Commuting Differential Operators of Rank 3 Associated to a Curve of Genus 2
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2012), 044, 11 pages Commuting Dif ferential Operatorsof Rank 3 Associated to a Curve of Genus 2
Dafeng ZUO †‡†
School of Mathematical Science, University of Science and Technology of China,Hefei 230026, P.R. China
E-mail: [email protected] ‡ Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences,P.R. China
Received March 12, 2012, in final form July 12, 2012; Published online July 15, 2012http://dx.doi.org/10.3842/SIGMA.2012.044
Abstract.
In this paper, we construct some examples of commuting differential opera-tors L and L with rational coefficients of rank 3 corresponding to a curve of genus 2. Key words: commuting differential operators; rank 3; genus 2
The study of the commutation equation[ L , L ] = 0of two scalar differential operators L = d n dx n + n − (cid:88) i =0 f i ( x ) d i dx i and L = d m dx m + m − (cid:88) j =0 g j ( x ) d j dx j , n < m, is one of the classical problems of the theory of ordinary differential equations.Burchnall and Chaundy in [1, 2, 3] have shown that “each pair of commuting operators L and L is connected by a nontrivial polynomial algebraic relation Q ( L , L ) = 0 ”. The equation Q ( z, w ) = 0 determines a smooth compact algebraic curve Ξ of finite genus g . For a generic point P ∈ Ξ, there exist common eigenfunctions ψ ( x, P ) on Ξ such that L ψ = λψ and L ψ = µψ .The dimension l of the space of these functions corresponding to P ∈ Ξ is called the rank of thecommuting pair ( L , L ). For simplicity, in this paper we denote “the commuting differentialoperators of rank l corresponding to a curve of genus g ” by “ ( l, g ) -operators” .Burchnall and Chaundy also made significant progress in solving the commutation equationfor relatively prime orders m and n . In this case, the rank l equals to 1. The study of this casewas completed by Krichever [11, 12], who also obtained explicit formulas of the function ψ andthe coefficients of L and L in terms of the Riemann Θ-function. Let us remark that there areseveral papers related to this case, for instance [5, 6, 23, 25, 28, 29].But for high rank case i.e. l >
1, it is much more complicated. In [10], the problem of classi-fying ( l, g )-operators was solved by reducing the computation of the coefficients to a Riemannproblem. In [13, 14] I.M. Krichever and S.P. Novikov developed a method of deforming theTyurin parameters on the moduli space of framed holomorphic bundles over algebraic curves.By using this method, in certain cases the Riemann problem can be avoided and they found a r X i v : . [ m a t h - ph ] J u l D. Zuoall (2 , , , , , σ -invariance to simplify the Krichever–Novikov system [14] and constructed some examples of (2 , , , g )-operators and (3 , g )-operators. Recently, an interesting paper is due to O.I. Mokhov in [22] who constructed examplesof (2 k, g )-operators and (3 k, g )-operators with polynomial coefficients for arbitrary genus g . Formore related results, please see [8, 9, 13, 15, 16, 25, 26, 27] and references therein.The aim of this paper is to construct examples of commuting differential operators L and L with rational coefficients of rank 3 corresponding to a curve of genus 2, which is different fromthose in [22]. In this section we want to construct (3,2)-operators. The first step is to use a σ -invariance, dueto A.E. Mironov [17], to simplify the Krichever–Novikov system (2). The second step is to solvethe simplified system by making a crucial hypothesis γ = γ, γ = aγ, γ = ¯ aγ, a = − √ i . The last step is to construct the commuting differential operators L and L . Let Γ be a curve of genus 2 defined in C by the equation w = z + c z + c z + c z + c z + c z + c . On the curve Γ, there is a holomorphic involution σ : Γ → Γ by σ ( z, w ) = ( z, − w ) , which has six fixed ramification points. It induces an action on the space of function by( σf )( x, P ) = f ( x, σ ( P )). Let us take q = (0 , √ c ) ∈ Γ. For a generic point P ∈ Γ thereexist common eigenfunctions ψ j ( x, P ), j = 0 , , q , of the opera-tors L and L . Without loss of generality, we assume that ψ j ( x, P ) are normalized by d i dx i ψ j ( x , P ) = δ ij , where x is a fixed point. Notice that on Γ − { q } , ψ j ( x, P ) are meromorphic and have six simplepoles at P , . . . , P independent of x . Let us consider the Wronskian matrix (cid:126) Ψ( x, P ; x ) = ψ ψ ψ ψ (cid:48) ψ (cid:48) ψ (cid:48) ψ (cid:48)(cid:48) ψ (cid:48)(cid:48) ψ (cid:48)(cid:48) of the vector-valued function (cid:126) Ψ( x, P ; x ), and (cid:126) Ψ x (cid:126) Ψ − = χ χ χ , (1)ommuting Differential Operators of Rank 3 Associated to a Curve of Genus 2 3where χ j = χ j ( x, P ) are independent of x and meromorphic functions on Γ with six poles at P ( x ) , . . . , P ( x ) coinciding with the poles of ψ j ( x, P ) at x = x . In a neighborhood of q , thefunctions χ j ( x, P ) have the form χ ( x, P ) = k + w ( x ) + O (cid:0) k − (cid:1) , χ ( x, P ) = w ( x ) + O (cid:0) k − (cid:1) ,χ ( x, P ) = O (cid:0) k − (cid:1) , (2)where k − is a local parameter near q . The expansion of χ j in a neighborhood of the pole P i ( x )has the form χ j ( x, P ) = − γ (cid:48) i ( x ) α ij ( x ) k − γ i ( x ) + d ij ( x ) + O ( k − γ i ( x )) , α i = 1 , (3)where k − γ i ( x ) is a local parameter near P i ( x ) for 1 ≤ i ≤ ≤ j ≤ Lemma 2.1 ([11]) . The parameters γ i ( x ) , α ij ( x ) and d ij ( x ) , ≤ i ≤ , ≤ j ≤ satisfy thesystem Eq[ i,
0] := α i ( x ) α i ( x ) + α i ( x ) d i ( x ) − α (cid:48) i ( x ) − d i ( x ) = 0 , Eq[ i,
1] := α i ( x ) − α i ( x ) + α i ( x ) d i ( x ) − α (cid:48) i ( x ) − d i ( x ) = 0 . (4) χ j ( x, P ) In this subsection, we discuss explicit forms of χ j ( x, P ) corresponding to the curve Γ defined by w = 1 + c z + c z + z . In order to do this, we assume that σχ ( x, P ) = χ ( x, P ) , σP s ( x ) = P s +3 ( x ) , s = 1 , , , (5)and γ = γ, γ = aγ, γ = ¯ aγ, a = − √ i . (6) Theorem 2.2.
Let γ be a solution of c γ + γ − − c γ (cid:48) = 0 , (7) then functions χ , χ , χ are given by the formulas χ ( x, P ) = − (cid:88) s =1 γ (cid:48) s z − γ s − (cid:88) s =1 γ (cid:48) s γ s = 3 z γ (cid:48) γ − z γ ,χ ( x, P ) = τ − (cid:88) s =1 G s γ (cid:48) s z − γ s + w ( z ) h z − γ )( z − γ )( z − γ ) , (8) χ ( x, P ) = τ z − (cid:88) s =1 H s γ (cid:48) s z − γ s − w ( z )( γ γ γ + zh )2 z ( z − γ )( z − γ )( z − γ ) , with G s , H s , τ , τ defined in (9) – (14) . Proof .
By using the σ -invariance of χ ( x, P ), we know γ s ( x ) = γ s +3 ( x ) , d s ( x ) = d s +3 , ( x ) , s = 1 , , . D. ZuoAccording to the properties of χ j ( x, P ) in (2), (3) and (5), we could assume that the functions χ j ( x, P ) are of the form in (8) with unknown functions G s = G s ( x ), H s = H s ( x ), τ r = τ r ( x )and h r = h r ( x ) for s = 1 , , r = 0 , χ ( x, P ) = 3 z γ (cid:48) γ − z γ , which yields that d i = − γ (cid:48) γ , i = 1 , . . . , . For simplicity we use the following notations a = 1 , a = a, a = ¯ a,a s +3 = a s , G s +3 = G s , H s +3 = H s , s = 1 , , . It follows from (3) that α s = H s + w ( a s γ ) h γ γ (cid:48) + a s w ( a s γ )6 γ (cid:48) , α s = G s − w ( a s γ ) h γ γ (cid:48) ,d s = τ a s γ + γ (cid:48) (cid:88) m =1 (1 − a s a s + m ) H s + m γ + ( h + 2( a s γ ) ) w ( a s γ ) − ( h + ( a s γ ) ) a s γw (cid:48) ( a s γ )6 γ ,d s = τ − G s +1 γ (cid:48) ( a s − a s +1 ) γ − G s +2 γ (cid:48) ( a s − a s +2 ) γ + ( a s γw (cid:48) ( a s γ ) − w ( a s γ )) h γ ,α s +3 ,r = σα sr , d s +3 , = σd s , r = 0 , , s = 1 , , . By substituting α ij and d ij into (4), we get twelve equationsEq[ i,
0] = 0 , Eq[ i,
1] = 0 , i = 1 , . . . , . We now try to solve these equations. Firstly, it follows fromEq[ s + 3 , − Eq[ s,
1] = 0 , s = 1 , , G s = h (cid:48) − h − ( a s γ ) h + γ (cid:48) γ − γ (cid:48)(cid:48) γ (cid:48) , s = 1 , , . (9)By using (9) and Eq[ s + 3 , − Eq[ s,
0] = 0, we get H s = ( h + ( a s γ ) ) h (cid:48) − h (cid:48) − a s γγ (cid:48) h − ( h + ( a s γ ) ) h (10) − h γ (cid:48) h γ + ( h + ( a s γ ) ) γ (cid:48)(cid:48) h γ (cid:48) , s = 1 , , . (11)Furthermore, by solvingEq[ s + 3 ,
1] + Eq[ s,
1] = 0 , Eq[ s + 3 ,
0] + Eq[ s,
0] = 0 , ommuting Differential Operators of Rank 3 Associated to a Curve of Genus 2 5we haveNeq[ s,
1] := − τ + h + 6 h ( a s γ ) + 6 a s γ − h h (cid:48) − a s γ ) h (cid:48) + 3 h (cid:48) h + 3 h (cid:48) − h (cid:48)(cid:48) + 9 a s γγ (cid:48) h + 3 h γ (cid:48) − h (cid:48) γ (cid:48) h γ + γ (cid:48)(cid:48) γ + γ (cid:48)(cid:48)(cid:48) γ (cid:48) − γ (cid:48) γ − γ (cid:48)(cid:48) γ (cid:48) − h γ (cid:48)(cid:48) h γ (cid:48) + h w ( a s γ )36 γ γ (cid:48) = 0 , s = 1 , , , and Neq[ s,
0] := − τ − a s γ + 4 h (cid:48)(cid:48) + 16 a s γγ (cid:48) + 21( a s γ ) h + (3 h (cid:48) + 9 a s γγ (cid:48) − h (cid:48)(cid:48) )( h + ( a s γ ) ) − h (cid:48) h (cid:48) − a s γγ (cid:48) h (cid:48) h + ( h + ( a s γ ) ) − h (cid:48) ( h + ( a s γ ) ) + 5 h (cid:48) ( h + ( a s γ ) )2 h + 6 h (cid:48) γ (cid:48) − h γ (cid:48)(cid:48) h γ + (3 h − h h (cid:48) ) γ (cid:48) h γ − ( h + ( a s γ ) ) γ (cid:48)(cid:48)(cid:48) h γ (cid:48) + ( h + ( a s γ ) ) h (cid:48) γ (cid:48)(cid:48) h γ (cid:48) + ( h + ( a s γ ) ) γ (cid:48)(cid:48) h γ (cid:48) + 3 h γ (cid:48) h γ − ( h + ( a s γ ) ) h w ( a s γ )18 γ γ (cid:48) , s = 1 , , . Let us remark that we have reduced twelve equations to six equationsNeq[ s,
0] = 0 , Neq[ s,
1] = 0 , s = 1 , , , with four unknown functions τ , τ , h and h .Let us take h = i ( − c − γ (cid:112) γ (cid:48) , h = i ( − ( γγ (cid:48)(cid:48) − γ (cid:48) )2 c √ γ (cid:48) . (12)From Neq[1 ,
1] = 0, we get τ = 4 γ (cid:48) − γγ (cid:48)(cid:48) γ + 4 γ (cid:48) γ (cid:48)(cid:48)(cid:48) − γ (cid:48)(cid:48) γ (cid:48) + i ( γ − √ c γ γ (cid:48) . (13)By using (13), we conclude that Neq[2 ,
1] = 0 and Neq[3 ,
1] = 0 always hold true.From the equation Neq[1 ,
0] = 0, we obtain τ = i ( γ − √ c γ − γ − i ( γ − γ (cid:48)(cid:48) √ c γ γ (cid:48) − i ( − c γ γ (cid:48) − i ( − ( γ − c γγ (cid:48) − γ (cid:48)(cid:48)(cid:48) γ + 10 γ (cid:48) γ (cid:48)(cid:48) γ − γ (cid:48) γ + γ (4) γ (cid:48) − γ (cid:48)(cid:48) γ (cid:48)(cid:48)(cid:48) γ (cid:48) + 3 γ (cid:48)(cid:48) γ (cid:48) − γ (cid:48)(cid:48) γγ (cid:48) . (14)By using (14), both Neq[2 ,
0] = 0 and Neq[3 ,
0] = 0 reduce to the same equation1 + c γ + γ − − c γ (cid:48) = 0 , which is exactly the equation (7). Thus we complete the proof of the theorem. (cid:4) D. ZuoGenerally, solutions of (7) are not useful for us to construct (3 , c = 2 or −
2, there are rational solutions. In what follows letus suppose c = − , c = − (cid:15) , (cid:15) < . The equation (7) is rewritten as1 − γ + γ + (cid:15)γ (cid:48) = 0 . (15)It is easy to check that when ( x + s ) + (cid:15) > γ = x + s (( x + s ) + (cid:15) ) , s ∈ C is a solution of (15). Without loss of generality, we set s = 0. In this case we would like towrite γ = γ ( x ; (cid:15) ). As a corollary of Theorem 2.2, we have Corollary 2.3.
Let γ ( x ; (cid:15) ) = x ( x + (cid:15) ) be a solution of (15) . Then we have χ ( x, P ) = 12 z − x ( (cid:15) + x )5832 + 10( z − κ + (cid:15) x z κ − w ( z ) + (cid:15) z κ − x w ( z )2 κz + 16 (cid:15) z κx ,χ ( x, P ) = 132 (cid:15) z − x [204 − z + 108 w ( z ) + (cid:15) z ]12 x κ , χ ( x, P ) = − (cid:15) z xκ , (16) where κ = ( (cid:15) + x ) z − x and w ( z ) = (cid:113) − z − (cid:15) z + z . By using (16), let us expand χ j ( x, P ) in a neighborhood of z = 0 χ ( x, P ) = 1 z + ζ − (cid:15) z + 2 (cid:15) x z + O (cid:0) z (cid:1) ,χ ( x, P ) = ζ + (cid:15) x z + O (cid:0) z (cid:1) , χ ( x, P ) = 3 (cid:15) x + O (cid:0) z (cid:1) , where ζ = 28 x − (cid:15) x + x ζ = 26 x . (17) Let Γ be a smooth curve of genus 2 defined by the equation w = 1 − z − (cid:15) z + z (18)on the ( z, w )-plane. Theorem 2.4.
The operator L corresponding to the meromorphic function λ = 1 + w ( z )2 z − on Γ with the unique pole at q = (0 , and L ψ = λψ has the form L = d dx + (cid:88) n =0 f n d n dx n , (19) where f = 152243 − x − (cid:15) x − (cid:15) x (cid:15) x x (cid:15) x (cid:15) x (cid:15) x x ,f = 58240 x + 55 (cid:15) x − x
243 + 5 (cid:15) x (cid:15) x x ,f = − x + 26 (cid:15) x − x
243 + (cid:15) x (cid:15) x x ,f = − (cid:15) x + 79 x
486 + (cid:15) x (cid:15) x x ,f = − x − (cid:15) x
243 + 16 x , f = − x + (cid:15) x
216 + x ,f = 384 x + (cid:15) x x , f = − x . (20) Proof .
By using (1), we have ψ (cid:48)(cid:48)(cid:48) j ( x, P ) = χ ( x, P ) ψ (cid:48)(cid:48) j ( x, P ) + χ ( x, P ) ψ (cid:48) j ( x, P ) + χ ( x, P ) ψ j ( x, P ) . (21)It follows from (21) that the equation L ψ j = λ ( z ) ψ j can be rewritten as Q ( x, z ) ψ j ( x, P ) + Q ( x, z ) ψ (cid:48) j ( x, P ) + Q ( x, z ) ψ (cid:48)(cid:48) j ( x, P ) = λ ( z ) ψ j . (22)According to the independence of χ ( x, P ), χ ( x, P ) and χ ( x, P ) at x = x , we conclude thatthe system (22) is equivalent to three equations Q ( x, z ) = λ ( z ) , Q ( x, z ) = 0 , Q ( x, z ) = 0 . By expanding Q j ( x, z ) at z = 0, we have0 = Q j ( x, z ) − δ j λ ( z ) = Q j, − z + Q j, − z + Q j + O ( z ) . Then by solving Q j, − s = 0 for s, j = 0 , ,
2, we get the coefficients of L given by f = − − ζ − (cid:15) + 32 x − ζ ζ (cid:48) ζ (cid:48) − − ζ ζ (cid:48)(cid:48) + 6 ζ (cid:48) ζ (cid:48)(cid:48) + 3 ζ (cid:48)(cid:48) ζ (cid:48)(cid:48) + 3 ζ (cid:48) ζ (cid:48)(cid:48)(cid:48) + ζ (cid:18) − (cid:15) − ζ ζ (cid:48) + 3 ζ (cid:48)(cid:48)(cid:48) (cid:19) + ζ (cid:48) ζ (cid:48)(cid:48)(cid:48) + 2 ζ ζ (4)1 − ζ (6)1 ,f = − x + 6 ζ (cid:48) + 9 ζ ζ (cid:48)(cid:48) + 12 ζ (cid:48) ζ (cid:48)(cid:48) + 9 ζ (cid:48) ζ (cid:48)(cid:48) + 3 ζ (cid:48)(cid:48) − ζ (3 ζ (cid:48) + ζ (cid:48)(cid:48) ) + 3 ζ ζ (cid:48)(cid:48)(cid:48) + 4 ζ (cid:48) ζ (cid:48)(cid:48)(cid:48) + ζ (cid:18) − (cid:15) − ζ − ζ ζ (cid:48) − ζ (cid:48) + 9 ζ (cid:48)(cid:48)(cid:48) + 2 ζ (4)2 (cid:19) − ζ (5)1 − ζ (6)2 ,f = 3 (cid:2) − ζ ζ (cid:48) + 5 ζ (cid:48) ζ (cid:48) + 5 ζ (cid:48) ζ (cid:48)(cid:48) − ζ ζ + 3 ζ ( ζ (cid:48) + ζ (cid:48)(cid:48) ) + ζ (5 ζ (cid:48)(cid:48) + 3 ζ (cid:48)(cid:48)(cid:48) ) − ζ (4)1 − ζ (5)2 (cid:3) ,f = (cid:15)
72 + 3 ζ − ζ + 9 ζ ζ (cid:48) + 9 ζ (cid:48) + 3 ζ (4 ζ (cid:48) + 5 ζ (cid:48)(cid:48) ) − ζ (cid:48)(cid:48)(cid:48) − ζ (4)2 , D. Zuo f = 15 ζ ζ (cid:48) − ζ (2( − ζ − ζ (cid:48) ) + 21 ζ (cid:48) ) − ζ (cid:48)(cid:48) − ζ (cid:48)(cid:48)(cid:48) ,f = 3 ζ − ζ (cid:48) − ζ (cid:48)(cid:48) , f = − ζ − ζ (cid:48) , f = − ζ . By substituting ζ and ζ in (17) into the above formula, we obtain explicit expressions of f j in (20). (cid:4) Next we want to look for a 12 th -order differential operator L = d dx + (cid:88) m =0 g m d m dx m , (23)such that [ L , L ] = 0. Let us sketch out our ideas and omit tedious computations. Thecommutation equation [ L , L ] = 0 is written as0 = (cid:34) d dx + (cid:88) n =0 f n d n dx n , d dx + (cid:88) m =0 g m d m dx m (cid:35) = (cid:88) k =0 W k ( f, g ) d k dx k , (24)which yields that W k ( f, g ) = 0 , k = 0 , . . . , . By using eleven equations W k ( f, g ) = 0, k = 8 , . . . ,
18, we could obtain explicit forms of g m = h m ( x ; ρ , . . . , ρ − m ) + ρ − m with integral constants ρ − m . The last eight equationswill determine some integral constants. For simplicity, we take all arbitrary parameters to bezero, and then obtain all coefficients g j as follows g = 45660160 x − (cid:15) x − x − (cid:15) x x (cid:15) x (cid:15) x (cid:15) x (cid:15) x x (cid:15) x (cid:15) x (cid:15) x x ,g = − x + 4928 (cid:15) x + 20048729 x − (cid:15) x (cid:15) x x (cid:15) x (cid:15) x (cid:15) x x ,g = 27758080 x − (cid:15) x + 2969 x − (cid:15) x (cid:15) x x (cid:15) x (cid:15) x (cid:15) x x ,g = − − x + 1028 (cid:15) x + 25 (cid:15) x (cid:15) x x (cid:15) x (cid:15) x (cid:15) x x ,g = 3395840 x + 271 (cid:15) x − x
243 + 193 (cid:15) x (cid:15) x x ,g = − x − (cid:15) x + 221 x
243 + (cid:15) x (cid:15) x x ,g = − (cid:15)
972 + 86464 x + 316 x
243 + (cid:15) x (cid:15) x x , ommuting Differential Operators of Rank 3 Associated to a Curve of Genus 2 9 g = − x + (cid:15) x
486 + 109 x , g = − x + (cid:15) x
108 + x ,g = 824 x + (cid:15) x x , g = − x . (25) Remark 2.5.
By analogy with the process of getting f j in (20), we could obtain the above g j in (25) by choosing another meromorphic function with a unique pole of order 4 at z = 0 on Γ µ ( z ) = 1 + w ( z )2 z − z . Remark 2.6.
One could find another operator L of order 15 from [ L , L ] = 0. Furthermore asin [17], the commutative ring of differential operators generated by L , L and L is isomorphicto the ring of meromorphic functions on Γ with the pole at q = (0 , According to the Burchnall–Chaundy’s correspondence in [1, 2, 3], for each pair of commutingoperators L and L there is a Burchnall–Chaundy curve defined by a minimal nontrivial poly-nomial Q ( z, w ) = 0 such that Q ( L , L ) = 0 (or Q ( L , L ) = 0). Obviously, the above curve Γdefined by (18) is not the Burchnall–Chaundy curve for L and L given in (19) and (24).Actually the corresponding Burchnall–Chaundy curve ˜Γ is given by w − (cid:15) w = z + z , that is to say, L − (cid:15) L = L + L . The curve ˜Γ has a cuspidal singularity at (0 , L and L correspond to thosemeromorphic functions on Γ λ = 1 + w ( z )2 z − , µ = 1 + w ( z )2 z − z defining a birational equivalence π : Γ → ˜Γ , π ( z, w ) = ( λ, µ ) . The inverse image of the cuspidal point is the point σ ( q ), where q = (0 , ∈ Γ. In order tomake π to be a morphism, we must complement ˜Γ at infinity by a cuspidal point of the type(3 , q . In summary by using a σ -invariance to simplify the Krichever–Novikov system, we have con-structed a pair of commuting differential operators L in (19) and L in (23) of rank 3 withrational coefficients corresponding to the singular curve ˜Γ, which is birationally equivalent tothe smooth curve Γ of genus 2.Let us remark that all of coefficients of L and L are polynomials with respect to theparameter (cid:15) . So if we take L = lim (cid:15) → L , L = lim (cid:15) → L , L , L ] = 0 , L = L + L . More precisely, we have L = L − , L = L − L , where L = d dx − x ddx − x + x . So, when (cid:15) = 0 this is a trivial example.How about the case (cid:15) (cid:54) = 0? Let us comment that in this case, by a direct verification there isnot such kind of L of order 3 commuting with L and L . Furthermore, according to the resultin [29], any rank one operator with rational coefficients whose second highest coefficient is zerohas the property that the limit as x goes to ∞ of the coefficients is zero. So, for example, theabsence of a d dx term in L and the x in the coefficient of its d dx term which means that L is not a rank 1 operator. Acknowledgments
The author is grateful to Andrey E. Mironov for bringing the attention to this project andhelpful discussions. The author also thanks referees’ suggestions and Alex Kasman for pointingsome errors in the first version of this paper, Qing Chen and Youjin Zhang for their constantsupports. This work is supported by “PCSIRT” and the Fundamental Research Funds for theCentral Universities (WK0010000024) and NSFC (No. 10971209) and SRF for ROCS, SEM.
References [1] Burchnall J.L., Chaundy T.W., Commutative ordinary differential operators,
Proc. Lond. Math. Soc. s2-21 (1923), 420–440.[2] Burchnall J.L., Chaundy T.W., Commutative ordinary differential operators,
Proc. R. Soc. Lond. Ser. A (1928), 557–583.[3] Burchnall J.L., Chaundy T.W., Commutative ordinary differential operators. II. The identity P n = Q m , Proc. R. Soc. Lond. Ser. A (1931), 471–485.[4] Dixmier J., Sur les alg`ebres de Weyl,
Bull. Soc. Math. France (1968), 209–242.[5] Dubrovin B.A., Periodic problems for the Korteweg–de Vries equation in the class of finite band potentials, Funct. Anal. Appl. (1975), 215–223.[6] Dubrovin B.A., Matveev V.B., Novikov S.P., Non-linear equations of Korteweg–de Vries type, finite-zonelinear operators, and Abelian varieties, Russ. Math. Surv. (1976), no. 1, 59–146.[7] Grinevich P.G., Rational solutions for the equation of commutation of differential operators, Funct. Anal.Appl. (1982), 15–19.[8] Kasman A., Darboux transformations from n -KdV to KP, Acta Appl. Math. (1997), 179–197.[9] Kasman A., Rothstein M., Bispectral Darboux transformations: the generalized Airy case, Phys. D (1997), 159–176, q-alg/9606018.[10] Krichever I.M., Commutative rings of ordinary linear differential operators,
Funct. Anal. Appl. (1978),175–185.[11] Krichever I.M., Integration of nonlinear equations by the methods of algebraic geometry, Funct. Anal. Appl. (1977), 12–26.[12] Krichever I.M., Methods of algebraic geometry in the theory of non-linear equations, Russ. Math. Surv. (1977), no. 6, 185–213. ommuting Differential Operators of Rank 3 Associated to a Curve of Genus 2 11 [13] Krichever I.M., Novikov S.P., Holomorphic bundles and nonlinear equations. Finite-gap solutions of rank 2, Dokl. Akad. Nauk SSSR (1979), 33–37.[14] Krichever I.M., Novikov S.P., Holomorphic bundles over Riemann surfaces and the Kadomtsev–Petviashviliequation. I,
Funct. Anal. Appl. (1978), 276–286.[15] Latham G.A., Previato E., Darboux transformations for higher-rank Kadomtsev–Petviashvili and Krichever–Novikov equations, Acta Appl. Math. (1995), 405–433.[16] Latham G., Previato E., Higher rank Darboux transformations, in Singular Limits of Dispersive Waves(Lyon, 1991), NATO Adv. Sci. Inst. Ser. B Phys. , Vol. 320, Plenum, New York, 1994, 117–134.[17] Mironov A.E., A ring of commuting differential operators of rank 2 corresponding to a curve of genus 2,
Sb.Math. (2004), 711–722.[18] Mironov A.E., Commuting rank 2 differential operators corresponding to a curve of genus 2,
Funct. Anal.Appl. (2005), 240–243.[19] Mironov A.E., On commuting differential operators of rank 2, Sib. `Elektron. Mat. Izv. (2009), 533–536.[20] Mironov A.E., Self-adjoint commuting differential operators and commutative subalgebras of the Weylalgebra, arXiv:1107.3356.[21] Mokhov O.I., Commuting differential operators of rank 3, and nonlinear equations, Math. USSR Izv. (1990), 629–655.[22] Mokhov O., On commutative subalgebras of the Weyl algebra that are related to commuting operators ofarbitrary rank and genus, arXiv:1201.5979.[23] Novikov S.P., The periodic problem for the Korteweg–de vries equation, Funct. Anal. Appl. (1974), 236–246.[24] Novikov S.P., Grinevich P.G., Spectral theory of commuting operators of rank two with periodic coefficients, Funct. Anal. Appl. (1982), 19–21.[25] Previato E., Seventy years of spectral curves: 1923–1993, in Integrable Systems and Quantum Groups(Montecatini Terme, 1993), Lecture Notes in Math. , Vol. 1620, Springer, Berlin, 1996, 419–481.[26] Previato E., Wilson G., Differential operators and rank 2 bundles over elliptic curves,
Compositio Math. (1992), 107–119.[27] Previato E., Wilson G., Vector bundles over curves and solutions of the KP equations, in Theta Functions –Bowdoin 1987, Part 1 (Brunswick, ME, 1987), Proc. Sympos. Pure Math. , Vol. 49, Amer. Math. Soc.,Providence, RI, 1989, 553–569.[28] Shabat A.B., Elkanova Z.S., Commuting differential operators,
Theoret. and Math. Phys. (2010), 276–285.[29] Wilson G., Bispectral commutative ordinary differential operators,
J. Reine Angew. Math.442