Physica D 31 (1988) 424-433 North-Holland, Amsterdam

COMMUTING PAIRS OF LINEAR ORDINARY DIFFERENTIAL OPERATORS OF ORDERS FOUR AND SIX

F. Alberto GRLINBAUM Department of Mathematics, University of California, Berkeley, CA 94720, USA

Received 23 December 1987 Revised manuscript received 14 January 1988 Communicated by H. Flaschka

We give an elementary and explicit description of all commuting linear ordinary differential operators of orders four and six. Our analysis gives an independent derivation of the results of Kxichever and Novikov in the "rank two" case.

1. Introduction

Given a linear ordinary differential operator defined locally by

Ln= ~ ai(x)D i, D=- d dx '

i=o

the generic situation is that its commutator in the class of differential operators consists of all constant coefficient polynomials in L n.

It is of interest to consider the case where this commutator has a richer structure. In a little known paper [1] I. Schur made a number of important discoveries, including the fact that this commutator is always a commutative ring. The classification of commutative rings of differential operators was attacked by Burchnall, Chaundy and Baker [2, 3] in a series of highly original papers. These papers went largely ignored, and interest in the subject was revived by Krichever [4] in connection with soliton-type equations.

If the ring contains two operators with relatively prime orders, the situation is well understood since [2, 4]. These pairs are parametrized by a finite number of parameters. The study of the cases where this condition is violated is much more complex.

The purpose of this note is to consider the simplest instance of this last situation, when the ring contains a pair of operators of orders four and six. More specifically: we are primarily interested in classifying commutative pairs L 4, L 6 which can be parametrized not only by a finite number of parameters but also by an arbitrary function of x. The discovery of this important difference with the case of relatively prime orders is one of the main points in [5-10]. From the results in [2] we know that a polynomial relation of

*Research supported by National Science Foundation grant DMS 84-03232 and ONR contract N00014-84-C-0159.

0167-2789/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

F.A. Grfinbaum / Commuting pairs of linear ordinary differential operators 425

the form

Q ( L 4 , L6) = 0

must hold, with Q(~, ~) a polynomial of degree six in ~ and four in ~. In [6-10] one assumes that the relation is given by the "elliptic curve"

2 3 L6 = L4 + gxL4 + g3.

It is however not hard to see that if one assumes that the ring generated by L 4 and L 6 has rank two, i.e. if all operators in the ring have even order, then we are dealing, after an affine transformation, with an elliptic curve as in [6-10]. For this definition of the rank and other useful tools, see [11]. Another useful reference for material taken up in section 5 is [12].

After completing this paper we learned of [13] which deals with meromorphic coefficients and an elliptic curve. A class of examples is given in [14].

Both [13] as well as the present paper have made use of computers. In our case the main use has been in obtaining the appropriate equations. We have used Vaxima, on a Vax 750 at Berkeley.

The main point of this paper is that the instances of commuting pairs L4, L6 that are parametrized by a infinite number of choices can be given an elementary and explicit representation. Moreover these are exactly the " r a n k two" cases.

We close the introduction with a description of the contents of this paper. Section 2 gives a set of equations equivalent to the condition [L4, L6] = 0. First we dispose of the simpler case when L4 is selfadjoint, and then we find the general solution of these equations under the restriction K o = Kt3 = 0. Section 3 considers the case K 0 * 0. Section 4 tackles the case K o = 0, KI3 6: 0. Section 5 shows that, in the nonselfadjoint case, "rank 2" situations arise exactly when K 0 = K13 = 0. Section 6 shows how the results in [6] are equivalent to those given in section 2.

2. The commutativity equations

Let c0(x), cl(x ), ¢2(X) be smooth functions of x, defined in the neighborhood of some fixed x 0. Consider the most general fourth order differential operator L 4. After a Liouville transformation L 4 can be given by

( L4-- D 2 + + 2 c x D + c ~ + C o ,

with ' denoting derivatives with respect to x and D -- d / d x .

One can consider, starting with Schur [1], the formal pseudo differential operator

and its n th power L~/4 -- D n + . . . .

It is easy to see that any operator of order six commuting with L 4 has to have a constant leading term. We set out to investigate all pairs L4, L 6 such that

[ L 6, L4] --- L 6 L 4 - L 4 L 6 = O.

426 F.A. Grfinbaum / Commuting pairs of linear ordinary differential operators

One knows, see [15, 16], that if (L~/4)+ denotes the differential operator part of L~/4 then any differential operator whose commutator with L 4 has order no higher than two is given by a linear combinat ion with constant coefficients of the (L] /4)+, n = 0,1, 2 , . . . .

Start with

5

Lr=2(Lr/4)+ + Es,(L~/')+ i~1

and assume, without loss of generality, that s 4 = s o = 0 . We get

[ L 6, L4] = a ( x ) D 2 + f l ( x ) D + "y(x).

The commutat ivi ty conditions become

~(x) =~(x)=~(~)=0.

A look at these equations reveals that if s5 ~ 0 one can solve for c v, c v and C VII in terms of lower order derivatives of these functions. If s5 = 0 then one can solve for ~v ..v ..v in terms of lower order derivatives, ~'1, ~'0~ v2

as long as s 3 ~= 0. Finally, if s 5 = s 3 = 0 one can solve for c v, c v, c~" in terms of lower derivatives, as long

a s s 1 ~ : 0 .

The implication of the analysis above is that unless

S 5 = S 3 ~ - - - S I = 0

the commutat ivi ty equations are equivalent to a system of the form X ' = f ( X ) where X ~ R" and the vector-valued function f satisfies at every point of R" a local Lipschitz condition insuring that, locally at

least, one can solve for Co, cx, c2 if enough derivatives of these functions are specified at an arbitrary point

X 0 •

Since we are interested in pairs parametrized by an arbitrary function, as different from those that are

parametrized by finite dimensional objects, we assume from now on that

+ + + ,

with s 2 an arbitrary constant. I t is also easy to see that the cases ruled out above give rank I pairs L4, Lr, i.e. the g.c.d, of the orders of

the operators in the algebra generated by L 4 and L 6 is one. Since s 2 is a constant, one can see that one can absorb s2 into Co(X), so from now on we take s2 = 0. We need to take a closer look at the now greatly simplified equations obtained from a ( x ) = f l ( x ) = "r(x)

--- 0. The equations are

t t

I ( q c ~ ) ' + 2(cxc2) - 6CoCx + c; '" + 6Ko = 0,

c g " _ ~ 2 + 5 c l c ? + ~ ( c ~ ) 2 - K 3 = O, n 3c ) + 2 2c0

I I I c~c~' - clcg' - 6c21c~ - {c~K o = O,

where K 0 and Ka are arbitrary constants.

EA. Griinbaum / Commuting pairs of linear ordinary differential operators 427

These equations are not a - 0, fl = 0, ~, = 0 respectively, but the two sets of equations are equivalent. More explicitly, the derivative of the left-hand side of I gives a(x). With K 0 an arbitrary constant, a -- 0 and I are thus equivalent. If one computes fl(x) - a '(x) one gets the derivative of the left-hand side of II. Thus as long as K a is an arbitrary constant a = fl = 0 is equivalent to I and II. Finally, if one computes - , / ( x ) - a " ( x ) / 4 + f l ' (x ) /2 and still subtracts the left-hand side of I multiplied by c~/4, one gets the left-hand side of III. In this fashion a = fl = ), = 0 is equivalent to I, II and III. This last set of equations is considerably simpler than the original ones.

If c x ---0 the three equations are reduced to only one equation. This is the "selfadjoint" case. If, moreover, c~ - 0 we get

L4-- ( D 2 + ~ ) 2 + Co,

L6 -- 2(D2 + -~)3 + 3co(D2 + -~) .

This case is rather uninteresting since L 4 and L 6 are functions of the operator D 2 + c2/2. This holds for arbitrary c 2. The "curve" is L 2 = 4L34 - 3c~L 4 - Co 3. This is our first instance of commuting pairs L4, L 6 parametrized by an arbitrary function.

If c I - 0 but c~ ¢ 0 then one can solve for c 2 in terms of the arbitrary function Co: the result is

K2 2K3co + c3o ct~" [c~'12 = + + ½ / c ; / ' (1)

with K 2 and K 3 arbitrary constants. Moreover, one shows that the algebraic relation between L 4 and L 6 is given by

L~ = 4L~ + 2K3L 4 K2 2 "

This is the second instance of commuting pairs L6, L 4 parametrized by an arbitrary function. Assume for the rest of this paper that cx is not identically zero, i.e. we are in the "nonselfadjoint" case.

To proceed further in this section, make the assumption that Ko = 0. The next section handles the case K o ~ 0. The general solution of (III) is then given by

CO=-g2-~ -g l lgaLg l2q -g l3 f c l f~ l . (2)

Here g(x) is given, up to a constant, by the condition

c 1 =g"

and KII, K12 , KI3 are arbitrary constants. In (2), and for the rest of the paper, the symbol f f denote an arbitrary antiderivative of f . Plugging (2) into (I) one can find the general solution for the resulting equation in c 2, namely

t t t 1 g14-1- 6g2K12 + 2g3K11- 2gglo_ g4..l_ g - 2 _ 2g,g,- /g 'fg Yg Y g-~

c2 ---- 2g,2 + 6K13 g,2 , (3)

428 F.A. Grfinbaum / Commuting pairs of linear ordinary differential operators

where Kxo and Kt4 are new arbitrary constants.

We can proceed further in the case when K13 = 0. The section after next deals with the case when K 0 = 0, Kx3 q= 0. It turns out that eq. (II) is now automatically solved when K 3 satisfies K14 = 2K 3 + KloKt l + 3Kx22.

As to the relation between L 6 and L 4 we get

L ~ = 4 L ] + ( K 1 4 - 3K~2- X x o X u ) L 4 + g ? o - 2 K l o g l l g 1 2 - 4 K ? 2 - g14 (g21 -4- 4Kl1 )

4 (4)

Notice that we can get any elliptic curve. The discussion above is summed up as follows.

Theorem. The following three families of pairs of commutative pairs (L4, L6) are parametrized by the choice of an arbitrary function

(a) c~ --- cl -- 0, c 2 arbitrary. (b) c x -- 0, c~ ¢ 0, c 2 given in terms of an arbitrary c o by (1) above. (c) c 1 ~ O, Co(X ) and c2(x ) given in terms of the arbitrary function g(x ) by (2) and (3) above with c x = g' .

It will be seen in the next two sections that these are the only cases parametrized by an infinite dimensional parameter space. Since the arbitrary constants K o and K13 satisfy K o = Kx3 = 0 one can say that these "'elementary" examples are highly exceptional.

3. Ko4=0

One solves (III) for c 2 to obtain

K 4 + 2c~c~ - 2clc 6' - 4c31 c 2 = 3K ° '

with K 4 a new arbitrary constant. Plugging this into (I),(II) one gets a linear system of equations in c~ v, c[ v whose solution is

3K0co I v = 12coKos ~ - K,(2c6' + 6c~) + Ko(6K 3 - 30c ,c ; ' - 15c; 2 + 9% 1)

"It.t2 vp -"-~o cl - 4c~cgc~ + 2clc~cg' + 4clcg 2 + 20c~cg + 24c~

and

~AoC 1,,.--2 iv = (36ctKg + 24CoC~Ko)s 2 _ Ka(6c~,Ko + 4clc o2 ,, + 12c~)

+ K 0 (c~ (12K 3 + 18c02) - 6c,c~c{" + (6ClC ~' - 30c~c~ ) c~' + 150c~c~'

+ 24clcd,, c~ ) _ 54K03 + 54CoCxKg _ 4ClC 02 ,'2C1,, -- 8CLC0C 0 2 , ,text

3 ,,, 8c31c~,2 + 40c~c6, + 48c 7. + 4ClC o +

Another approach consists in looking at the equation obtained by plugging c 2 into I. This gives a linear

F.A. Grfinbaum / Commuting pairs of linear ordinary differential operators 429

equation for c o with coefficients that depend on c 1. Solving for c o and inserting into the equation obtained by replacing c 2 into (II) gives one a nonlinear equation for cv

Thus under the condition K 0 * 0, the equations [L4, L6] = 0 (and the restrictions s I = s 3 = s 5 = 0) lead once again to a first order system of ordinary differential equations, in this case for the vector (c o, c~, c~', c~", Cl, c~, c~', c{" ). Since a local Lipschitz condition is clearly satisfied we obtain a local solution Co(X), ct (x ) for any choice of eight constants. The total number of constants to be chosen in order to completely determine this family of examples is eleven since K0, K3, Ka are arbitrary.

4. Ko= 0,Kl3* 0

Put c 1 = g ' (g an arbitrary smooth function) while c o and c 2 are given by

Co = _g2 + Kl lg + K12 + K13k(x) ,

KI4 + 6K12g 2 + 2 K n g 3 - 2Ktog - 2g'g " + g,,2 _ g4 c 2 = 2g,2

with the functions k ( x ) and h(x) a pair of solutions of

1 + g, k " = k ' ~ , ~

and

+ 6K13h(x),

(2')

(3,)

I t t t P

h

respectively. One can get rid of k ( x ) by noticing that the second expression gives

c'h' " = h" + 3 " " + 2Cih. (5) C 1 el

Using the first expression in the form

1 cl/ c-7' (6)

we see that I and III are satisfied exactly when the integral terms in c o and c 2 are Klak(x ) and 6K13h(x ) respectively for a pair of functions h(x), k ( x ) such that k ( x ) is given by (5) while h(x) satisfies the equation

h Iv= -(2c21c~h " + (8c21c{ ' - 9clc~Z)h"+ (7c?c;" - lSclC~C{'+ 9c~3)h '

+ (2c~c~ - 6c~c~c{" - 2cxc~ '2 + 6c~Zc{')h - c~)/c~. (7)

Using this expression we can obtain expressions for h v, h vI and h vn in terms of h, h ' , h" , h " and the function c 1.

430 F.A, Grfmbaum / Commuting pairs of linear ordinary differential operators

Now we plug the expressions for Co, c x and c 2 into the conditions I = II = III -- 0. Using the expression for k we get that the "new" i vanishes.

__Going into the expression for the "new" II"~ with (4) as well as the values of h TM and h v we get that III = 0.

Now we come to the expression for the "new" II. We first notice that if K13 = 0 then II vanishes. Dividing through by Kt3 and plugging k as well as h rv, h v, h va, h vn we obtain an expression for h ' in terms of g.

We can now compute h", h ' " and h Iv in terms of g and replace all these expressions into (5). The resulting expression is of the form

2Klag'2Ah + B(g, g',..., gVH) ___ 0,

where A is given by

( d g ] 2 dSg dg d2g d4g d--x] dx 5 3 d x dx 2 dx 4

dg(d g t 3t dx dx 3 ] + ~ dx 2] dx 3

and where the second summand, B, depends on g and several of its derivatives. If the coefficient in front of h(x) vanishes we are led into an inconsistent set of equations, otherwise we

can solve for h in terms of g. We can now compute a second expression for h ' in terms of g. Equating this expression with the one

obtained above gives a differential equation for g of the form

gVni = G (g , g ' , g " . . . . . gVli).

One can now reverse the steps and see that every time that g satisfies this equation, and A does not vanish, we have a solution of the commutativity equations.

Since a Lipschitz condition is satisfied locally we conclude that g (x) is determined by an appropriate number of constants.

5. Rank 2 exactly when g 0 - gt3 -- 0

Useful insight comes from considering the operator K given by

K - - I + k l ( x ) D -1 + k 2 ( x ) D - 2 + . . .

and such that

K-tL4K = D 4.

A convenient reference for this is [12]. Since this implies that

r-lL6l¢= 0 6 - K - t ( L 6/4) _ K

- D 6 - ( f l D - 1 + f 2 D - 2 + • • • ),

F.A. Griinbaum / Commuting pairs of linear ordinary differential operators 431

it is now easy to see that

½K-t[L6, L4]K= 4fx'D 2 + (6fi" + 4f2')D + (4fi'" + 6f2" + 4f3')I + "'" •

Either of the last two expressions make it clear that L 4 and L 6 commute exactly when fl, f2, f3 are constants.

The relation between the coefficients f,. and those in the expansion

( L 6 / 4 ) _ - l l D - t + 12D-2+ 13D-3 + . . .

is given by

Ix= A , 12=/2, 1 3 = / 3 + A k ~ - f { k 1.

We are now in a position to interpret the vanishing of the constant K 0 and g13 in a conceptual manner, pursuing a suggestion of G. Wilson.

From the fact that

3c;c c;'" c2c7 11 = - cl 4 4 4 T + ~2C0C1'

we get that fl ' = 0 exactly when I holds, and furthermore fx = 0 exactly when K o = O.

As sume now that K o = 0 and that [L 4, L6] -- 0. Then (2) gives

K13 -- 2Cl 3 + c~'c 1 - C~C~.

A simple computation shows that

c 2 I{' CoC l, ' clc ~' c 3 -K13/4 13+11"-2 + -2- + 1 2 = 4 4 2 =

and thus when K o = 0, the vanishing of Kla is equivalent to that o f l 3.

Summarizing: we have K o = 0 and K13 = 0 exactly when

fx =A =0.

These conditions, and the commutativity of L 4, L6, can be used to prove inductively that

f 2 , +1= 0, n = 0 , 1 , 2 . . . . .

It is now elementary to see that these are exactly the conditions required to insure that no polynomial in L4, L 6 would have odd order, i.e. the rank of the pair L4, L 6 is two.

6. The formulas of Krichever, Novikov and Grinevieh

I owe to G. Wilson and E. Previato the observation that the expressions for c 0, q , c 2 given above contain those given in [6-10].

432 F.A. Gr~nbaum / Commuting pairs of linear ordinary differential operators

Let ~ z ) , ~(z) and °(z) be the usual functions of Weierstrass, corresponding to a choice of constants g2,g3. For an arbitrary function c(x) and a number ~/0 put ),x(x) = "to + c(x), ~2(x) = "to - c(x). Set

o ( v x , v~,0) = ~ ( ~ 2 - v~) + ~ ( ~ ) - ~(~:).

In terms of these functions, the expression for L 4 given in [10] amounts to choosing

Co(X) ~--" --go(] t l ) -- go(V2),

c2 = ½\_C7 ] ( c " ] 2 c".c "7 2c '21 + 4c, ,~(. t l , . t2,0) + 2c,2[,e(),1, Y2,0) _ 02(~1, Y2,0)]

The fourth term on the right-hand side corrects a minor misprint in [10]. We will quote below some identities for elliptic functions. They will show that the case K o = K13 = 0 agrees with the expressions in [101.

Start from

o(a + b)o(a - b) o(2a) g o ( a ) - go(b) = - o : ( a ) o ~ ( b ) , o ' ( a ) = - g o ' ( a ) .

This makes it possible to prove

go(V~) _ go(Vx) = go'(V0)go'(c(x))

(go(Vo) - go(c(x))) ~

A consequence of this is that if we put, up to an additive constant,

go'(~0) g ( x ) = go(Vo) - go(c(x) )

w e get

g ' ( x ) = c'(~(~2) - go(~x))

and thus our expressions for c x agree. Furthermore, the addition theorem implies

go(a + b) + go(a - b) -- (2go(a)go(b) + gz/Z)(go(a) + go(b)) + g3

(go(a) - go(o)) ~

This can be used to prove that our formulas for c o agree if

Kx2 = - 2go('to),

6go2(~o) + ~ / 2 KH = ~'(~0)

F.A. Grfinbaum / Commuting pairs of linear ordinary differential operators 433

The correspondence is completed by using

l_ ga:(b) - ~o'(a) = ~(b + a) - ~(b) - ~ ( a ) , 2 ga(b)-ga(a)

which in our set-up gives

d x log(h°(72) - ~o(~/1)) = - 2c'O(~q, ~'2,0).

This can be used to show that the choice

K10 = - 2~0' (7o), K14 = 0

in our formulas, produces the results given in [5]. Finally, recall that (4) forces K 3 = g2/2 and that (II) gives Ki4 = 2K 3 + KoK n + 3K22 . This shows that

the cases covered in section 2 and those in [10] agree: they can be parametrized by (K10, Kn, K12, g(x)) or equivalently by (g2, g3, 70, c(x)). The value of Kx4 can be fixed arbitrarily by adding a constant to g(x).

References

[1] I. Schur, Ober vertauschbare lineare DifferentialausdnTxke, Sitz. Berliner Math. Ges. 4 (1905) 2-8. See also Ges. Abhandlungen, I (Springer, Berlin, 1973), pp. 170-176.

[2] J.L. Burchnall and T.W. Chatmdy, Commutative ordinary differential operators, Proc. London Math. Soc. 21 (2) (1922) 420-440.

[3] H.F. Baker, Note on the foregoing paper . . . . Proc. Roy. Soc. A 118 (1928) 584-593. [4] I.M. Krichever, Methods of algebraic geometry in the theory of nonlinear equations, Russ. Math. Surveys 32 (6) (1977) 183-208. [5] I.M. Krichever, Commutative rings of ordinary linear differential operators, Funct. Anal. Appl. 12 (1978) 175-185. [6] I.M. Krichever and S.P. Novikov, Holomorphic bundles over Riemann surfaces and the K-P equation, Funct. Anal. Appl. 12

(1978) 276-286. [7] I.M. Krichever and S.P. Novikov, Holomorphic bundles and nonlinear equations. Finite zone solutions of rank 2, Sov. Math.

Dokl. 20 (1979) 650-654. [8] I.M. Krichever and S.P. Novikov, Holomorphic bundles over algebraic curves and nonlinear equations, Russ. Math. Surveys 35

(6) (1980) 53-79. [9] S.P. Novikov and P.G. Grinevieh, Spectral theory of commuting operators of rank two with periodic coefficients, Funct. Anal.

Appl. (1982) 19-20. [10] P.G. Grinevich, Rational solutions for the equations of commutation of differential operators, Funct. Anal. Appl. (1982) 15-19. [11] G. Wilson, Algebraic curves and soliton equations, Giornate di Geometrica, Rome, June 1984 (Birkl~user, Basle, 1985). [12] G. Wilson, Commuting flows and conservation laws for Lax equations, Math. Proc. Cambridge Phil. Soc. 86 (1979) 131. [13] P. Dehomoy, Operateurs differentiels et courbes elliptiques, Compositio Math. 43 (1981) 71-99. [14] J. Dixmier, Sur les algebres de Weyl, Bull. Soc. Math. France 96 (1968) 209-242. [15] I.M. Gelfand and L. Dikii, Fractional powers of operators and Hamiltonlan systems, Funct. Anal. Appl. 10 (1976) 259-273. [16] G. Segal and G. Wilson, Loop groups and equations of KdV type, IHES publications.