Noncommutative Painlevé equations and systems of Calogero type
aa r X i v : . [ m a t h - ph ] D ec Noncommutative Painlev´e equations and systems of Calogero type
M. Bertola †‡♣ , M. Cafasso ♦ , V. Rubtsov ♦ † Department of Mathematics and Statistics, Concordia University1455 de Maisonneuve W., Montr´eal, Qu´ebec, Canada H3G 1M8 ‡ SISSA/ISAS, via Bonomea 265, Trieste, Italy ♣ Centre de recherches math´ematiques, Universit´e de Montr´ealC. P. 6128, succ. centre ville, Montr´eal, Qu´ebec, Canada H3C 3J7 ♦ LAREMA, Universit´e d’Angers2 Boulevard Lavoisier, 49045 Angers, France.
Abstract
All Painlev´e equations can be written as a time–dependent Hamiltonian system, and as such they admit a naturalgeneralization to the case of several particles with an interaction of Calogero type (rational, trigonometric or elliptic).Recently, these systems of interacting particles have been proved to be relevant in the study of β –models. An almost twodecade old open question by Takasaki asks whether these multi-particle systems can be understood as isomonodromicequations, thus extending the Painlev´e correspondence. In this paper we answer in the affirmative by displaying explicitlysuitable isomonodromic Lax pair formulations. As an application of the isomonodromic representation we provide aconstruction based on discrete Schlesinger transforms, to produce solutions for these systems for special values of thecoupling constants, starting from uncoupled ones; the method is illustrated for the case of the second Painlev´e equation. Contents β –models and open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 D6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.5 Painlev´e III D7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.6 Painlev´e III D8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.7 Painlev´e II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.8 Painlev´e I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Marco.Bertola@ { concordia.ca, sissa.it } [email protected] [email protected] A case study: the second Calogero–Painlev´e system 16
The celebrated Painlev´e–Calogero correspondence [18, 29] consists in the remarkable observation that all Painlev´e equationscan be written in the particular form ¨ q = − V ( q ; t ) for some function V depending explicitly on both the dependent and independent variables. In other words, they all can bethought of as systems with a physical Hamiltonian H ( p, q ; t ) := p V ( q ; t ) (and standard symplectic form), describing the motion of a particle in a time–dependent potential.For the sixth Painlev´e equation (PVI), this result is due to Manin [18] (an expression of the sixth Painlev´e equation interms of elliptic functions is already present in the original work of Painlev´e, [23]) and the Hamiltonian is written as H = p − X i =0 g i ℘ ( q + ω i ) , (1.1)where ℘ is the Weierstrass function and the four parameters { g i , i = 0 , . . . , } are in correspondence with the parametersin the PVI equation (see the subsection 3.1 for their explicit expression). The four coefficients ω i reads ( ω , ω , ω , ω ) = (0 , / , − (1 + τ ) / , τ / , where τ is the modular parameter and plays the role of the independent time variable t .It was Levin and Olshanetsky [17] that pointed out that the Hamiltonian (1.1) corresponds to the rank–one case ofInozemtsev’s extensions [8] of the Calogero-Moser systems, when one considers τ as a parameter and not as the independentvariable. Takasaki, in [29], found the Calogero form of each of the Painlev´e equations by observing that they all can bededuced from the Okamoto’s ones [21] by some explicit canonical transformation. Moreover, he extended the Calogero–Painlev´e correspondence to what he called “multi–component” Painlev´e equations. More precisely, he proved that thereexist some canonical transformations between: • The Inozemstev’s multi–component systems and their degenerations (from PVI to PI). • A multi–component generalization of the Okamoto polynomials.For the reader’s convenience, we report here the list of the Hamiltonians of what we call “Calogero-Painlev´e” systems. ˜ H V I : n X j =1 p j X ℓ =0 g ℓ ℘ ( q j + ω ℓ ) ! + g X j = k ℘ ( q j − q k ) + ℘ ( q j + q k ) ! . ˜ H V : n X j =1 p j − α sinh ( q j / − β cosh ( q j /
2) + γt q j ) + δt q j ) ! ++ g X j = k (( q j − q k ) /
2) + 1sinh (( q j + q k ) / ! . H IV : n X j =1 p j − (cid:16) q j (cid:17) − t (cid:16) q j (cid:17) − t − α ) (cid:16) q j (cid:17) + β (cid:16) q j (cid:17) − ! + g X j = k q j − q k ) + 1( q j + q k ) ! . ˜ H III : n X j =1 p j − α q j + βt − q j − γ q j + δt − q j ! + g X j = k (cid:0) ( q j − q k ) / (cid:1) . ˜ H II : n X j =1 p j − (cid:16) q j + t (cid:17) − αq j ! + g X j = k q j − q k ) . ˜ H I : n X j =1 p j − q j − tq j ! + g X j = k q j − q k ) . (1.2)In the concluding remarks of his paper, Takasaki stated that “ a central issue will be to find an isomonodromic descriptionof the multi–component Painlev´e equations. If such an isomonodromic description does exist, it should be related to a newgeometric structure ”. The main result of this paper is exactly the description of this isomonodromic formulation: Theorem 1.1.
All the Hamiltonian systems in the Takasaki list (1.2) have an isomonodromic formulation in terms of a n × n Lax pair, where n is the number of particles. In each case the dynamical variables appear as the eigenvalues of an n × n matrix, which we denote hereafter q , evolvingin accordance to a matrix (i.e. non-commutative ) version of the corresponding Painlev´e equation. Our result is constructiveand the Lax pairs are explicitly written. They are obtained by Hamiltonian reduction `a la Kazhdan-Konstant-Sternberg[15] on the Lax systems for the matrix Painlev´e equations recently written by Kawakami [15] except for Painlev´e II, wherethe Lax pair is closer to the standard Flaschka–Newell Lax pair [6]. These are particular examples of the so–called simplylaced isomonodromy systems introduced by P. Boalch in [ ? ]. More precisely, they correspond to hyperbolic Dinkyn diagramsobtained by adding one leg to the affine diagrams associated to the corresponding “scalar” Painlev´e equations.An implicit result of the isomonodromic representation is the remarkable property (which is subsumed by the namingconvention in the literature but was never proved), that all the equations satisfy the Painlev´e property, namely, the solutions q ( t ) have only movable poles when considered as functions of the complex time t (note, however, that the eigenvalues of q ( t ) in general are not meromorphic functions, only their symmetric polynomials are).A second important observation is that the space of initial data of each equation is identifiable with a suitable manifoldof (generalized) monodromy data, which is an algebraic variety that can be easily written in explicit form. We do so hereonly in the case of the second Painlev´e system (Section 4, and particularly Theorem 4.2) but the construction is clearlygeneral, with the due modifications. This manifold plays, in this setting, a role similar to the one played by the completedCalogero–Moser space C n (the adelic Grassmannian of [33]) in the “classical” setting (see Remark 4.5).The structure of the paper is as follows: in the Section 2 we explain the general structure of the construction, while inSection 3 we provide details for each of the equations. The Section 4 focuses on the case of the second Painlev´e equation.Here we start studying its Stokes phenomena in a more general setting (instead of a matrix-valued Painlev´e II equation westudy the equation with values in an arbitrary non–commutative algebra, as in [26]) and we relate it to the quantization ofthe monodromy manifold of the Flashka–Newell Lax pair for the second Painlev´e equation [19]. Finally, in the subsection4.2, we show how to use discrete Schlesinger transformations to construct solutions of the second Calogero–Painlev´e system,out of n “decoupled” solution of the second Painlev´e equation.3 .1 Quantization, β –models and open questions Zabrodin and Zotov, in [34, 35], provided a quantized version of the Calogero–Painlev´e correspondence. Namely, theyproved that for each of the Painlev´e equation it exists a Lax pair such that the first component ψ ( z ; t ) of its eigenfunctionsatisfies the equation ∂ t ψ ( z ; t ) = (cid:16) ˜ H ( z, ∂ z ) − ˜ H ( q, p ) (cid:17) ψ ( z ; t ) , (1.3)where ˜ H here indicates any of the Hamiltonian in the Takasaki’s list, in the case n = 1 . It would be interesting to extendthis quantum Calogero–Painlev´e correspondence to the higher rank cases n > .The quantum Painlev´e equations have interesting applications in the theory of β –models, which are statistical modelsgeneralising (unitary-invariant) random matrices (which correspond to β = 2 ). For instance, as the fluctuations of thelargest eigenvalue of a random matrix (under suitable assumptions) is governed by the Hasting Mc–Leod solution of thesecond Painlev´e equation [30], in the same way the position of the largest particle in a β –ensemble is governed by the β –dependent quantum Painlev´e II equation β ∂∂s + ∂ ∂ξ + ( s − ξ ) ∂∂ξ ! F ( ξ ; s ) = 0 , (1.4)as discovered in [4]. The equation (1.4), indeed, for β = 2 , is of the same type of (1.3), except for the fact that a differentHamiltonian H := p − ( q + t ) p had been used, and that F ( ξ ; s ) = e K ψ ( ξ ; s ) , where K is the (indefinite) integral of the Hamiltonian. In [27], Rumanovprovided an important step in extending the quantum Calogero-Painlev´e correspondence to general values of β . Namely,consider an arbitrary Lax pair (cid:18) ∂ ξ − (cid:20) L ( ξ ; s ) L + ( ξ ; s ) L − ( ξ ; s ) L ( ξ ; s ) (cid:21)(cid:19) (cid:20) F ( ξ ; s ) G ( ξ ; s ) (cid:21) = 0 , (1.5) (cid:18) ∂ s − (cid:20) B ( ξ ; s ) B + ( ξ ; s ) B − ( ξ ; s ) B ( ξ ; s ) (cid:21)(cid:19) (cid:20) F ( ξ ; s ) G ( ξ ; s ) (cid:21) = 0 . Following [27] we impose that β ∈ N is an even integer and that the ratio of B + and L + has the following form B + L + := b + ( ξ ; s ) = − β β X k =1 ξ − Q k ( s ) . (1.6)The result of Rumanov can be summarized as follows: Theorem 1.2 ([27]) . If the poles { Q k ( s ) , k = 1 , . . . , β/ } of b + satisfy the equation β Q k = − Q k ( s − Q k ) + (cid:18) β − (cid:19) − β/ X j = k Q k − Q j ) (1.7) then there exists a Lax pair as in (1.5) such that F is a solution of (1.4). Up to a shift on the parameters of the equation.
4s an interesting parallel, one might say that Rumanov’s result is a sort of analogue of the famous theorem by Krichever[16] stating that the poles of a rational function of the KP equation evolve according to the “classical” (i.e. withoutpotential) Calogero system.Up to a rescaling of the variables, the equation (1.7) is nothing but the Hamiltonian dynamics induced by ˜ H II inTakasaki’s list (1.2) Hence, from the point of view of applications to β –models, the results presented in this paper (andin particular in Section 4) provide an essential step to make Rumanov’s Lax pair effective. Indeed, the Riemann–Hilbertrepresentation of the solutions of (1.7) is needed in order to apply the Deift-Zhou non–linear steepest descent method to(1.5) and try to prove, for instance, the conjectured tail asymptotics of the β –Tracy–Widom distribution (see [5]).From a more theoretical point of view, the study of the monodromy manifold of the second Painlev´e system gives thefirst concrete realization of the “quantization” of the Stokes manifolds, in a sense demystifying the concept since it becomessimply a (structured) manifold of generalized monodromy manifold in the usual sense. The real leap to the quantum settingis provided by passing to operator-valued Stokes’ parameters (see Remark 4.4), which should be pursued on a more analyticalfooting. On the other hand the quantization (this time in the sense of Reshetikhin, [25]) of simply laced isomonodromicsystems had been recently described, in full generality, in [24]. We plan to study in subsequent works the precise form ofthe monodromy manifold for the other Calogero–Painlev´e systems, as well as their deformation quantization. As explained in the introduction, our aim is to give an isomonodromic formulation of the Calogero–Painlev´e Hamiltoniansystems found by Takasaki [29]. This result is achieved by performing a reduction `a la
Kazdan-Konstant-Sternberg on somematrix-valued Lax pairs for the corresponding Painlev´e equations. For most of the cases, the Lax pairs we used are the onesfound by Kawakami [13, 14], with the exception of the second Painlev´e equation where we used the one found in [2],giving an isomonodromic formulation of the “fully non–commutative” second Painlev´e equation introduced in [26].For all the cases the starting point is a Lax system of type ∂∂z Φ( z ; t ) = A ( z ; q , q − , p , t )Φ( z ; t ) ,∂∂t Φ( z ; t ) = B ( z ; q , q − , p , t )Φ( z ; t ) , (2.1)where the pair of matrices A, B are × block matrices with blocks of arbitrary size n . A and B depend rationally on the spectral parameter z ∈ CP and their entries are polynomials in the n × n matrices { q , q − , p } . The dependence on t isboth implicit and explicit. If a joint solution of (2.1) exists, then necessarily the matrices A, B satisfy the zero curvatureequations ≡ [ ∂ t − B, ∂ z − A ] ⇐⇒ ∂ t A − ∂ z B + [ A, B ] ≡ . (2.2)These conditions are satisfied if A and B have the special form listed in Section 3; for the time being we point out thecommon features to all the cases here below.1. The isomonodromic equations for the matrices p and q can be expressed in a Hamiltonian form with Hamiltonianfunction of the form Tr H ( p , q , q − ) , where H is a non-commutative polynomial depending on the indicated variables.For all the cases the Hamiltonian H is a polynomial depending just on p and q except for the case PIII D8 , where it islinear in q − . These provide non–commutative versions of the Okamoto polynomial Hamiltonians. The matrices p , q are canonically conjugate with respect to the symplectic structure ω ( p , q ) := Tr (d p ∧ d q ) = d θ, θ := Tr p d q = X j,k p k,j d q j,k . (2.3) Strictly speaking, for the ramified cases (PIII D6 , PIII D7 , PIII D8 , PI), the Lax pairs of Kawakami are written for blocks of size × , but it iseasy to realise that Kawakami’s formulas hold more generally for matrices of arbitrary size, and even for general non-commutative algebras.
5. The resulting equations for p , q have the form ˙ q = A ( q , p , t ) , ˙ p = B ( q , p , t ) (2.4)with A , B polynomials in q (and q − for PIII D8 ) and A is of degree at most in p and B is of degree at most in p . (2.5)3. The “angular momenta” M ( p , q ) := [ p , q ] = pq − qp (2.6)are conserved quantities for the equations (2.4) and the level-sets of M are the coadjoint orbits of the group GL n ( C ) .This conservation law is a consequence of the invariance of the Hamiltonian under simultaneous conjugations q C q C − , p C p C − , and the fact that the adjoint action of GL n ( C ) is symplectic.4. Reinforcing the previous point, in all cases the commutator [ p , q ] has an additional interpretation in terms of the (gen-eralized) monodromy data associated to the z –ODE in (2.1), which are conserved by the isomonodromic deformation(i.e. the t –equation in (2.1)). Remark 2.1.
In all the cases illustrated below, the Lax pairs can be considered in the more general setting in which p , q and t are elements of a general non–commutative unital algebra, equipped with a derivation ∂ t such that ∂ t t = (see [26]),and with t in the center of the algebra. In particular, the features 2,3 and 4 are still valid, even if 2 has to be verified bydirect computations and it does not have an interpretation in terms of symplectic actions. This more general setting willbe very useful in the Section 4, where the Stokes manifold associated to the (non–commutative) second Painlev´e equationis studied in relation with the quantization proposed in [19, 20], see Remark 4.4. The independence of [ p , q ] on t allows us to write reduced equations for the eigenvalues of q on special coadjoint orbits M ( p , q ) ; this reduction follows an idea that first appeared in [15]. Lemma 2.2. [15] Let p , q be matrices satisfying [ p , q ] = ig ( − v T v ) , with v := (1 , . . . , (2.7) and assume that q is diagonalizable. Then there exists an invertible matrix C such that: • C diagonalises q : q = C X C − , X = diag( x , . . . , x n ) . • The matrices X and Y := C − p C satisfy the same commutation relation as q and p : [ Y, X ] = ig ( − v T v ) . (2.8) In particular, for every j = k , j, k ∈ { , . . . , n } , Y j,k = igx j − x k . (2.9) Remark 2.3.
The diagonal entries of Y are not determined by the equation (2.7), in the sequel they will be denoted y , . . . , y n . Proof.
Let ˆ C be any diagonalising matrix for q . From (2.7) we deduce (conjugating both sides) that h ˆ C − p ˆ C, X i = ig (cid:16) − ˆ C − v T v ˆ C (cid:17) = ig (cid:0) − a T b (cid:1) , (2.10)where b = v ˆ C and a = v ˆ C − T . In the (matrix) equation above, the diagonal elements on the left hand side are zero, andthen we deduce that a j = b − j . Let A := diag( a , . . . , a n ) . Then (cid:26) a = vA = v ˆ C − T b = vA − = v ˆ C ⇐⇒ (cid:26) v = v ( ˆ CA ) − T v = v ˆ CA. (2.11)Since ˆ C is defined up to right multiplication by any invertible diagonal matrix A , we have that C := ˆ CA also diagonalises q and satisfies the conditions of the Lemma. (cid:4) The reduction to the eigenvalues now proceeds as follows; let C = C ( t ) , det C ≡ , be the diagonalizing matrix ofLemma 2.2 and introduce the new wave function Ψ( z ; t ) := (cid:20) C C (cid:21) Φ( z ; t ) . The following proposition is an immediate consequence of the standard formulas for gauge transformations together withthe definition of the matrices X and Y . Proposition 2.4.
The wave function Ψ( z ; t ) is an eigenfunction of the Lax system ∂∂z Ψ( z ; t ) = A ( z ; X, X − , Y, t )Ψ( z ; t ) ∂∂t Ψ( z ; t ) = (cid:16) B ( z ; X, X − , Y, t ) − F ( X, X − , Y, t ) (cid:17) Ψ( z ; t ) (2.12) where F := ( C − ˙ C ) ⊗ . Consequently, the isomonodromic equations (2.4) become ˙ X = A ( X, Y, t ) + [
X, F ] , (2.13) ˙ Y = B ( X, Y, t ) + [
Y, F ] . (2.14)It is also possible to express the entries of F just in terms of the eigenvalues { x , . . . , x n } of X , as explained in thefollowing lemma. Lemma 2.5.
Let C be the conjugating matrix satisfying the conditions of Lemma 2.2 and F = C − ˙ C . Then the entries of F are given by ( x i − x j ) F i,j = (cid:16) [ A ( X, Y, t ) , X ] (cid:17) i,j , i = j, (2.15) F j,j = − X k : k = j F j,k + K, (2.16) K := 1 n X ℓ,m : ℓ = m F ℓ,m (2.17) and they are rational functions of ( x , . . . , x n ) only. roof : We start from (2.13). Taking the commutator [ ˙
X, X ] , we get the equation h X, [ X, F ] i = [ A ( X, Y ) , X ] (2.18)from which we deduce, since X is diagonal, the formula (2.15).Since A is a polynomial of first degree in Y (see (2.5)), and X is diagonal, the commutator in (2.15) does not contain thediagonal entries of Y . On the other hand, the off–diagonal entries of Y are expressed by (2.9) and therefore F j,k , j = k arerational functions of ( x , . . . , x n ) only.In order to find the diagonal terms of F , we take the derivative of the commutator dd t [ X, Y ] = [ ˙
X, Y ] + [ X, ˙ Y ] = 0 . Using (2.13) and (2.14) we then obtain A ( X, Y ) , X ] + [ Y, B ( X, Y )] + (cid:16)(cid:2) [ X, F ] , Y (cid:3) + (cid:2) X, [ Y, F ] (cid:3)(cid:17) . Now, we know that also [ q , p ] is constant and therefore [ ˙ q , p ] + [ q , ˙ p ] = 0 ; using (2.4) we then obtain [ A ( q , p ) , p ] + [ p , B ( q , p )] = 0 . (2.19)By conjugating the equation (2.19) with C we obtain that also [ A ( X, Y ) , X ] + [ Y, B ( X, Y )] = 0 . Hence we conclude that (cid:2) [ X, F ] , Y (cid:3) + (cid:2) X, [ Y, F ] (cid:3) = − (cid:2) [ Y, X ] , F (cid:3) = [ ig ( v T v ) , F ] . (2.20)The off–diagonal entries of the equation (2.20) give the linear system of equations F i,i + X j = i F i,j − F k,k − X j = k F j,k = 0 , i, k = 1 , . . . , n ; i = k, which has (2.16), (2.17) for solution. (cid:4) The details of the expressions of F in terms of X depend on the specific case considered, as will be seen in Section3. At the general level, however, it already follows that the eigenvalues of X evolve according to a “Calogero-Moser” typesystem, in the sense specified by the following proposition: Proposition 2.6.
The equations (2.13), (2.14) are Hamiltonian with respect to the symplectic structure P ni =1 dy i ∧ dx i .Moreover, they yield a closed differential system of the second order for the eigenvalues of X , with poles along the diagonals x j = x k , j = k . Proof :
The fact that the equations (2.13), (2.14) are Hamiltonian comes from the fact that the equations (2.4) for q and p are Hamiltonian and the action q C q C − , p C p C − is symplectic.From (2.13) and the fact that A is of first degree in Y , we realise that the diagonal entry y j := Y j,j is an expressioninvolving only x j , ˙ x j : y j = W ( x j , ˙ x j ) . (2.21)At this point Y given by (2.9) is completely determined by X, ˙ X and then the equation (2.14) yields a closed differentialsystem of second order for the eigenvalues of X . The poles along the diagonals appear as a consequence of the formula(2.9). (cid:4) Finally, when necessary (in all the cases but Painlev´e I and II), following [29], in the next sections we provide the explicitcanonical change of variable { x j , y i } → { q j , p j } necessary to transform the equations (2.13),(2.14) in the dynamicalequations for the Hamiltonian functions in the Takaksaki’s list.8 The isomonodromic formulation of the Calogero-Painlev´e systems
Following the scheme introduced in the previous Section, we explicitly display, for each of the Calogero–Painlev´e systems,an isomonodromic formulation. More specifically, for each case, we report: • The explicit expression of the Lax system (2.1). • The matrix Hamiltonian related to the equations (2.4). • The explicit expression of F . • The Hamiltonian related to the equations (2.13), (2.14). • When necessary, the change of coordinates necessary to go from the Hamiltonian of the previous point to the one ofthe Calogero-Painlev´e system (see [29]).
We start from the Fuchsian system of spectral type nn, nn, nn, n n − as written by Kawakami (up to a renaming of theconstants) [13] ∂ Φ ∂z = A z + A z − A t z − t ! Φ ,∂ Φ ∂t = − A t z − t + B ! Φ , (3.1)where the matrices are explicitly given by A := − − θ t q t −
10 0 , A := − qp + 12 ( k + θ ) 1( θ − qp ) qp + 14 ( k − θ ) qp + 12 ( k − θ ) ,A t := qp − θ − q tt ( − θ + pq ) p − pq , B := t (cid:16) [ q , p ] + − θ (cid:17) + θ q − [ qp , q ] + t ( t −
1) 0 − θ p + pqp . Here and below, the entries of the matrices are ( n × n ) blocks. When we write scalars (i.e. , k, t, . . . ) we mean that thisscalar is to be multiplied by the identity matrix. θ , θ , θ t and k are free parameters while θ = θ + θ t + θ . The resultingHamiltonian equations for p and q are given by ˙ q = A ( q , p )˙ p = B ( q , p ) , (3.2)with t ( t − A ( q , p ) := − θ t + ( θ + θ t ) q + ( θ + θ ) t q − θ q − qpq + t [ p , q ] + − [ t p , q ] + + [ qpq , q ] + (3.3) t ( t − B ( q , p ) := 14 ( k − θ ) − ( θ + θ t ) p − ( θ + θ ) t p + θ [ q , p ] + − t p ++ t [ q , p ] + + p (2 q − q ) p − [ q , pqp ] + . t ( t − H =Tr (cid:16) qpqpq − t pq p + t pqp − pqpq − θ qpq + t ( θ + θ ) pq + ( θ + θ t ) pq − θ t p − ( k − θ ) q (cid:17) . (3.4)The matrix F reads F := diag( f , . . . , f n ) + igt ( t − ( t − x i ) x i ( x i −
1) + ( t − x j ) x j ( x j − x i − x j ) + x i + x j − ! ni = j =1 , (3.5)leading to the multi–particle Hamiltonian t ( t − H V I := n X i =1 h x i ( x i − x i − t ) y i − (cid:16) θ t x i ( x i −
1) + θ x i ( x i − t ) + θ ( x i − x i − t ) (cid:17) y i + 14 (cid:16) θ − k (cid:17) x i ++ g X j 1) + θ x i ( x i − t ) + θ ( x i − x i − t ) (cid:17) y i + 14 (cid:16) θ − k + 60 g ( n − (cid:17) x i ++ (4 g ) X j 7→ − θ ), we construct the n × n bare solution in block form as Φ ( z ; A , B , C ; θ ) = diag h φ ( j )11 ( z ; a ( j ) , b ( j ) , c ( j ) ; ( − ) r θ ) i nj =1 diag h φ ( j )12 ( z ; a ( j ) , b ( j ) , c ( j ) ; ( − ) r θ ) i nj =1 diag h φ ( j )21 ( z ; a ( j ) , b ( j ) , c ( j ) ; ( − ) r θ ) i nj =1 diag h φ ( j )22 ( z ; a ( j ) , b ( j ) , c ( j ) ; ( − ) r θ ) i nj =1 (4.35)so that each block is a diagonal matrix. Because of (4.14), the matrix Φ has the following asymptotic behaviour at z = ∞ (cf. also [7]) Φ ∼ (cid:0) + O ( z − ) (cid:1) z e i (cid:16) z + tz (cid:17)b σ , (4.36)where we emphasized that the exponents of formal monodromy at ∞ are all zero. The main result of this subsection is thefollowing: Proposition 4.6. For any couple ( r = ig, n ) as in (4.32) or (4.33) there exists a polynomial matrix R ( z ) such that Φ( z ) := R ( z )Φ ( z ) is a solution of the Lax system (3.47) with [ p , q ] = r (1 − v T v ) . Hence, in particular, the solution q of (4.31) is givenexplicitely in terms of Φ using (4.14). Proof: Let K be the diagonalizing matrix of − vv t : K − ( − vv t ) K = diag(1 − n, , . . . , , (4.37)and consider e Φ ( z ) := ( K⊗ )Φ ( z ; A , B , C ; θ )( K⊗ ) − , which is still a solution of (3.47) with all the exponents of the formal monodromy at infinity equal to (but non–diagonalStokes parameters). Thanks of the classical work of Jimbo Miwa and Ueno [10] (see also [3, 1]) it is known that there existsa polynomial matrix e R ( z ) of degree rn such that the matrix e Φ( z ; A , B , C ; θ ) := e R ( z ) e Φ ( z ; A , B , C ; θ ) (4.38)22as exponents of formal monodromy at ∞ equal to r diag(1 − n, , . . . , ⊗ . In other words, at infinity we have that e Φ( z ; A , B , C ; θ ) = (cid:0) + O ( z − ) (cid:1) z r diag(1 − n, ,..., ⊗ e i (cid:16) z + tz (cid:17)b σ . (4.39)Indeed the construction of e R ( z ) amounts to a large but finite set of linear equations; it can be constructed iteratively interms of “elementary” Schlesinger transformations that shift by +1 and − exactly two exponents of formal monodromy.The resulting formulas are completely explicit (in terms of the coefficients of the z − –expansion of Φ ) but extremely large,even for the simplest case n = 2 , r = ± . Finally, in order to replace z r diag(1 − n, ,..., in the equation above with z ig ( − v T v ) ,we simply define Φ( z ; A , B , C ; θ ) := ( K⊗ ) − e Φ( z ; A , B , C ; θ )( K⊗ ) . (4.40)In particular, we have that the matrix R ( z ) to be found is equal to R ( z ) := ( K⊗ ) − e R ( z )( K⊗ ) . (4.41) (cid:4) As a Hamiltonian system, the particle–particle interaction in (4.31) is attractive and hence from a dynamical point ofview it would be not clear whether there exist global solutions ~x ( t ) . Indeed the Hamiltonian is not bounded below. However,we have constructed explicit solutions starting from solutions of the single–particle equations; these solutions q ( t ) necessarilyhave the Painlev´e property (they are ultimately very complicated rational expressions in the single–particle solutions andtheir derivatives) and hence, even if they may have poles for some real values of t , they are globally defined; of course thisdoes not prevent the eigenvalues to collide for some values of t . Acknowledgements. We thank P. Boalch for pointing out the relations between this work and his article [ ? ], and G.Rembado for some interesting discussions on the deformation quantization of simply laced isomonodromy systems.The three authors acknowledge the support of the project IPaDEGAN (H2020-MSCA-RISE-2017), grant number 778010.The research of M.B was supported in part by the Natural Sciences and Engineering Research Council of Canada grantRGPIN-2016-06660 and by the FQRNT grant ”Applications des syst`emes int´egrables aux surfaces de Riemann et aux espacesde modules”. The research of M.C. and V.R. was partially supported by a project “Nouvelle ´equipe” funded by the regionPays de la Loire. V. R. acknowledges the support of the Russian Foundation for Basic Research under the grants RFBR 18-01-00461 and 16-51-53034-GFEN. M.C. and V.R. thank the Centre “Henri Lebesgue” ANR-11-LABX-0020-01 for creatingan attractive mathematical environment. 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