RIMS Kôyûrou Bessatsu 4x (20XX), 000 000 Semi-formal solution and monodromy of some confluent hypergeometric equations Dedicated to Professor Taashi AOKI for his sixtieth birthday By MasafumiYoshino Abstract This paper studies the monodromy of some class of confluent hypergeometric equations. By using the convergent semi-formal solution introduced in [1] we will show the concrete formula of the monodromy for some class of confluent hypergeometric equations. 1. Introduction In this note we will study the monodromy of some class of confluent hypergeometric equations which can be written in a Hamiltonian system. In [1] it was shown that monodromy in the class of formal power series can be calculated if one uses semi-formal solutions in expressing monodromy. We will use the convergent semi-formal solutions defined by first integrals of the Hamiltonian system which are identical to the ones given in [1]. More precisely, a convergent semi-formal solution is defined in terms of sufficiently many functionally independent first integrals. There are similarities between our idea and the so-called KAM theory. The definition of the monodromy via first integrals enables us to calculate the monodromy in an elementary way. We will give examples for which one can calculate the monodromy concretely. We hope that our method may be extended to more general class of equations in a future paper. This paper is organized as follows. In section 2 we study the convergent semi-formal solutions. In section 3 we introduce a class of confluent hypergeometric system written Received?, 200x, Accepted??, 200x. 2000 Mathematics Subject Classification(s): Primary 34M35; Secondary 34M25, 34M40 Key Words: convergent semi-formal solution, confluent hypergeometric equation, monodromy Partially supported by Grant-in-Aid for Scientific Research (No. 20540172), JSPS, Japan. Hiroshima University, Hiroshima 739-8526, Japan. c 201X Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.
2 Masafumi Yoshino in a Hamiltonian form. In section 4 we construct functionally independent first integrals and calculate the monodromy for a certain example. 2. Semi-formal solution via first integrals Let n 2 and σ 1 be integers. Consider the Hamiltonian system (2.1) 2σ dq z dz = ph(z, q, p), 2σ dp z dz = qh(z, q, p), where q =(q 2,...,q n ), p =(p 2,...,p n ), and where H(z, q, p) is analytic in z C in some neighborhood of the origin and entire in (q, p) C n 1 C n 1. We note that, by taing q 1 = z as a new unnown function (2.1) is written in an equivalent form for the Hamiltonian function H, H(q 1,q,p 1,p):=p 1 q 2σ 1 + H(q 1,q,p) (2.2) q 1 = H p1 = q1 2σ, p 1 = H q1 = 2σp 1 q1 2σ 1 q1 H(q 1,q,p), q = p H = p H(q 1,q,p), ṗ = q H = q H(q 1,q,p). The solution of (2.1) is given in terms of that of (2.2) by taing q 1 = z as an independent variable. Semi-formal solution. We define the semi-formal solution of (2.1) following [1]. Let O( S 0 ) be the set of holomorphic functions on S 0, where S 0 is the universal covering space of the punctured dis of radius r, S 0 = { z <r}\0 for some r > 0. The (2n 2)-vector ˇx(z, c) of formal power series of c (2.3) ˇx(z, c) := ˇx ν (z)c ν =ˇx 0 (z)+x(z)c + ˇx ν (z)c ν ν 0 ν 2 is said to be a semi-formal solution of (2.1) if ˇx ν (O( S 0 )) 2n 2 and (q(z, c),p(z, c)) := ˇx(z, c) is the formal power series solution of (2.1). As for the properties of the semiformal series (2.3) we refer to [1]. Here X(z) isa(2n 2) square matrix with component belonging to O( S 0 ). If X(z) is invertible, then we say that (q(z, c),p(z, c)) is a complete semi-formal solution. We say that a semi-formal solution is a convergent semi-formal solution (at the origin) if the following condition holds. For every compact set K in S 0 there exists a neighborhood U such that the formal series converges for q 1 K and c U. The semi-formal solution at the general point z 0 C is defined similarly. Monodromy function. We consider (2.1). Let z 0 be any point in C and let q and p be semi-formal solutions of (2.1) around z 0. We define the monodromy function v(c) around z 0 by (2.4) (q, p)((z z 0 )e 2πi + z 0,v(c)) = (q, p)(z, c),
Semi-formal solution and monodromy 3 where v(c) =(v j (c)). The existence of v(c) is proved in [1]. If we denote the linear part of v(c) bym 1 c, then by considering the linear part of the monodromy relation we have X((z z 0 )e 2πi + z 0 )=X(z)M. Hence M is the so-called monodromy factor. In the following we will show that the convergent semi-formal solutions of (2.1) can be obtained by solving certain system of nonlinear equations given by first integrals. We consider (2.2). Given functionally independent first integrals H(q 1,q,p 1,p) and ψ j ψ j (q 1,q,p)(j =1, 2,...,2n 2) of (2.2), where the functional independentness means that there exists a neighborhood V of the origin of (q, p, p 1 ) such that the matrix (2.5) t ( q,p,p1 H, q,p,p1 ψ j ) j 1,2,...,2n 2 has full ran 2n 1on(q 1,p 1,q,p) S 0 V. We assume that every coefficient of ψ j expanded in the power series of q, p is holomorphic with respect to q 1 on S 0. Let the point (q 1,0,p 1,0,q 0,p 0 ) and the values c j,0 (j =1, 2,...,2n 2) satisfy that (2.6) H(q 1,0,p 1,0,q 0,p 0 )=0,ψ j (q 1,0,q 0,p 0 )=c j,0, (j =1, 2,...,2n 2). For c j = c j + c j,0, c =( c 1,..., c 2n 2 ) C 2n 2 we consider the system of equations of p 1,q and p (2.7) H(q 1,p 1,q,p)=0, ψ j (q 1,q,p)=c j, (j =1, 2,...,2n 2). If (2.7) has a solution, then we denote it by q q(q 1,c), p p(q 1,c), p 1 p 1 (q 1,c). We see that q, p and p 1 are holomorphic functions of q 1 in S 0 and c in some neighborhood of the origin if we assume (2.5). We have Theorem 2.1. Suppose that H(q 1,q,p 1,p) and ψ j ψ j (q 1,q,p)(j =1, 2,..., 2n 2) be functionally independent. Assume (2.6). Then the solution of (2.7) gives the convergent complete semi-formal solution (q(z, c),p(z, c)) (q 1 = z) of (2.1) provided q or p is not a constant function. Proof. Define q =(q 1,q), p =(p 1,p) and write G (j) := ψ j. For the sae of simplicity we write q and p instead of q and p, respectively. By assumption and the implicit function theorem q, p and p 1 are convergent semi-formal series in some neighborhood of c = c 0. In order to show that they are the solution of (2.2) we differentiate (2.7) with respect to the time variable. Then we have (2.8) qh q +ṗh p =0, qg (j) q +ṗg (j) p =0, (j =1, 2,...,2n 2), where q = dq/dt and so on. Because G (j) is the first integral it follows that (2.9) H p G (j) q H q G (j) p =0, (j =1, 2,...,2n 2).
4 Masafumi Yoshino By assumption on the functional independentness we see, from (2.8) and (2.9), that the vectors ( q, ṗ) and (H p, H q ) are contained in some two dimensional plane Π. Note that these vectors are orthogonal to (H q,h p ) 0. If(H q,h p ) Π, then there exists c(t) such that q = c(t)h p, ṗ = c(t)h q. In order to show that the assertion holds in case (H q,h p ) Π, we note that the orthogonal projection of (H q,h p )toπ,( H q, H p )does not vanish by the assumption on (2.5). By (2.8) we have that q H q +ṗ H p = 0. On the other hand, by (2.9) (H p, H q ) is orthogonal to (H q,h p ) ( H q, H p ). Since (H p, H q ) is orthogonal to (H q,h p ), it follows that H p Hq H q Hp = 0. Hence we have the same assertion. If q 2 + ṗ 2 0, then c(t) does not vanish. Because H p,h q O( S 0 ) do not vanish similutaneously, we see that c(t) O( S 0 ). If we introduce s by ṡ = c(t), then (2.10) dq/ds = H p, dp/ds = H q. We will remove the assumption q 2 + ṗ 2 0. Ifq and p are not a constant function, then either q or p does not vanish except for a discrete set because they are analytic functions. Hence, by continuity we see that q and p satisfy (2.10). We note that the invertibility of X in (2.3) is verifed because (2.5) has a full ran. If we come bac to the original notation, then by definition and the relation between (2.1) and (2.2) (q(z, c),p(z, c)) (z = q 1 ) is the convergent semi-formal solution of (2.1). 3. Confluent hypergeometric equation We consider a class of hypergeometric system introduced by Oubo (cf. [2]) (3.1) (z C) dv dz = Av, where C is a diagonal matrix and A is a constant matrix. The system has only regular singular points on C { }. Set v = t (q, p) C n and assume that C and A are bloc diagonal matrices (3.2) C = diag(λ 1, Λ 1 ),A= diag(a 1, t A 1 ) where Λ 1 and A 1 are n 1 square diagonal and constant matrices, respectively such that (3.3) (z Λ 1 )A 1 = A 1 (z Λ 1 ), z C. Define (3.4) H := (z Λ 1 ) 1 p, A 1 q.
Semi-formal solution and monodromy 5 Then one can write (3.1) in the Hamiltonian form dq dz = H dp (3.5) p(z, q, p), dz = H q(z, q, p). We will introduce the irregular singularity by the confluence of singularities. Let λ j (j =2,...,n) be the diagonal elements of Λ 1. We assume λ j 0 for all j. Tae nonempty sets J and J such that J J = {2, 3,...,n} and λ i λ j for every i J and j J. Without loss of generality one may assume J = {2, 3,...,n 0 } for some n 0 2. We merge all regular singular points z = λ j (j J ) to the infinity. First, by setting z =1/ζ in (3.5) we have (3.6) ζ 2 dq dζ =(1 ζ Λ 1) 1 A 1 q, ζ 2 dp dζ = t A 1 ( 1 ζ Λ 1) 1 p. Subsitute ζ = ε 1 η in (3.6). Replace λ ν with ελ ν if ν J and multiply the μ-th row of A 1 with ε 1 if μ J. Then we let ε 0. Define the diagonal matrix A by A := diag (A 1,...,A n ) where A ν is given by λ 1 ν if ν J and (η 1 λ μ ) 1 if μ J, respectively. Then we obtain (3.7) η 2 dq dη = AA 1q, η 2 dp dη = t A 1 Ap. We will write (3.7) in a Hamiltonian form. Set η = q 1, and define H by (3.8) H(q 1,p 1,q,p):=p 1 q 2 1 A(q 1)A 1 q, p. One can easily see that q = η 2 dq dη and ṗ = η2 dp dη. Because AA 1q = H p and t A 1 Ap = H q, one easily sees that (3.7) is equivalent to the Hamiltonian system with the Hamiltonian function (3.8). If λ j s are mutually different, then it follows from (3.3) that A 1 is a diagonal matrix. Denote the diagonal entries of A 1 by τ j. Then we have (3.9) H(q 1,p 1,q,p)=p 1 q 2 1 + n j=2 τ j λ j q j p j + j J τ j λ 2 j 4. Calculation of monodromy q j p j q 1 λ 1 j In this section we will calculate the monodromy for the Hamiltonian (3.9) via first integrals. We assume that λ j s are mutually different. First, we construct first integrals of the Hamiltonian vector field (4.1) χ H := q1 2 2q 1 p 1 q 1 p 1 j J n ( ) τ j + q j p j + λ j=2 j q j p j j J τ j λ 2 j τ j λ 2 j q j p j (q 1 λ 1 j ) 2 p 1 1 q 1 λ 1 j ( q j q j p j. p j ).
6 Masafumi Yoshino For =2,...,n we will construct the first integrals in the form q w (q 1 ). We see that w satisfies ( q 2 1 + τ + τ ) 1 q 1 λ λ 2 q 1 λ 1 w =0 if J (4.2) ( q1 2 + τ ) w =0 if J. q 1 λ Hence we have (4.3) ( ) τ q 1 w (q 1 )= q 1 λ 1 if J ( ) τ exp if J. λ q 1 Next we consider the first integrals w := p u (q 1 ). By (4.1) the equation χ H w =0 can be written in the form q 1 2 d τ τ j δ,j (4.4) dq 1 λ λ 2 j J j q 1 λ 1 u (q 1 )=0, j where δ,j is the Kronecer s delta, namely δ,j =1if = j and =0 if otherwise. By ( ) τ ( ) solving the equation we have u (q 1 )= q1 q 1 λ if J, and = exp τ 1 λ q 1 if J. Hence we have (4.5) u (q 1 )=w (q 1 ) 1, =2,...,n. By (4.3) and (4.5) we have the first integrals ψ j (j =1, 2,...,2n 2) (4.6) { q j+1 w j+1 (q 1 ) (j =1, 2,...,n 1) ψ j = p j n+2 w j n+2 (q 1 ) 1 (j = n, n +1,...,2n 2). Summing up the above we have Theorem 4.1. Assume λ j 0for all j and that λ j s are mutually different. Then the Hamitonian vector field (4.1) has 2n 1 functionally independent first integrals H and ψ j s (j =1, 2,...,2n 2) given by (4.6). We will determine monodromy using first integral. We tae the convergent non constant semi-formal solution q(q 1,c), p(q 1,c) and p 1 (q 1,c) defined by (2.7). The monodromy function v(c) around z 0 is defined by (2.4). In view of the argument in section 2, we will study the monodromy around the origin z 0 = 0 or around z 0 = λ 1 for some J. Note that λ 1 is a regular singular point of the our equation which remains unchanged under the confluence procedure.
Semi-formal solution and monodromy 7 First we consider the case z 0 = 0. In order to determine the monodromy function v(c), we first note H(q 1 e 2πi,p 1,q,p)=H(q 1,p 1,q,p). On the other hand, for 1 j n 1 we have (4.7) ψ j (q 1 e 2πi,q,p)=q j+1 w j+1 (q 1 e 2πi )= { e 2πiτ j+1 q j+1 w j+1 (q 1 )=c j e 2πiτ j+1 if j +1 J = q j+1 w j+1 (q 1 )=c j if j +1 J. If n j 2n 2, then we have (4.8) ψ j (q 1 e 2πi,q,p)=q j n+2 w j n+2 (q 1 e 2πi ) 1 = { e 2πiτ j n+2 p j n+2 w j n+2 (q 1 ) 1 = c j e 2πiτ j n+2 if j n +2 J = p j n+2 w j n+2 (q 1 ) 1 = c j if j n +2 J. We define v(c) =(v j (c)) j by c j e 2πiτ j+1 if 1 j n 1, j+1 J c j if 1 j n 1, j+1 J (4.9) v j (c) = c j e 2πiτ j n+2 if n j 2n 2, j n +2 J c j if n j 2n 2, j n +2 J. Similarly we define ṽ(c) =(ṽ j (c)) j by the right-hand side of (4.9) with τ j+1 and τ j n+2 in (4.9) replaced by τ j+1 δ,j+1 and τ j n+2 δ,j n+2, respectively. Here δ,j+1 and δ,j n+2 are Kronecer s delta. Let q and p satisfy (2.7) with ψ j s given by (4.6). Then we easily see that (4.10) H(q 1 e 2πi,p 1,q,p)=0, ψ j (q 1 e 2πi,q,p)=v j (c), 1 j 2n 2. By the uniqueness of semi-formal solution we obtain q(q 1 e 2πi,v(c)) = q(q 1,c) and p(q 1 e 2πi,v(c)) = p(q 1,c). This implies that v(c) is the monodromy function as desired. In the case of other regular singular points we may argue in the same way as in the case of the origin. Thus we have proved Theorem 4.2. Assume λ j 0for all j and that λ j s are mutually different. Then the monodromy functions v(c) around the origin and λ 1 ( J) corresponding to the semi-formal solution of (2.1) defined by (2.7) are given by (4.9) and ṽ(c), respectively. Acnowledgement The author expresses sincere thans to the anonymous referee for reading the paper carefully and maing important comments.
8 Masafumi Yoshino References [1] Balser, W., Semi-formal theory and Stoes phenomenon of nonlinear meromorphic systems of ordinary differential equations, Formal and analytic solutions of differential and difference equations, Banach Center Publications, 97 (2012), 11-28. [2] Oubo, K., On the Group of Fuchsian Equations, Toyo Metropolitan University seminary note. 1987.