s ∇ b,a C + s ∇ b,a C = 0. By(8) and (9), ∇ b,a C and ∇ b,a C are linearly independent, hence s = s = 0.This shows (i). The proof of (i) also shows that (ii) holds. Theorem 7.3.
In addition to the canonical relations stated in Theorem 7.1, theangle variables ( θ n ) ≤ n ≤ N − satisfy for any ≤ n, m ≤ N − on M \ ( D n ∪ D m ) { θ n , θ m } J = 0 . (116) Proof.
Let 1 ≤ n, m ≤ N −
1. By continuity, it suffices to prove the identity(116) for ( b, a ) ∈ M \ (cid:0) S N − l =1 D l (cid:1) . Let ( b, a ) be an arbitrary element in M \ (cid:0) S N − l =1 D l (cid:1) . Recall that Iso( b, a ) denotes the set of all elements ( b ′ , a ′ ) in M with spec( Q b ′ ,a ′ ) = spec( Q b,a ),Iso ( b, a ) = { ( b ′ , a ′ ) ∈ M : ∆( · , b ′ , a ′ ) = ∆( · , b, a ) } . Then Iso( b, a ) is a torus contained in M \ (cid:0) S N − l =1 D l (cid:1) , and as all eigenvalues of Q ( b, a ) are simple, its dimension is N −
1. By Lemma 7.2, at any point ( b ′ , a ′ ) ∈ Iso( b, a ), the vectors ( J ∇ b ′ ,a ′ I k ) ≤ k ≤ N − are linearly independent. Using theformula (115) for the gradient of I k , one sees that, by Corollary 6.2, for any µ ∈ R , 1 ≤ k ≤ N − h∇ b ′ ,a ′ ∆ µ , J ∇ b ′ ,a ′ I k i = − π Z Γ n { ∆ µ , ∆ λ } J c p ∆ λ − dλ = 0 . Hence for any ( b ′ , a ′ ) ∈ Iso( b, a ),( J ∇ b ′ ,a ′ I k ) ≤ k ≤ N − ∈ T b ′ ,a ′ Iso ( b, a ) , and therefore these vectors form a basis of T b ′ ,a ′ Iso ( b, a ).To prove the identity (116), we apply the Jacobi identity { F, { G, H } J } J + { G, { H, F } J } J + { H, { F, G } J } J = 04 to the functions I k , θ n and θ m . Since by Theorem 7.1, { I k , θ n } J = δ kn , weobtain { I k , { θ n , θ m } J } J = 0 on M \ N − [ l =1 D l ! for any 1 ≤ k ≤ N − . It then follows by the above considerations that ∇ b ′ ,a ′ { θ n , θ m } J is orthogonalto T b ′ ,a ′ Iso ( b, a ) for all ( b ′ , a ′ ) ∈ Iso( b, a ), i.e. { θ n , θ m } J is constant 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 exists a unique element ( b ′ , a ′ ) ∈ Iso( b, a ) satisfying µ k ( b ′ , a ′ ) = λ k ( b, a ) for all 1 ≤ k ≤ N −
1. The claimed identity (116) thenfollows from Proposition 6.7.
Proof of Theorem 1.1.
By Theorem 3.5 and Theorem 4.2, the action and anglevariables introduced in Definitions 3.1 and 4.1, respectively, have the claimedanalyticity properties. The canonical relations among these variables have beenverified in Theorem 7.1 and Theorem 7.3, and the relations { 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 and C areCasimir functions. It remains to show that the actions Poisson commute withthe Toda Hamiltonian. To this end note that that the Hamiltonian H can bewritten as H = 12 N X n =1 b n + N X n =1 a n = 12 tr ( L ( b, a ) ) = 12 N X j =1 ( λ + j ) where ( λ + j ) ≤ j ≤ N are the N eigenvalues of L ( b, a ). Recall that on the denseopen subset M \ ∪ N − k =1 D k of M , the λ + i ’s (1 ≤ i ≤ N ) are simple eigenvaluesand hence real analytic. It then follows by (115) that for any 1 ≤ n ≤ N − { H, I n } J = N X i =1 λ + i { λ + i , I n } J = − N X i =1 λ + i π Z Γ n { λ + i , ∆ λ } J c p ∆ λ − dλ = 0 , where for the latter identity we used Corollary 6.3. Hence for any 1 ≤ n ≤ N − { H, I n } J = 0 on M \ ∪ N − k =1 D k . By continuity it then follows that { H, I n } J = 0 everywhere on M . Proof of Corollary 1.2.
Since for any β ∈ R and α > M β,α is a submanifold of M of dimension 2( N − N − M β,α . For any given ( b, a ) ∈ M β,α let π β,α denote theorthogonal projection T b,a M → T b,a M β,α . Then the gradient of the restriction 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 thevectors ( π β,α ∇ b,a I n ) n ∈ K are linearly independent. As M β,α \ ∪ N − k =1 D k is densein M β,α , it then follows that ( I n | M β,α ) ≤ n ≤ N − are functionally independent.5Finally, as C and C are Casimir functions of {· , ·} J , it follows that for any( 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 restrictions I n | M β,α of I n (1 ≤ n ≤ N −
1) are in involution.
A Proof of Lemma 3.7
In this Appendix, we prove Lemma 3.7. It turns out that the proof in ([1],p. 601-602) of the special case where the parameter α in (1) equals 1 can beadapted for arbitrary values. Proof of Lemma 3.7.
Let ( b, a ) be an arbitrary element in M and 1 ≤ n ≤ N − I n = π R λ n +1 λ n arcosh | ∆( λ ) | dλ and use ddt arcosh ( t ) = √ t − to obtain I n = 1 π Z λ n +1 λ n Z | ∆( λ ) | / √ t − dt dλ. Since the integrand of the inner integral is nonincreasing, we estimate it frombelow by its value at | ∆( λ ) | . This leads to I n ≥ π Z λ n +1 λ n p | ∆( λ ) | − p | ∆( λ ) | + 2 dλ. (117)We will show below that for λ n ≤ λ ≤ λ n +1 p | ∆( λ ) | − p | ∆( λ ) | + 2 ≥ √ λ − λ n p λ n +1 − λλ N − λ . (118)We then substitute (118) into the integral (117) and split the integration intervalinto two equal parts, I n ≥ π λ N − λ Z τ n λ n √ λ − λ n p λ n +1 − λλ N − λ dλ, where τ n = ( λ n + λ n +1 ) /
2. For λ n ≤ λ ≤ τ n we estimate the quantity λ n +1 − λ from below by γ n /
2, yielding I n ≥ π λ N − λ Z τ n λ n r γ n p λ − λ n dλ = 13 π ( λ N − λ ) γ n . It remains to verify (118). Recall that λ n and λ n +1 are either both periodicor both antiperiodic eigenvalues of L . If λ n and λ n +1 are periodic eigenvalues,we have ∆( λ ) ≥ λ n ≤ λ ≤ λ n +1 , i.e. | ∆( λ ) | = ∆( λ ). In order to makewriting easier, let us assume that N is even - the case where N is odd is treated6 B PROOF OF THEOREM ?? in the same way. Then by (19), λ and λ N are periodic eigenvalues of L andthus for any λ n ≤ λ ≤ λ n +1 , the left side of (118) can be estimated frombelow by p ∆( λ ) − p ∆( λ ) + 2 = s ( λ − λ n )( λ − λ n +1 )( λ − λ )( λ − λ N − ) · R ≥ √ λ − λ n p λ n +1 − λλ N − λ · R, where R ≡ R ( λ ) = s λ − λ λ − λ · · · λ − λ n − λ − λ n − λ n +4 − λλ n +2 − λ · · · λ N − λλ N − − λ . (119)As each of the the fractions under the square root in (119) can be estimated frombelow by 1, for any λ n ≤ λ ≤ λ n +1 it follows that R ( λ ) ≥ λ n , λ n +1 ],leading to the claimed estimate (118). B Proof of Theorem 3.9
In this Appendix we prove Theorem 3.9 using estimates derived in [1]. Let ( b, a )be in M β,α with β ∈ R and α > Proposition B.1.
For any ( b, a ) ∈ M β,α with β ∈ R , α > arbitrary and any ≤ n ≤ N , λ n ( b, a ) − λ n − ( b, a ) ≤ παN . (120)Before proving Proposition B.1 we show how to use it to prove Theorem 3.9. Proof of Theorem 3.9.
We begin by adding up the inequalities (60) and get N − X n =1 γ n ≤ π ( λ N − λ ) N − X n =1 I n ! . (121)Note that λ N − λ = N − X n =1 γ n + N X n =1 ( λ n − λ n − ) . By the estimate of Proposition B.1 we get for any ( b, a ) ∈ M β,α λ N − λ ≤ πα + N − X n =1 γ n which we substitute into (121) to yield N − X n =1 γ n ≤ απ N − X n =1 I n ! + 3 π N − X n =1 γ n ! N − X n =1 I n ! . ab ≤ ǫ a + 1 ǫ b ( a, b ∈ R , ǫ > a = P N − n =1 γ n , b = P N − n =1 I n , and ǫ = π ( N − , one gets N − X n =1 γ n ≤ π α N − X n =1 I n ! + 3 π π ( N − N − X n =1 γ n ! + 3 π ( N − N − X n =1 I n ! . As (cid:16)P N − n =1 γ n (cid:17) ≤ ( N − (cid:16)P N − n =1 γ n (cid:17) , one then concludes that12 N − X n =1 γ n ≤ π α N − X n =1 I n ! + 9 π N − N − X n =1 I n ! , (122)which is the claimed estimate (61).To prove Proposition B.1 we first need to make some preparations. Notethat for an element of the form ( b, a ) = ( β N , α N ) one has, by Lemma 2.6, λ n ( β N , α N ) − λ n − ( β N , α N ) = 2 α (cid:16) cos ( n − πN − cos nπN (cid:17) = 4 α sin (2 n − π N sin π N< παN . Hence to prove Proposition B.1 it suffices to show that for any ( b, a ) ∈ M β,α and any 1 ≤ n ≤ Nλ n ( b, a ) − λ n − ( b, a ) ≤ λ n ( − β N , α N ) − λ n − ( − β N , α N ) . (123)To this end, following [15] (cf. also [7]), we introduce the conformal map δ ( λ ) := ( − N Z λλ ˙∆( µ ) p − ∆ ( µ ) dµ, (124)where the sign of the square root is chosen such that for µ < λ , p − ∆ ( µ ) haspositive imaginary part. It is defined on the upper half plane U := { Im z > } and its image is the spike domainΩ( b, a ) := { x + iy : 0 < x < N π, y > } \ N − [ n =1 T n where for 1 ≤ n ≤ N − T n denotes the spike T n := ( nπ + it : 0 < t ≤ arcosh ( − N + n ∆( ˙ λ n )2 !) . B PROOF OF THEOREM ?? To see that δ ( U ) = Ω( b, a ), note that for any ( b, a ) ∈ M and λ ∈ U thediscriminant ∆( λ ) and the function δ ( λ ) are related by the formula∆( λ ) = 2( − N cos δ ( λ ) . (125)To prove (125), recall that for − < t <
1, one has ddt arccos t = + √ − t .This formula remains valid for any t in C \ (( −∞ , − ∪ [1 , ∞ )). Thus δ ( λ ) = ( − N Z λλ ˙∆( µ ) p − ∆ ( µ ) dµ = Z λλ ddµ arccos (cid:18) ( − N ∆( µ )2 (cid:19) . Since by (19), ∆( λ ) = 2( − N , we then get δ ( λ ) = arccos (cid:18) ( − N ∆( µ )2 (cid:19) (cid:12)(cid:12)(cid:12) λλ = arccos (cid:18) ( − N ∆( λ )2 (cid:19) , (126)leading to formula (125) and the claimed statement that δ ( U ) = Ω( b, a ).The map δ can be extended continuously to the closure { Im z ≥ } of theupper half plane. This extension, again denoted by δ , is 2-1 over each nontrivialspike T n and 1-1 otherwise. Since the n -th spike T n is the image under δ ofthe n -th gap ( λ n , λ n +1 ), all spikes are empty iff all gaps are collapsed. ByLemma 2.6, all gaps are collapsed for ( b, a ) = ( − β N , α N ), hence Ω( b, a ) ⊂ Ω( − β N , α N ) for any ( b, a ) ∈ M β,α .Note that λ n ( b, a ) − λ n − ( b, a ) = δ − ( nπ − ) − δ − (( n − π +) = Z ∞−∞ u ( n ) ( δ ( λ )) d λ, (127)where u ( n ) : Ω( b, a ) → R , z u ( n ) ( z ; b, a ) is the harmonic measure of the opensubset (( n − π, nπ ) of ∂ Ω( b, a ) (see e.g. [6] for the notion of the harmonicmeasure).We need two lemmas from complex and harmonic analysis, respectively. Lemma B.2.
For ( b, a ) = ( − β N , α N ) with β ∈ R , α > arbitrary, the map δ ( λ ) defined by (124) is given by δ ( λ ) = N arccos (cid:18) − λ + β α (cid:19) . (128) For arbitrary ( b, a ) in M β,α and ξ ∈ R , the following asymptotic estimate holdsas η → ∞ δ ( ξ + iη ) = N arccos (cid:18) − ξ + iη + β α (cid:19) + O ( η − ) , (129) locally uniformly in ξ .Proof of Lemma B.2. In view of the formulas (51) and (52) for the fundamen-tal solutions y and y for ( b, a ) = ( − β N , α N ), the discriminant ∆( λ ) =9∆( λ, − β N , α N ) is given by∆( λ ) = y ( N, λ ) + y ( N + 1 , λ )= − sin( ρ ( N − ρ + sin( ρ ( N + 1))sin ρ = 2 cos( ρN ) , where π < ρ < π is determined by cos ρ = λ + β α . Hence∆( λ ) = 2 T N (cid:18) λ + β α (cid:19) (130)where for any z ∈ U , T N ( z ) = cos( N arccos z ) . (131)Actually, T N ( z ) is a polynomial in z of degree N , referred to as Chebychevpolynomial of the first kind. Substituting (130) into (126), we obtain δ ( λ ) = arccos (cid:18) ( − N ∆( λ )2 (cid:19) = arccos (cid:18) ( − N T N (cid:18) λ + β α (cid:19)(cid:19) . The claimed identity (128) now follows from the elementary symmetry T N ( z ) = ( − N T N ( − z ) ∀ z ∈ C . Now let ( b, a ) ∈ M β,α . The asymptotic estimate (129) follows by compar-ing the polynomials ∆( λ ) correponding to ( b, a ) and the one corresponding to( − β N , α N ). By (17) (and the discussion following it), in both cases,∆( λ ) = α − N λ N (cid:0) N βλ − + O ( λ − ) (cid:1) as | λ | → ∞ . This implies that ∆ b,a ( λ ) = ∆ − β N ,α N ( λ ) · (1 + O ( λ − )) , hence by (128) and (130) δ b,a ( λ ) = arccos (cid:18) ( − N ∆ b,a ( λ )2 (cid:19) = arccos (cid:18) ( − N − β N ,α N ( λ ) · (1 + O ( λ − )) (cid:19) . = arccos (cid:18) ( − N T N (cid:18) λ + β α (cid:19) · (1 + O ( λ − )) (cid:19) = arccos (cid:18) T N (cid:18) − λ + β α (cid:19) · (1 + O ( λ − )) (cid:19) . Substituting formula (131) for T N , one then concludes that δ b,a ( λ ) = arccos (cid:18) cos (cid:18) N arccos (cid:18) − λ + β α (cid:19)(cid:19) · (1 + O ( λ − )) (cid:19) = N arccos (cid:18) − λ + β α (cid:19) + O ( λ − ) , (132)0 B PROOF OF THEOREM ?? where in the last step we used that arccos z = − i log( z + i + √ − z ) for any z in C \ (( −∞ , − ∪ [1 , ∞ )). Lemma B.3.
Let u : Ω( b, a ) → R be a bounded harmonic function such that thenontangential limit of u ( z ) on ∂ Ω( b, a ) has compact support, and let U ( λ ) := u ( δ ( λ )) , where δ ( λ ) is the function defined by (124). Then for almost every t ∈ R , the limit U ( t ) := lim η → U ( t + iη ) exists and is integrable, and Z ∞−∞ U ( t ) dt = lim x →∞ πα sinh (cid:16) xN (cid:17) u (cid:18) N π ix (cid:19) . (133) Proof of Lemma B.3.
Again by Fatou’s theorem, for a.e. t the (nontangen-tial) limit lim η → U ( t + iη ) exists, since U is a bounded harmonic function on { Im ( z ) > } . Since u is bounded on Ω( b, a ) and its nontangential limit to ∂ Ωhas compact support, U ( t ) is bounded and compactly supported and thus inparticular integrable. For λ = ξ + iη , one then has the Poisson representation U ( ξ + iη ) = ηπ Z ∞−∞ U ( t )( t − ξ ) + η , (134)and by dominated convergence we conclude that Z ∞−∞ U ( t ) dt = lim η →∞ πη U ( ξ + iη ) . (135)(In particular, the limit in the latter expression exists.) In order to computethe right hand side of (135), let ξ + iη be given by ξ + iη = δ − (cid:18) N π ix (cid:19) , for x sufficiently large. Then U ( ξ + iη ) = u (cid:0) Nπ + ix (cid:1) , and by (132), it followsthat π iN x = arccos (cid:18) − ( ξ + β ) + iη α (cid:19) + O (cid:0) ( ξ + iη ) − (cid:1) ( x → ∞ ) . (136)Taking the cosine of both sides of (136), multiplying by − α and using thatcos( π + it ) = − i sinh t for t ∈ R , we obtain2 iα sinh xN = (( ξ + β ) + iη ) (cid:0) O (cid:0) ( ξ + iη ) − (cid:1)(cid:1) ( x → ∞ ) . Hence, as x → ∞ , ξ = O (1) , η = 2 α sinh xN + O (1) . (137)Substituting (137) into (135) leads to the claimed formula (133). EFERENCES Proof of Proposition B.1.
Let ( b, a ) ∈ M β,α . Besides the harmonic measure u ( n ) of the set E := (( n − π, nπ ) ⊂ ∂ Ω( b, a ) we also consider the harmonic measure u ( n ) β,α of E ⊂ ∂ Ω( − β N , α N ); note thatΩ( − β N , α N ) = { x + iy | < x < N π, y > } and hence Ω( b, a ) ⊆ Ω( − β N , α N ). According to [6], both u ( n ) and u ( n ) β,α satisfythe hypotheses of Lemma B.3. Let us recall the monotonicity property of theharmonic measures u ( z, E, Ω) with respect to Ω (see e.g. [6]): If Ω ⊆ Ω , E ⊂ ∂ Ω ∩ ∂ Ω , and u ( z, E, Ω i ) ( i = 1 ,
2) denotes the harmonic measure of E ⊂ ∂ Ω i , then for any z ∈ Ω , u ( z, E, Ω ) ≤ u ( z, E, Ω ). Apply this generalprinciple to Ω := Ω( b, a ) and Ω := Ω( − β N , α N ) to get u ( n ) ( x ) ≤ u ( n ) β,α ( x ) . (138)Writing 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 conclude that λ n ( b, a ) − λ n − ( b, a ) = Z ∞−∞ U ( n ) ( λ ) d λ = lim x →∞ πα sinh (cid:16) xN (cid:17) u ( n ) (cid:18) N π ix (cid:19) ≤ lim x →∞ πα sinh (cid:16) xN (cid:17) u ( n ) β,α (cid:18) N π ix (cid:19) = Z ∞−∞ U ( n ) β,α ( λ ) d λ = λ n ( − β N , α N ) − λ n − ( − β N , α N ) . This completes the proof of the estimate (120) and therefore of PropositionB.1.
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EFERENCES Institut f¨ur Mathematik, Universit¨at Z¨urich, Winterthurerstrasse190, CH-8057 Z¨urich, Switzerland
E-mail address: [email protected]
Institut f¨ur Mathematik, Universit¨at Z¨urich, Winterthurerstrasse190, CH-8057 Z¨urich, Switzerland