Elliptic solutions to the KP hierarchy and elliptic Calogero-Moser model
aa r X i v : . [ n li n . S I] F e b Elliptic solutions to the KP hierarchy and ellipticCalogero-Moser model
V. Prokofev ∗ and A. Zabrodin † Moscow Institute of Physics and Technology, Dolgoprudny, Institutskyper., 9, Moscow region, 141700, Russia Skolkovo Institute of Science and Technology, 143026 Moscow, RussianFederation National Research University Higher School of Economics, 20Myasnitskaya Ulitsa, Moscow 101000, Russian Federation Institute of Biochemical Physics, Kosygina str. 4, 119334, Moscow,Russia, Russian FederationFebruary 2021
Abstract
We consider solutions of the KP hierarchy which are elliptic functions of x = t .It is known that their poles as functions of t move as particles of the ellipticCalogero-Moser model. We extend this correspondence to the level of hierarchiesand find the Hamiltonian H k of the elliptic Calogero-Moser model which governs thedynamics of poles with respect to the k -th hierarchical time. The Hamiltonians H k are obtained as coefficients of the expansion of the spectral curve near the markedpoint in which the Baker-Akhiezer function has essential singularity. Contents t ∗ [email protected] † [email protected] The spectral curve 106 Dynamics in higher times 117 Calculation of the Hamiltonians 148 Rational and trigonometric limits 16Acknowledgments 19References 19
The investigation of dynamics of poles of singular solutions to nonlinear integrable equa-tions was initiated in the seminal paper [1], where it was shown that poles of elliptic andrational solutions to the Korteweg-de Vries and Boussinesq equations move as particlesof the integrable many-body Calogero-Moser system [2, 3, 4, 5] with some restrictionsin the phase space. As it was proved in [6, 7], this connection becomes most naturalfor the more general Kadomtsev-Petviashvili (KP) equation, in which case there are norestrictions in the phase space for the Calogero-Moser dynamics of poles.The KP equation is the first member of an infinite hierarchy of consistent integrableequations with infinitely many independent variables (times) t = { t , t , t , . . . } (the KPhierarchy). In [8], Shiota has shown that the correspondence between rational solutions tothe KP equation and the Calogero-Moser system with rational potential can be extendedto the level of hierarchies: the evolution of poles with respect to the higher times t k of theKP hierarchy was shown to be governed by the higher Hamiltonians H k = tr L k of theintegrable Calogero-Moser system, where L is the Lax matrix. Later this correspondencewas generalized to trigonometric solutions of the KP hierarchy (see [9, 10]).A natural generalization of rational and trigonometric solutions are elliptic (doubleperiodic in the complex plane) solutions. Elliptic solutions to the KP equation3 u t t = (cid:16) u t − uu x − u xxx (cid:17) x (1)(where x = t ) were studied by Krichever in [11], where it was shown that poles x i of theelliptic solutions u = − N X i =1 ℘ ( x − x i ) + 2 c (2)as functions of t move according to the equations of motion¨ x i = 4 X k = i ℘ ′ ( x i − x k ) (3)of the Calogero-Moser system of particles with the elliptic interaction potential ℘ ( x i − x j )( ℘ is the Weierstrass ℘ -function). Here dot means derivative with respect to the time t .2ee also the review [12]. The Calogero-Moser system is Hamiltonian with the Hamiltonian H = X i p i − X i 1. This means that theevolution in t is simply a shift of x : u k ( x, t ) = u k ( x + t , t , t , . . . ).An equivalent formulation of the KP hierarchy is through the zero curvature (Zakha-rov-Shabat) equations ∂ t n A m − ∂ t m A n + [ A m , A n ] = 0 . (9)The simplest nontrivial equation (1) is obtained for u = u at m = 2, n = 3.3 common solution to the KP hierarchy is provided by the tau-function τ = τ ( x, t ).The coefficient functions u k of the Lax operator can be expressed through the tau-function. For example, u ( x, t ) = u ( x, t ) = ∂ x log τ ( x, t ) . (10)The whole hierarchy is encoded in the bilinear relation [14, 15] I ∞ e ( x − x ′ ) z + ξ ( t ,z ) − ξ ( t ′ ,z ) ( e − D ( z ) τ ( x, t ))( e D ( z ) τ ( x ′ , t ′ )) dz = 0 (11)valid for all x, x ′ , t , t ′ , where ξ ( t , z ) = X k ≥ t k z k and D ( z ) is the differential operator D ( z ) = X k ≥ z − k k ∂ t k . (12)The integration contour is a big circle around infinity separating the singularities comingfrom the exponential factor from those coming from the tau-functions.Let us point out an important corollary of the bilinear relation. Applying the operator D ′ ( µ ) = − X k ≥ µ − k − ∂ t k to (11) and putting x = x ′ , t = t ′ after that, we obtain − X k ≥ I ∞ µ − k − z k ( e − D ( z ) τ ) ( e D ( z ) τ ) dz + I ∞ D ′ ( µ )( e − D ( z ) τ ) ( e D ( z ) τ ) dz = 0or 12 πi I ∞ zµ ( z − µ ) ( e − D ( z ) τ ) ( e D ( z ) τ ) dz = D ′ ( µ ) ∂ x τ τ − D ′ ( µ ) τ ∂ x τ. Taking the residues in the left hand side, we get the equation( e D ( µ ) τ ) ( e − D ( µ ) τ ) τ = 1 − D ′ ( µ ) ∂ x log τ. (13)The zero curvature equations (9) are compatibility conditions of the auxiliary linearproblems ∂ t k ψ = A k ψ (14)for the wave function ψ = ψ ( x, t , z ) depending on the spectral parameter z . In particular,at k = 2 we have the equation ∂ t ψ = ∂ x ψ + 2 uψ. (15)One can also introduce the adjoint wave function ψ ∗ satisfying the adjoint equation (14): − ∂ t k ψ ∗ = A † k ψ ∗ , (16)where the † -operation is defined as ( f ( x ) ◦ ∂ nx ) † = ( − ∂ x ) n ◦ f ( x ). In [14, 15] it is shownthat the wave functions can be expressed through the tau-function in the following way: ψ ( x, t , z ) = e xz + ξ ( t ,z ) e − D ( z ) τ ( x, t ) τ ( x, t ) , (17)4 ∗ ( x, t , z ) = e − xz − ξ ( t ,z ) e D ( z ) τ ( x, t ) τ ( x, t ) . (18)Note that in terms of the wave functions the equation (13) can be written in the form ∂ t m ∂ t log τ ( x, t ) = res ∞ (cid:16) z m ψ ( x, t , z ) ψ ∗ ( x, t , z ) (cid:17) , (19)where res ∞ is defined as res ∞ ( z − n ) = δ n . The ansatz for the tau-function of elliptic (double-periodic in the complex plane) solutionsto the KP hierarchy is τ = e Q ( x, t ) N Y i =1 σ ( x − x i ( t )) , (20)where Q ( x, t ) = c ( x + t ) + ( x + t ) A ( t , t , . . . ) + B ( t , t , . . . )with a constant c , a linear function A ( t , t , . . . ) = A + X j ≥ a j t j (21)and some function B ( t , t , . . . ). In (20) σ ( x ) = σ ( x | ω, ω ′ ) = x Y s =0 (cid:16) − xs (cid:17) e xs + x s , s = 2 ωm + 2 ω ′ m ′ with integer m, m ′ is the Weierstrass σ -function with quasi-periods 2 ω , 2 ω ′ such that Im( ω ′ /ω ) > 0. It isconnected with the Weierstrass ζ - and ℘ -functions by the formulas ζ ( x ) = σ ′ ( x ) /σ ( x ), ℘ ( x ) = − ζ ′ ( x ) = − ∂ x log σ ( x ). The monodromy properties of the function σ ( x ) are σ ( x + 2 ω ) = − e η ( x + ω ) σ ( x ) , σ ( x + 2 ω ′ ) = − e η ′ ( x + ω ′ ) σ ( x ) , (22)where the constants η = ζ ( ω ), η ′ = ζ ( ω ′ ) are related by ηω ′ − η ′ ω = πi/ 2. The roots x i are assumed to be all distinct. Correspondingly, the function u = ∂ x log τ is an ellipticfunction with double poles at the points x i : u = − N X i =1 ℘ ( x − x i ) + 2 c. (23)The poles depend on the times t , t , t , . . . . The dependence on t is especially simple:since the solution must depend on x + t , we have ∂ t x i = − µ ) be the difference operator∆( µ ) = e D ( µ ) + e − D ( µ ) − . (24)Substituting the ansatz (20) into equation (13), we get: e G +( x + t )∆( µ ) A Y i σ ( x − e D ( µ ) x i ) σ ( x − e − D ( µ ) x i ) σ ( x − x i )5 1 + 2 cµ − − D ′ ( µ ) A − X k D ′ ( µ ) x k ℘ ( x − x k ) , where G = 2 cµ − + µ − ( e D ( µ ) − e − D ( µ ) ) A + ∆( µ ) B. The right hand side is an elliptic function of x with periods 2 ω , 2 ω ′ . Therefore, for the lefthand side be also an elliptic function of x with the same periods the following relationshave to be satisfied: exp (cid:16) − η ∆( µ ) X k x k + 2 ω ∆( µ ) A (cid:17) = 1exp (cid:16) − η ′ ∆( µ ) X k x k + 2 ω ′ ∆( µ ) A (cid:17) = 1from which it follows that∆( µ ) A = 2 n ′ η − nη ′ , ∆( µ ) X k x k = 2 n ′ ω − nω ′ with integer n, n ′ . The right hand sides do not depend on µ . Expanding the equalities in powers of µ , onesees that the left hand sides are O ( µ − ) as µ → ∞ , therefore, n = n ′ = 0 and we have∆( µ ) A = 0 , ∆( µ ) X k x k = 0 . (25)The first equation is satisfied if A is a linear function of times as in (21). The secondequation means that (1 − e − D ( µ ) ) X i x i = − (1 − e D ( µ ) ) X i x i . (26)Note that the functions (17), (18) with τ as in (20) are double-Bloch functions , i.e.,they satisfy the monodromy properties ψ ( x + 2 ω ) = Bψ ( x ), ψ ( x + 2 ω ′ ) = B ′ ψ ( x ) withsome Bloch multipliers B , B ′ . Any non-trivial double-Bloch function (i.e. not an expo-nential function) must have poles in x in the fundamental domain. The Bloch multipliersof the function (17) are B = e ω ( z − α ( z )) − ζ ( ω )( e − D ( z ) − P i x i , B ′ = e ω ′ ( z − α ( z )) − ζ ( ω ′ )( e − D ( z ) − P i x i , (27)where α ( z ) = 2 cz − + X j ≥ a j j z − j (28)with the constants a j entering (21). Equation (26) means that the Bloch multipliers ofthe adjoint wave function ψ ∗ are B − and B ′ − .Let us introduce the elementary double-Bloch function Φ( x, λ ) defined asΦ( x, λ ) = σ ( x + λ ) σ ( λ ) σ ( x ) e − ζ ( λ ) x (29)( ζ ( λ ) is the Weierstrass ζ -function). The monodromy properties of the function Φ areΦ( x + 2 ω, λ ) = e ζ ( ω ) λ − ζ ( λ ) ω ) Φ( x, λ ) , x + 2 ω ′ , λ ) = e ζ ( ω ′ ) λ − ζ ( λ ) ω ′ ) Φ( x, λ ) , so it is indeed a double-Bloch function. The function Φ has a simple pole at x = 0 withresidue 1: Φ( x, λ ) = 1 x + α x + α x + . . . , x → , where α = − ℘ ( λ ), α = − ℘ ′ ( λ ). We will often suppress the second argument of Φwriting simply Φ( x ) = Φ( x, λ ). We will also need the x -derivative Φ ′ ( x, λ ) = ∂ x Φ( x, λ ).Equations (17), (18) and (20) imply that the wave functions ψ , ψ ∗ have simple polesat the points x i . One can expand the wave functions using the elementary double-Blochfunctions as follows: ψ = e xk + t ( k − z )+ ξ ( t ,z ) X i c i Φ( x − x i , λ ) (30) ψ ∗ = e − xk − t ( k − z ) − ξ ( t ,z ) X i c ∗ i Φ( x − x i , − λ ) (31)(this is similar to expansion of a rational function in a linear combination of simplefractions). Here c i , c ∗ i are expansion coefficients which do not depend on x and k is anadditional spectral parameter. Note that the normalization of the functions (17), (18)implies that c i and c ∗ i are O ( λ ) as λ → 0. One can see that (30) is a double-Blochfunction with Bloch multipliers B = e ω ( k − ζ ( λ ))+2 ζ ( ω ) λ , B ′ = e ω ′ ( k − ζ ( λ ))+2 ζ ( ω ′ ) λ (32)and (31) has Bloch multipliers B − and B ′ − . These Bloch multipliers should coincidewith (27).Therefore, comparing (27) with (32), we get2 ω ( k − ζ ( λ ) − z + α ( z )) + 2 ζ ( ω ) (cid:16) λ + ( e − D ( z ) − X i x i (cid:17) = 2 πin, ω ′ ( k − ζ ( λ ) − z + α ( z )) + 2 ζ ( ω ′ ) (cid:16) λ + ( e − D ( z ) − X i x i (cid:17) = 2 πin ′ with some integer n, n ′ . Regarding these equations as a linear system, we obtain thesolution k − z + α ( z ) − ζ ( λ ) = 2 n ′ ζ ( ω ) − nζ ( ω ′ ) ,λ + ( e − D ( z ) − X i x i = 2 nω ′ − n ′ ω. Shifting λ by a suitable vector of the lattice spanned by 2 ω , 2 ω ′ , one gets zeros in theright hand sides of these equalities, so we can write k = z − α ( z ) + ζ ( λ ) ,λ = (1 − e − D ( z ) ) X i x i . (33)These two equations for three variables k, z, λ determine the spectral curve. Below we willobtain another description of the spectral curve as the spectral curve of the Calogero-Moser system (given by the characteristic polynomial of the Lax matrix L ( λ ) for the7alogero-Moser system). It appears in the form R ( k, λ ) = 0, where R ( k, λ ) is a polyno-mial in k whose coefficients are elliptic functions of λ (see below in section 5). These co-efficients are integrals of motion in involution. The spectral curve in the form R ( k, λ ) = 0appears if one excludes z from the equations (33). Equivalently, one can represent thespectral curve as a relation connecting two variables z and λ : R ( z − α ( z ) + ζ ( λ ) , λ ) = 0 . (34)Let us write the second equation in (33) as the expansion in powers of z : λ = − X m ≥ z − m ˆ h m X, X := X i x i , (35)where ˆ h k are differential operators of the formˆ h m = − m ∂ t m + higher order operators in ∂ t , ∂ t , . . . , ∂ t m − . (36)For example, the first few areˆ h = − ∂ t , ˆ h = 12 ( ∂ t − ∂ t ) , ˆ h = 16 ( − ∂ t + 3 ∂ t ∂ t − ∂ t ) . As is explained above, the coefficients in the expansion (35) are integrals of motion, i.e., ∂ t j ˆ h m X = 0 for all j, m . It then follows from the equation ∂ t x i = − h m that ∂ t j ∂ t X = 0 and ∂ t j ∂ t X = 0. A simple inductiveargument then shows that ∂ t j ∂ t m X = 0 for all j, m . This means that − ˆ h m X = m ∂ t m X and X is a linear function of the times: X = X i x i = X − N t + X m ≥ V m t m (37)with some constants V m (velocities of the “center of masses” of the points x i multipliedby N ). Therefore, the second equation in (33) can be written as λ = D ( z ) X i x i = − N z − + X j ≥ z − j j V j . (38)In what follows we will show that H m = − m +1 V m +1 are Hamiltonians for the dynamicsof the poles in t m , with H being the standard Calogero-Moser Hamiltonian. t u in the linear problem (15) ∂ t ψ − ∂ x ψ − uψ = 0 (39)is an elliptic function of x of the form (23). Therefore, one can find solutions which aredouble-Bloch functions of the form (30). 8he next procedure is standard after the work [11]. We substitute u in the form(23) and ψ in the form (30) into the left hand side of (39) and cancel the poles at thepoints x = x i . The highest poles are of third order but it is easy to see that they cancelidentically. It is a matter of direct calculation to see that the conditions of cancellationof second and first order poles have the form c i ˙ x i = − kc i − X j = i c j Φ( x i − x j ) , (40)˙ c i = ( k − z + 4 c − α ) c i − X j = i c j Φ ′ ( x i − x j ) − c i X j = i ℘ ( x i − x j ) , (41)where dot means the t -derivative. Introducing N × N matrices L ij = − δ ij ˙ x i − (1 − δ ij )Φ( x i − x j ) , (42) M ij = δ ij ( k − z + ℘ ( λ ) + 4 c ) − δ ij X k = i ℘ ( x i − x k ) − − δ ij )Φ ′ ( x i − x j ) , (43)we can write the above conditions as a system of linear equations for the vector c =( c , . . . , c N ) T : L ( λ ) c = k c ˙ c = M ( λ ) c . (44)Differentiating the first equation in (44) with respect to t , we arrive at the compatibilitycondition of the linear problems (44): (cid:16) ˙ L + [ L, M ] (cid:17) c = 0 . (45)The Lax equation ˙ L + [ L, M ] = 0 is equivalent to the equations of motion of the ellipticCalogero-Moser system (see [12] for the detailed calculation). Our matrix M differsfrom the standard one by the term δ ij ( k − z ) but it does not affect the compatibilitycondition. It follows from the Lax representation that the time evolution is an isospectraltransformation of the Lax matrix L , so all traces tr L m and the characteristic polynomialdet( L − kI ), where I is the unity matrix, are integrals of motion. Note that the Laxmatrix is written in terms of the momenta p i as follows: L ij = − δ ij p i − (1 − δ ij )Φ( x i − x j ) . (46)A similar calculation shows that the adjoint linear problem for the function (31) leadsto the equations c ∗ T L ( λ ) = k c ∗ T ˙ c ∗ T = − c ∗ T M ( λ ) (47)with the compatibility condition c ∗ T (cid:16) ˙ L + [ L, M ] (cid:17) = 0.9 The spectral curve The first of the equations (44) determines a connection between the spectral parameters k, λ which is the equation of the spectral curve: R ( k, λ ) = det (cid:16) kI − L ( λ ) (cid:17) = 0 . (48)As it was already mentioned, the spectral curve is an integral of motion. The matrix L = L ( λ ), which has an essential singularity at λ = 0, can be represented in the form L = V ˜ LV − , where matrix elements of ˜ L do not have essential singularities and V is thediagonal matrix V ij = δ ij e − ζ ( λ ) x i . Therefore, R ( k, λ ) = N X m =0 R m ( λ ) k m , where the coefficients R m ( λ ) are elliptic functions of λ with poles at λ = 0. The functions R m ( λ ) can be represented as linear combinations of the ℘ -function and its derivatives.Coefficients of this expansion are integrals of motion. Fixing values of these integrals, weobtain via the equation R ( k, λ ) = 0 an algebraic curve Γ which is a N -sheet covering ofthe initial elliptic curve E realized as a factor of the complex plane with respect to thelattice generated by 2 ω , 2 ω ′ . Example ( N = 2):det × (cid:16) kI − L ( λ ) (cid:17) = k + k ( p + p ) + p p + ℘ ( x − x ) − ℘ ( λ ) . Example ( N = 3): det × (cid:16) kI − L ( λ ) (cid:17) = k + k ( p + p + p )+ k (cid:16) p p + p p + p p + ℘ ( x ) + ℘ ( x ) + ℘ ( x ) − ℘ ( λ ) (cid:17) + p p p + p ℘ ( x ) + p ℘ ( x ) + p ℘ ( x ) − ℘ ( λ )( p + p + p ) − ℘ ′ ( λ ) , where x ik = x i − x k .In a neighborhood of λ = 0 the matrix ˜ L can be written as˜ L = − λ − ( E − I ) + O (1) , where E is the rank 1 matrix with matrix elements E ij = 1 for all i, j = 1 , . . . , N . Thematrix E has eigenvalue 0 with multiplicity N − N .Therefore, we can write R ( k, λ ) in the form R ( k, λ ) = det (cid:16) kI + λ − ( E − I ) + O (1) (cid:17) = (cid:16) k + ( N − λ − − f N ( λ ) (cid:17) N − Y i =1 ( k − λ − − f i ( λ )) , (49)where f i are regular functions of λ at λ = 0: f i ( λ ) = O (1) as λ → 0. This means thatthe function k has simple poles on all sheets at the points P j ( j = 1 , . . . , N ) of the curve10 located above λ = 0. Its expansion in the local parameter λ on the sheets near thesepoints is given by the multipliers in the right hand side of (49): k = λ − + f j ( λ ) near P j , j = 1 , . . . , N − ,k = − ( N − λ − + f N ( λ ) near P N . (50)The N -th sheet is distinguished, as it can be seen from (50). As in [11], we call it theupper sheet. Note that equations (33), (38) imply k ( λ ) = − N − λ + O (1) as λ → , so the expansion (38) is the expansion of λ ( z ) on the upper sheet of the spectral curvein a neighborhood of the point P N . Our basic tool is equation (19). Substituting τ ( x, t ) in the form (20) and ψ , ψ ∗ in theform (30), (31) in it, we have: X i ∂ t m x i ℘ ( x − x i ) + C ( t , t , . . . ) = res ∞ z m X i,j c i c ∗ j Φ( x − x i , λ )Φ( x − x j , − λ ) . (51)Equating the coefficients in front of the second order poles at x = x i , we obtain ∂ t m x i = res ∞ (cid:16) z m c ∗ i c i (cid:17) = res ∞ (cid:16) z m c ∗ T E i c (cid:17) , (52)where E i is the diagonal matrix with 1 at the place ii and zeros otherwise. At m = 1this reads res ∞ (cid:16) z c ∗ T E i c (cid:17) = − ∞ (cid:16) z c ∗ T c (cid:17) = − N. Summing the equations (52) over i , we getres ∞ (cid:16) z m c ∗ T c (cid:17) = ∂ t m X i x i . It then follows from these equations that( c ∗ T c ) = − N/z + X m z − m − ∂ t m X i x i = − λ ′ ( z ) . (53)The absence of terms with non-negative powers of z in the right hand side (which wouldnot change the residue) follows from the above mentioned fact that c and c ∗ are O ( λ ) = O ( z − ) as z → ∞ . The last equality in (53) follows from (38). Equation (53) is animportant non-trivial relation which will allow us to identify the Hamiltonians for thehigher flows t m . 11ow let us note that according to (46) E i = − ∂ p i L . Therefore, we can continue thechain of equalities (52) as follows: ∂ t m x i = res ∞ (cid:16) z m c ∗ T E i c (cid:17) = − res ∞ (cid:16) z m c ∗ T ∂ p i L c (cid:17) = − ∂ p i res ∞ (cid:16) z m c ∗ T L c (cid:17) + res ∞ (cid:16) z m ∂ p i c ∗ T L c (cid:17) + res ∞ (cid:16) z m c ∗ T L∂ p i c (cid:17) = − ∂ p i res ∞ (cid:16) z m c ∗ T L c (cid:17) + res ∞ (cid:16) z m ∂ p i c ∗ T k c (cid:17) + res ∞ (cid:16) z m c ∗ T k∂ p i c (cid:17) = − ∂ p i res ∞ (cid:16) z m c ∗ T L c (cid:17) + ∂ p i res ∞ (cid:16) z m c ∗ T k c (cid:17) − res ∞ (cid:16) z m ∂ p i k c ∗ T c (cid:17) = − ∂ p i res ∞ (cid:16) z m c ∗ T ( L − kI ) c (cid:17) − res ∞ (cid:16) z m ∂ p i k c ∗ T c (cid:17) = res ∞ (cid:16) z m λ ′ ( z ) ∂ p i k (cid:17) . Here ∂ p i k = ∂ p i k ( λ, I ) (cid:12)(cid:12)(cid:12) λ =const , where I is the full set of integrals of motion. From (33) wesee that ∂ p i k = (1 − α ′ ( z )) ∂ p i z ( λ, I ) (cid:12)(cid:12)(cid:12) λ =const . (54)We consider z as an independent variable, so we can write0 = dzdp i = ∂ p i z (cid:12)(cid:12)(cid:12) λ =const + ∂ λ z (cid:12)(cid:12)(cid:12) I =const ∂ p i λ or ∂ p i z = − ∂ p i λλ ′ ( z ) . Therefore, we have the first set of the Hamiltonian equations ∂ t m x i = − res ∞ (cid:16) z m (1 − α ′ ( z )) ∂ p i λ (cid:17) = ∂ p i H m , (55)where the Hamiltonian H m = H ( α ) m + 2 cH ( α ) m − + m − X j =2 a j H ( α ) m − j − (56)is the linear combination of the Hamiltonians H ( α ) m = − res ∞ (cid:16) z m λ ( z ) (cid:17) (57)with constant coefficients. The latter implicitly depend on α ( z ) through the parametriza-tion of the spectral curve (34).In their turn, the Hamiltonians H ( α ) m are linear combinations of the basic Hamiltonians H m defined at α ( z ) = 0 by H m = − res ∞ (cid:16) z m λ ( z ) (cid:17) , (58)where λ ( z ) is defined through the equation of the spectral curve R ( z + ζ ( λ ) , λ ) = det (cid:16) ( z + ζ ( λ )) I − L ( λ ) (cid:17) = 0 . (59)12hen λ ( z ) = λ ( z − α ( z )) = λ ( z ) − α ( z ) λ ′ ( z ) + 12 α ( z ) λ ′′ ( z ) + . . . and so we see that the Hamiltonians (57) are indeed linear combinations of the H m ’swith constant coefficients.The remaining set of Hamiltonian equations can be obtained by differentiating (52)with respect to t and using (44), (47):2 ∂ t m p i = ∂ t m ˙ x i = res ∞ (cid:16) z m ˙c ∗ T E i c (cid:17) + res ∞ (cid:16) z m c ∗ T E i ˙c (cid:17) = res ∞ (cid:16) z m c ∗ T [ E i , M ] c (cid:17) Now, it is a matter of direct verification to see that[ E i , M ] = 2 ∂ x i L. (60)Therefore, we can write ∂ t m p i = res ∞ (cid:16) z m c ∗ T ∂ x i L c (cid:17) . Repeating the transformations presented above in detail, we have: ∂ t m p i = res ∞ (cid:16) z m c ∗ T c ∂ x i k (cid:17) = − res ∞ (cid:16) z m λ ′ ( z ) ∂ x i k (cid:17) The same argument as above shows that ∂ x i k = (1 − α ′ ( z )) ∂ x i z ( λ, I ) (cid:12)(cid:12)(cid:12) λ =const (61)and ∂ x i z = − ∂ x i λλ ′ ( z ) . Therefore, we obtain the second set of Hamiltonian equations for the dynamics of poles: ∂ t m p i = res ∞ (cid:16) z m (1 − α ′ ( z )) ∂ x i λ (cid:17) = − ∂ x i H m . (62)Let us find H m explicitly in terms of H m for the case when a j = 0, c = 0. In this case α ( z ) = 2 cz − and we have λ ( z ) = λ ( z ) − X j,n ≥ (2 c ) n n + j − n ! H j − z − j − n , (63) H m = H ( α ) m + 2 cH ( α ) m − and from (63) we see that H ( α ) m = H m + [ m/ X j =1 (2 c ) j m − jj ! H m − j . Therefore, H m = H m + [ m/ X j =1 (2 c ) j " m − jj ! + m − j +1 j − ! H m − j . (64)In particular, H = H + 6 cH which agrees with the result of the paper [12].13 Calculation of the Hamiltonians In order to find the Hamiltonians explicitly, we use the description of the spectral curvegiven in the paper [16]: N X j =0 I j T N − j ( k | λ ) = 0 , (65)where T j ( k | λ ) are polynomials in k of degree N such that ∂ k T n ( k | λ ) = nT n − ( k | λ ) (66)and I j are integrals of motion. The first few are I = 1 ,I = X i p i ,I = X ′ (cid:18) p i p j + 12! ℘ ( x ij ) (cid:19) ,I = X ′ (cid:18) p i p j p k + 12! p i ℘ ( x jk ) (cid:19) ,I = X ′ p i p j p k p l + 12! · p i p j ℘ ( x kl ) + 12 · (2!) ℘ ( x ij ) ℘ ( x kl ) ! ,I = X ′ p i p j p k p l p r + 12! · p i p j p r ℘ ( x kl ) + 12 · (2!) p r ℘ ( x ij ) ℘ ( x kl ) ! , (67)where P ′ means summation over distinct indices. Recalling the equation of the spectralcurve in terms of z and λ , let us also introduce S n ( z | λ ) = T n ( z + ζ ( λ ) | λ ), then ∂ z S n ( z | λ ) = nS n − ( z | λ ) . (68)For example, T ( k | λ ) = k − ℘ ( λ ) k − ℘ ′ ( λ ) k − ℘ ′′ ( λ ) − ℘ ( λ )) k − ℘ ′ ( λ ) ℘ ( λ ) . We have ζ ( λ ) = 1 λ − g λ · · O ( z − ) , where g = 60 X s =0 s , s = 2 mω + 2 m ′ ω ′ , m, m ′ ∈ Z . Expanding S ( z | λ ) in z using the above formula for T , we get: S ( z | λ ) = z + 5 z λ − g (cid:16) λ + 5 z + 5 z λ + 103 z λ + 16 z λ (cid:17) + O ( z − ) . z = 1, deg λ = − 1, then deg g = 4,deg S n = n . Note also that in the rational limit g = 0 and the equation of the spectralcurve becomes linear in λ − (see below in the next section). This can be only in thecase if S n ( z | λ ) = z n − nz n − λ − in the rational limit (the coefficient is found from thecondition (68)).In the non-degenerate case we have S n ( z | λ ) = z n − nz n − λ + g O ( z n − ) (69)or S n ( z | λ ) = z n − nz n − λ + g A n z n − + g B n I z n − + O ( z n − ) , (70)where A n and B n are some constant coefficients. ( I comes from the expansion λ = − N z − − I z − + O ( z − ).) Therefore, we can write S N ( z | λ ) = z N − N z N − λ + g ( A N z N − + B N I z N − ) + O ( z N − ) ,S N − ( z | λ ) = z N − − ( N − z N − λ + g A N − z N − + O ( z N − ) ,S N − j ( z | λ ) = z N − j − ( N − j ) z N − j − λ + O ( z N − ) , j = 2 , , , . (71)and the equation of the spectral curve acquires the form z N + X i =1 I i z N − i + g A N z N − + g ( A N − + B N ) I z N − + O ( z N − )= − λ (cid:16) N z N − + X i =1 ( N − i ) I i z N − i (cid:17) . Expressing λ as a function of z from here, we have: H = − I ,H = I − I ,H = − I + 3 I I − I ,H = I − I I + 2 I + 4 I I − I + const ,H = − I + 5 I I − I I + 5 I I − I I + 5 I I − I + g KI , (72)15here K = ( N + 1) A N − N ( A N − + B N ), or, explicitly, H = − X i p i ,H = X i p i − X i = j ℘ ( x ij ) ,H = − X i p i + 3 X i = j p i ℘ ( x ij ) ,H = X i p i − X i = j p i p j ℘ ( x ij ) − X i = j p i ℘ ( x ij )+ X i = j ℘ ( x ij ) + 2 X ′ ℘ ( x ij ) ℘ ( x jk ) + const ,H = − X i p i + 5 X i = j ( p i + p i p j ) ℘ ( x ij ) − X i = j p i ℘ ( x ij ) − X ′ p i ℘ ( x ij ) ℘ ( x ik ) − X ′ p i ℘ ( x ij ) ℘ ( x jk ) + const · X i p i . (73)These are indeed the Hamiltonians of the elliptic Calogero-Moser model. It is easy to seethat they satisfy the property H m − = − m X i ∂ p i H m . (74)Indeed, we have λ ( z ) = − N z − + X m ≥ z − m m V m = − N z − + X m ≥ z − m − H m so V m = ∂ t m X i x i = X i ∂ p i H m = − mH m − . One can see that the higher Hamiltonians will consist from the principal part andother terms as follows: H n = ( − n X | µ | = n C nµ I µ + g X | ν | = n − B nν I ν + . . . , (75)where the first sum is taken over Young diagrams µ of n = | µ | boxes, I µ = I µ I µ . . . I ℓ ( µ ) ,where ℓ ( µ ) is the number of non-empty rows of the diagram µ and C nµ is the matrix ofthe transition from the basis of elementary symmetric polynomials to the basis of powersums. In the rational limit ω , ω → ∞ , σ ( λ ) = λ , Φ( x, λ ) = ( x − + λ − ) e − x/λ and the equationof the spectral curve becomesdet (cid:16) L rat − ( E − I ) λ − − ( z + λ − ) I (cid:17) = 0 , (76)16here ( L rat ) ij = − δ ij p i − − δ ij x i − x j (77)is the Lax matrix of the rational Calogero-Moser model. Rewriting the equation of thespectral curve in the form det I − E λ − L rat − zI ! = 0and using the property det( I + Y ) = 1 + tr Y for any matrix Y of rank 1, we get λ = − tr (cid:16) E zI − L rat (cid:17) = − X n ≥ z − n − tr L n rat , (78)where we use the well known property tr( EL n rat ) = tr( L n rat ). So the Hamiltonians are H m = tr L m rat which agrees with Shiota’s result [8].The trigonometric limit is more tricky. Let πi/γ be period of the trigonometric (orhyperbolic) functions (the second period tends to infinity). The Weierstrass functions inthis limit become σ ( x ) = γ − e − γ x sinh( γx ) , ζ ( x ) = γ coth( γx ) − γ x. The tau-function for trigonometric solutions is τ ( x, t ) = N Y i =1 (cid:16) e γx − e γx i ( t ) (cid:17) , (79)so we should consider τ ( x, t ) = N Y i =1 σ ( x − x i ) e γ ( x − x i ) + γ ( x + x i ) . (80)Wit this choice, equation (33) acquires the form k = z + ζ ( λ ) + γ λ or k = z + γ coth( γλ ) . (81)The trigonometric limit of the function Φ( x, λ ) isΦ( x, λ ) = γ (cid:16) coth( γx ) + coth( γλ ) (cid:17) e − γx coth( γλ ) . Therefore, the equation of the spectral curve can be written in the formdet (cid:16) W / LW − / + γ (1 − coth( γλ ))( E − I ) − ( z + γ coth( γλ )) I (cid:17) = 0 , (82)where W = diag( w , w , . . . , w N ) and L ij = − δ ij p i − (1 − δ ij ) γ sinh( γ ( x i − x j )) = − δ ij p i − γ (1 − δ ij ) w / i w / j w i − w j (83)is the Lax matrix of the trigonometric Calogero-Moser model. Here and below we usethe notation w i = e γx i . 17fter the transformations similar to the rational case equation (82) can be broughtto the form γ (1 − coth( γλ ))tr " W − / EW / zI − ( L − γI ) = 1or λ = 12 γ log " − γ tr W − / EW / zI − ( L − γI ) ! . (84)Applying the formula det( I + Y ) = 1 + tr Y for any matrix Y of rank 1 in the oppositedirection, we have λ = 12 γ log det " I − γW − / EW / zI − ( L − γI ) . (85)Now we are going to use the identity[ L, W ] = 2 γ ( W / EW / − W ) (86)which can be easily checked. With the help of this identity, we can transform (85) asfollows:2 γλ = log det I − W − LW zI − ( L − γI ) + LzI − ( L − γI ) − γzI − ( L − γI ) ! = log det " I − γzI − ( L − γI ) ! zI − ( L − γI ) zI − ( L + γI ) × I − W − LW zI − ( L − γI ) + LzI − ( L − γI ) − γzI − ( L − γI ) ! = log det " I − γzI − ( L − γI ) ! I − W − LW zI − ( L + γI ) + LzI − ( L + γI ) ! = log det " I − γzI − ( L − γI ) ! zI − ( L + γI ) (cid:16) zI − ( L + γI ) − W − LW + L (cid:17) = log det zI − ( L + γI ) zI − ( L − γI ) . Therefore, we get λ = 12 γ tr (cid:16) log( I − z − ( L + γI )) − log( I − z − ( L − γI )) (cid:17) = − γ tr X m ≥ z − m m (cid:16) ( L + γI ) m − ( L − γI ) m (cid:17) (87)and H m = 12 γ ( m + 1) tr (cid:16) ( L + γI ) m +1 − ( L − γI ) m − (cid:17) (88)which agrees with the result of paper [10]. 18 cknowledgments We thank I. Krichever for illuminating discussions. 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