New York Journal of Mathematics New York J. Math.

New York J. Math. 8(2002)9–30.

On Commuting Matrix Differential Operators

Rudi Weikard

Abstract. If the differential expressionsP and Lare polynomials (over C) of another differential expression they will obviously commute. To have a P which does not arise in this way but satisfies [P, L] = 0 is rare. Yet the question of when it happens has received a lot of attention since Lax presented his description of the KdV hierarchy by Lax pairs (P, L). In this paper the question is answered in the case where the given expression L has matrix- valued coefficients which are rational functions bounded at infinity or simply periodic functions bounded at the end of the period strip: if Ly = zy has only meromorphic solutions then there exists aP such that [P, L] = 0 while P andLare not both polynomials of any other differential expression. The result is applied to the AKNS hierarchy where L=JD+Qis a first order expression whose coefficientsJ andQare 2×2 matrices. It is therefore an elementary exercise to determine whether a given matrixQwith rational or simply periodic coefficients is a stationary solution of an equation in the AKNS hierarchy.

Contents

1. Introduction 10

2. The rational case 11

3. The simply periodic case 16

4. Application to the AKNS system 17

5. The lemmas 19

Appendix A. The theorems of Halphen and Floquet 26

Appendix B. Higher order systems of differential equations 27

Appendix C. Wasow’s theorem 28

References 29

Received November 8, 2000.

Mathematics Subject Classification. 34M05, 37K10.

Key words and phrases. Meromorphic solutions of differential equations, KdV-hierarchy, AKNS-hierarchy, Gelfand-Dikii-hierarchy.

Research supported in part by the US National Science Foundation under Grant No. DMS- 9970299.

ISSN 1076-9803/02

9

1. Introduction

Consider the differential expression L=Q₀ dⁿ

dxⁿ +· · ·+Q_n.

When does a differential expressionP exist which commutes withL? This question has drawn attention for well over one hundred years and its relationship with com- pletely integrable systems of partial differential equations has led to a heightened interest in the past quarter century. A recent survey [10] by F. Gesztesy and myself tries to capture a part of that story and might be consulted for further references.

IfL=d²/dx²+qthe problem is related to the Korteweg-de Vries (KdV) hierar- chy which, according to Lax [17], can be represented as the hierarchy of equations qt= [P2n+1, L] forn= 0,1, . . ., whereP2n+1 is a certain differential expression of order 2n+ 1. The stationary solutions of these equations give rise to commuting differential expressions and play an important role in the solution of the Cauchy problem of the famous KdV equation (the casen= 1). Relying on a classical result of Picard [21], Gesztesy and myself discovered in [8] that, whenqis an elliptic func- tion, the existence of an expressionP_2n+1 which commutes withLis equivalent to the property that for allz∈Call solutions of the equationLy =zy are meromor- phic functions of the independent variable. This discovery was since generalized to cover certain 2×2 first order systems with elliptic coefficients (see [9]) and scalar n-th order equations with rational and simply periodic coefficients (see [24]).

According to the famous results of Burchnall and Chaundy in [1] and [2] a commuting pair of scalar differential expressions is associated with an algebraic curve and this fact has been one of the main avenues of attack on the problems posed by this kind of integrable systems (Its and Matveev [13], Krichever [14], [15], [16]). For this reason such differential expressions or their coefficients have been called algebro-geometric.

In this paper I will consider the case where the coefficients Q0, . . ., Qn of L arem×mmatrices with rational or simply periodic entries. First let us make the following definition.¹

Definition 1. A pair (P, L) of differential expressions is called a pair of nontrivially commuting differential expressions if [P, L] = 0 while there exists no differential expressionAsuch that both P andLare in C[A].

I will give sufficient conditions for the coefficientsQj which guarantee the exis- tence of aP such that (P, L) is a nontrivially commuting pair when mn is larger than one.² Theorem1covers the rational case while Theorem2covers the periodic case. These results are then applied to the AKNS hierarchy to obtain a character- ization of all rational and simply periodic algebro-geometric AKNS potentials (see Theorem3).

The main ingredients in the proofs are generalizations of theorems by Halphen [12] and Floquet [6], [7] which determine the structure of the solutions ofLy =zy.

1The definition is motivated by the following observation. The expressionsP andLcommute if they are both polynomials of another differential expressionA, i.e., ifP, L∈C[A]. Note that this does not happen in the case discussed above, i.e., whenL=d²/dx²+qandPis of odd order, unlessqis constant.

2Whenm=n= 1 and [P, L] = 0 thenP is necessarily a polynomial ofL.

The original theorems cover the scalar case. The generalizations, which are quoted in AppendixA, are proven in [11] and [25], respectively.

Algebro-geometric differential expressions with matrix coefficients have attracted a lot of attention in the past. The papers by Cherednik [3], Dickey [4], Dubrovin [5], van Moerbeke [19], Mumford [20], and Treibich [22] form a (rather incomplete) list of investigations into the subject.

The organization of the paper is as follows: Sections2and 3contain the state- ments and proofs of Theorems 1 and 2, respectively. Section 4 contains a short description of the AKNS hierarchy as well as Theorem3and its proof. The proofs of Theorems 1and 2rely on several lemmas which do not specifically refer to one or the other case. These lemmas are stated and proved in Section 5. Finally, for the convenience of the reader, three appendices provide the statements of the theo- rems of Halphen and Floquet, a few facts about higher order systems of differential equations, and the statement of a theorem of Wasow on the asymptotic behavior of solutions of a system of first order differential equations depending on a parameter.

Before we actually get started let us agree on a few pieces of notation. IfFis a field we denote byF[x] the ring of polynomials with coefficients inF and by F(x) the associated quotient field. The ring ofj×kmatrices with entries inFis denoted by F^j×k. The letter A represents the field of algebraic functions in one variable over the complex numbers. The symbol1denotes an identity matrix. Occasionally it is useful to indicate its dimension by a subscript as in 1_k. Similarly0and0_j×k denote zero matrices. Polynomials are to be regarded as polynomials overCunless the contrary is explicitly stated.

2. The rational case

Theorem 1. Let Lbe the differential expression given by Ly =Q₀y⁽ⁿ⁾+Q₁y⁽ⁿ⁻¹⁾+· · ·+Q_ny.

Suppose that the following conditions are satisfied:

1. Q0, . . . , Qn∈C(x)^m×m are bounded at infinity.

2. Q0 is constant and invertible.

3. The matrix

B(λ) =λⁿQ₀+λⁿ⁻¹Q₁(∞) +· · ·+Q_n(∞) is diagonalizable (as an element of A^m×m).

4. There are linearly independent eigenvectors v1, . . . , vm∈A^m of B such that limλ→∞vj(λ), j = 1, . . . , m, exist and are linearly independent eigenvectors of Q0. In particularQ0 is diagonalizable.

If mn > 1 and if, for allz ∈C, all solutions of Ly =zy are meromorphic, then there exists a differential expressionP with coefficients inC(x)^m×msuch that(P, L) is a pair of nontrivially commuting differential expressions.

Note that Conditions 3 and 4 are automatically satisfied if all eigenvalues ofQ₀ are algebraically simple.

Proof. Without loss of generality we will assume that Q0 is diagonal. Lemma 1 gives a large class of differential expressions P which commute with L. Our goal

is therefore to check the hypotheses of Lemma 1. After that we will address the question of finding aP which commutes nontrivially with L.

LetM=C(x). Forj= 0, . . . , nletQ∞,j =Qj(∞) and let the functionτbe the identity. The eigenvectors ofB, which are linearly independent as elements ofA^m, become linearly dependent (as elements ofC^m) for at most finitely many values of λ, since the determinant of the matrix whose columns are these eigenvectors is an algebraic function. Conditions1–3of Lemma1are then satisfied. Next we have to constructU such that Conditions4 and5are also satisfied.

Let the characteristic polynomial ofB be given by det(B(λ)−z) =

ν j=1

fj(λ, z)^mj

where the f_j ∈C[λ, z] are pairwise relatively prime. Denote the degree of f_j(λ,·) (which does not depend on λ) by k_j. According to Lemma 2 we may choose λ among infinitely many values such that:

1. B(λ) is diagonalizable.

2. (f₁. . . f_ν)(λ,·) has k₁+· · ·+k_ν distinct roots.

3. ifz_j,k(λ) is a root off_j(λ,·), thenλis a simple root of (f₁. . . f_ν)(·, z_j,k(λ)).

Until further notice we will think of this value of λas fixed and, accordingly, we will typically suppress the dependence onλof the quantities considered.

Let z_j,k be a root of f_j(λ,·), i.e., an eigenvalue of B(λ) of multiplicity m_j. The equation Ly = zy is equivalent to a first-order system ψ = A(z,·)ψ where A(z, x) ∈ C^mn×mn remains bounded as x tends to infinity. By Lemma 3 the characteristic polynomial ofA(z,∞) is a constant multiple of_ν

j=1f_j(λ, z)^mj and henceλis an eigenvalue of A(z_j,k,∞) of algebraic multiplicitym_j. But Lemma 3 implies also that the geometric multiplicity of λ is equal to m_j. Theorem 2.4 of [11] (quoted in Appendix A), which is a generalization of a theorem of Halphen, guarantees then the existence ofm_j linearly independent solutions

ψj,k, (x) =Rj,k, (x) exp(λx), = 1, . . . , mj

of ψ = A(z_j,k,·)ψ where the components of R_j,k, are rational functions. The common denominatorqof these components is a polynomial inxwhose coefficients are independent ofλandzj,ksince the poles of the solutions ofLy=zymay occur only at points where one of the coefficient matricesQj has a pole. Moreover,qmay be chosen such that the entries ofqQj are polynomials for allj∈ {1, . . . , n}.

TheRj,k, ,= 1, . . . , mj, may have poles at infinity whose order can be deter- mined from asymptotic considerations. We denote the largest order of these poles, i.e., the largest degree of the numerators of the components of theRj,k, bysand perform the substitution

ψ(x) =exp(λx) q(x)

s j=0

α_jx^s−j.

This turns the equationψ =A(z,·)ψinto the equivalent equation

s+s

=0

x^s+s−

j+k=

{(s−j)q_k−1−Γ_k(λ, z)}α_j = 0 (1)

wheres = deg(q) and where the Γk and qk are defined respectively by q(x)A(z, x) + (q(x)−λq(x)) = ^s

k=0

Γ_k(λ, z)x^s−k and q(x) = ^s

k=0

q_kx^s−k

(quantities whose index is out of range are set equal to zero). Equation (1) repre- sents a system of (s+s+ 1)mnlinear homogeneous equations for the (s+ 1)mn unknown components of the coefficients α_j and is thus equivalent to the equation A(λ, z)β = 0 whereAis an appropriate (s+s+1)mn×(s+1)mnmatrix andβis a vector with (s+1)mncomponents comprising all components of all theαj. Lemma4 applies to the equationA(λ, z)β = 0 with R=C[λ] andg(λ, z) = det(B(λ)−z).

We therefore conclude that there are polynomialsβ1, . . . , β_(s+1)mninC[λ, z]^(s+1)mn (some of which might be zero) such that

β1(λ, zj,k), . . . , βmj(λ, zj,k)

are linearly independent solutions of A(λ, z _j,k)β = 0 for k = 1, . . . , k_j and j = 1, . . . , ν. Hence

ψ_j,k, (x) =exp(λx)

q(x) (x^s1_mn, . . . , x⁰1_mn)β (λ, z_j,k).

Using next that fj(λ, zj,k) = 0 and the fact thatz^kj has a constant nonvanishing coefficient infj(λ,·) we obtain thatψj,k, can be expressed as

ψj,k, (x) = exp(λx)

q(x) (x^s1mn, . . . , x⁰1mn)

kj−1 r=0

β ,j,r(λ)z_j,k^r

where theβ _,j,rare elements ofC[λ]^(s+1)mn. (They are independent of the subscript k.) The firstmcomponents of eachψ_j,k, form a solutiony_j,k, ofLy=z_j,ky. One obtains

y_j,k, (x) = exp(λx)

kj−1 r=0

γ _,j,r(λ, x)z^r_j,k, whereγ _,j,r(λ,·)∈C(x)^mandγ _,j,r(·, x)∈C[λ]^m.

Now define

S_j= (γ_1,j,0, . . . , γ_1,j,kj−1, γ_2,j,0, . . . , γ_mj,j,kj−1),

Vj =





1 · · · 1

... ...

z_j,1^kj−1 · · · z^k_j,k^j−1_j



,

Zj =⊕^m_r=1^j Vj, and Yj(λ, x) =Sj(λ, x)Zjexp(λx).

The matrixYj is am×mjkj matrix. Themj columns whose index is equal tok modulokj are the linearly independent solutions ofLy =zj,ky whose asymptotic behavior is given by exp(λx). Finally we define them×mmatrices

S(λ, x) = (S₁(λ, x), . . . , S_ν(λ, x)), Z=⊕ⁿ_j=1Z_j, and Y(λ, x) = (Y1(λ, x), . . . , Yν(λ, x)) =S(λ, x)Zexp(λ, x).

We now study the asymptotic behavior of Y(λ, x) as λ tends to infinity. By Lemma 3 the matrixA(zj,k,∞) is diagonalizable and there is a positive integerh such that the eigenvalues ofA(zj,k,∞) are given by

µj,k, =λ

σj,k, ,0+ ∞ r=1

σj,k, ,rλ^−r/h

, = 1, . . . , mn

where the numbers σj,k, ,0 are different from zero. Define the diagonal matrices Mr= diag(σj,k,1,r, . . . , σj,k,mn,r) and order the eigenvalues in such a way that, for r= 0, . . . , h−1,

Mr=

σj,k,1,r1pr 0

0 Σj,k,r

,

where p₀ ≥ p₁ ≥ · · · ≥ p_h−1 and where σ_j,k,1,r is not an eigenvalue of Σ_j,k,r. Moreover, require that the m_j eigenvalues which are equal to λ are first. Then we have σj,k,1,0 = 1, σj,k,1,1 = · · · = σj,k,1,h−1 = 0, and ph−1 ≥ mj. There are p0eigenvalues which are asymptotically equal toλand there areph−1 eigenvalues which differ fromλby a function which stays bounded as λtends to infinity.

To each eigenvalueµj,k, we have an eigenvectoruj,k, of the form

u_j,k, =





 v_j,k, µ_j,k, v_j,k,

...

µⁿ⁻¹_j,k, vj,k,





,

wherev_j,k, is an appropriate eigenvector ofB(µ_j,k, ) associated with the eigenvalue z_j,k, and can, by assumption, be chosen to be holomorphic at infinity. DefineT_j,k to be (mn)×(mn) matrix whose columns are the vectorsu_j,k,1,. . . ,u_j,k,mn. Then T_j,k(λ) is invertible at and near infinity. Let

A˘_j,k(λ, x) =λ⁻¹T_j,k⁻¹A(z_j,k, x)T_j,k=λ⁻¹diag(µ_j,k,1, . . . , µ_j,k,mn) +λ⁻¹X_j,k(λ, x) where, according to Lemma5,Xj,k(λ, x) is bounded asλtends to infinity. Further- more,Xj,k(λ, x) tends to zero asxtends to infinity. Hence, given aδ >0, there is anx0(δ) and a numberr(δ) such thatXj,k(λ, x)< δwhenever|x−x0(δ)| ≤r(δ).

The matrix ˘Aj,k satisfies now the assumptions of Lemma 6 with ρ = λ^−1/h, Ω = {x : |x−x₀(δ)| < r(δ)}, and S = {ρ : 0 < |ρ| < ρ₀} for some suitable constant ρ₀. The matrix Γ is the upper left p_h−1×p_h−1 block of M_h and hence diagonal. The matrix ∆(x) is the upper left p_h−1×p_h−1 block of X_j,k(∞, x).

Hence Lemma6guarantees the existence ofp_h−1linearly independent solutions for λy= ˘A_j,ky whose asymptotic behavior is given by

exp(Γ(x−x0))(1ph−1+ Υ(x)) 0_{(mn−ph−1)×ph−1}

exp(λ, x).

(2)

Moreover, given any ε >0 there is aδ > 0 such thatΥ(x) < εfor all x∈ {x:

|x−x₀(δ)| < r(δ)}. Since the first m_j entries in the diagonal of Γ are zero we obtain that the asymptotic behavior of the firstm_j columns of matrix (2) is given by

E_j,k(x) =



 1_mj + Υ_1,1(x)) exp(Γ_2,2(x−x₀))Υ_2,1(x)

0_{(mn−ph−1)×mj}



exp(λ, x)

where Υ1,1 and Υ2,1 are the upper leftmj×mj block and the lower left (ph−1− mj)×mj block of Υ, respectively, and where Γ2,2 is the lower right (ph−1−mj)× (ph−1−mj) block of Γ.

We have now arrived at the following result: themj columns ofYj whose index is equal tok modulokj have asymptotic behavior whose leading order is given by the firstmrows ofTj,kEj,k(x)CwhereCis an appropriate constant and invertible m_j×m_j matrix. By choosing the eigenvectors u_j,k,1, . . ., u_j,k,mj (which are all associated with the eigenvalueµ_j,k,1=λ) appropriately we may assume thatC=1.

Hence, up to terms of a negligible size, the linearly independent eigenvectors ofQ₀ are the columns ofYexp(−λx) =SZ whenλandxare large. This implies thatY is invertible.

Similarly, considering the differential expressionL_∞ defined by (L_∞y)(x) =Q₀y⁽ⁿ⁾(x) +Q₁(∞)y⁽ⁿ⁻¹⁾(x) +· · ·+Q_n(∞)y(x) we obtain the invertible matrices

S_∞(λ) = (S_∞,1(λ), . . . , S_∞,ν(λ))

and Y∞(λ, x) = (Y∞,1(λ, x), . . . , Y∞,ν(λ, x)) =S∞(λ)Z(λ) exp(λx)

whereZis as before. Them_j columns ofY_∞,j whose index is equal tokmodulok_j are those solutions ofL_∞y=z_j,ky whose asymptotic behavior is given by exp(λx).

Note that S_∞ is x-independent, since the matrices A(z_j,k,∞) are diagonalizable.

Furthermore, since L_∞(vexp(λx)) = (B(λ)v) exp(λx), the columns of S_∞Z are eigenvectors ofB(λ), which, to leading order asλtends to infinity, are eigenvectors ofQ₀.

Letd∈C[λ] be such thatdS(·, x)S∞(·)⁻¹becomes a polynomial (at leastd(λ) = det(S∞(λ)) will do). Then we may define matricesUj∈C(x)^m×mby the equation

g j=0

λ^g−jUj(x) =d(λ)S(λ, x)S∞(λ)⁻¹ and a differential expression

U = g j=0

Uj(x)D^g−j. Then, obviously,

U(S∞(λ)Zexp(λx)) =d(λ)S(λ, x)Zexp(λx) =d(λ)Y(λ, x).

Since Y Y_∞⁻¹ is close to the identity when λand xare sufficiently large we obtain thatU₀is invertible and hence that Conditions4and5 of Lemma1 are satisfied.

Applying Lemma1gives now the existence of a nonempty setF of polynomials such that the differential expression P defined by P U = UDf(L_∞) commutes with Lwhen f ∈F. Assume thatP and L commute trivially. Then, by the first statement of Lemma7,Q₀is a multiple of the identity andPandLare polynomials of a unique first order differential expression G=D+G1 where G1 ∈ C(x)^m×m is bounded at infinity and where G1(∞) is a multiple of the identity. Let y be a solution of Gy = gy where g ∈ C and let ϕ be the polynomial such that L = ϕ(G). Then y satisfies also Ly = ϕ(g)y and hence every solution of Gy =gy is meromorphic. By applying what we just proved to the expression Grather than

L we know that we also have a differential expression U and a nonempty set F of polynomials such that, when f ∈ F, the differential expression P defined by P U = UCf(D+G1(∞)) commutes with G and hence with L for all matrices C∈C^m×m. The second statement of Lemma7 shows thatP is not a polynomial ofG, ifCis not a multiple of the identity. Hence, in this case, (P, L) is a nontrivially

commuting pair.

3. The simplyperiodic case

Iff is an ω-periodic function we will usef^∗ to denote the one-valued function given byf^∗(t) =f(_2πi^ω log(t)). Conversely, if a functionf^∗is givenf(x) will refer to f^∗(exp(2πix/ω)). We say that a periodic functionf is bounded at the ends of the period strip iff^∗is bounded at zero and infinity. A meromorphic periodic function which is bounded at the ends of the period strip can not be doubly periodic unless it is a constant. The functionf is a meromorphic periodic function bounded at the ends of the period strips if and only iff^∗is a rational function bounded at zero and infinity. For more information on periodic functions see, e.g., Markushevich [18], Chapter III.4.

The field of meromorphic functions with periodωwill be denoted byPω. Theorem 2. Let Lbe the differential expression given by

Ly =Q0y⁽ⁿ⁾+Q1y⁽ⁿ⁻¹⁾+· · ·+Qny.

Suppose that the following conditions are satisfied:

1. Q0, . . . , Qn∈P^m×m_ω are bounded at the ends of the period strip.

2. Q0 is constant and invertible.

3. The matrix

B(λ) =λⁿQ0+λⁿ⁻¹Q^∗₁(∞) +· · ·+Q^∗_n(∞) is diagonalizable (as an element of A^m×m).

4. There are linearly independent eigenvectors v₁, . . . , v_m∈A^m of B such that lim_λ→∞v_j(λ), j = 1, . . . , m, exist and are linearly independent eigenvectors of Q₀. In particularQ₀ is diagonalizable.

If mn > 1 and if, for allz ∈C, all solutions of Ly =zy are meromorphic, then there exists a differential expression P with coefficients in P^m×m_ω such that (P, L) is a pair of nontrivially commuting differential expressions.

Proof. The proof of this theorem is very close to that of Theorem 1. We record the few points where more significant deviations exist. For notational simplicity we will assume thatω= 2π.

Lemma 1 is now used with M = P_ω, Q_∞,j = Q^∗_j(∞), and τ(x) = e^ix. As before we have to construct the expressionU: The role of Halphen’s theorem (or, more precisely, Theorem 2.4 of [11]) is now played by Theorem 1 in [25] (quoted in AppendixA), which is a variant Floquet’s theorem. We have therefore the existence ofmj linearly independent functions

ψj,k, (x) =R^∗_j,k, (e^ix) exp(λx), = 1, . . . , mj

where the components ofR^∗_j,k, are rational functions. The substitution y(x) = exp(λx)

q(e^ix) s j=0

αje^ix(s−j)

turns the equationy =A(zj,k,·)y into a system of linear algebraic equation with mj linearly independent solutions. This way one shows as before that

ψj,k, (x) = exp(λx)

q(e^ix) (e^six1mn, . . . ,e^ix1mn,1mn)

kj−1 r=0

β ,j,r(λ)z^r_j,k

where the β _,j,r are elements of C[λ]^(s+1)mn. Doing this for k = 1, . . . , k_j and for j = 1, . . . , ν and selecting the firstm components of all the resulting vectors provides once more an m×m matrices S, Z, and Y = SZexp(λx). Again the entries ofS are polynomials with respect toλbut now they are rational functions with respect to e^ix. By considering the constant coefficient expression

L_∞=Q₀ dⁿ

dxⁿ +· · ·+Q^∗_n(∞)

one obtains also matrices S∞ and Y∞ =S∞Zexp(λx) and U is defined as before through a multiple ofS(λ, x)S∞(λ)⁻¹. The investigation of the asymptotic behavior ofY andY_∞asλtends to infinity, which leads to proving the invertibility ofU₀, is unchanged as it did not use the special structure of theQ_j, except that one should choose exp(ix₀) large rather than x₀ large.

Finally, the argument that it is possible to pick, among all expressions commuting withL, an expression which does not commute trivially remains unchanged.

4. Application to the AKNS system

LetL=Jd/dx+Q(x),where J =

i 0 0 −i

and Q(x) =

0 −iq(x) ip(x) 0

. Note thatJ²=−12 and thatJQ+QJ = 0.

The AKNS hierarchy is then a sequence of equations of the form Q_t= [P_n+1, L], n= 0,1,2, . . .

where P_n+1 is a differential expression of order n+ 1 such that [P_n+1, L] is a multiplication operator. For this to happenP_n+1 has to be very special. It can be recursively computed in the following way: Let

Pn+1=

n+1

=0

(kn+1− (x) +vn+1− (x)J+Wn+1− (x))L ,

where the k_j and v_j are scalar-valued and where theW_j are 2×2 matrices with vanishing diagonal elements. Requiring that [P_n+1, L] is a multiplication operator yieldsk_j= 0 forj = 0, . . . , n+ 1 and the recursion relations

W0= 0 v_j1₂=W_jQ+QW_j, W_j+1= 1

2J(W_j−2v_jQ_j), j= 0, . . . , n+ 1.

This gives finally

[P_n+1, L] = 2v_n+1JQ−JW_n+1 . The first few AKNS equations are

Q_t=−c₀Q+ 2c₁JQ, Qt=−c0

2J(Q−2Q³)−c1Q+ 2c2JQ, Qt=c0

4(Q−6Q²Q)−c1J(Q−2Q³)−c2Q+ 2c3JQ.

Here we are interested in the stationary solutions of AKNS equations. Therefore we make the following definition.

Definition 2. Supposep and qare meromorphic functions. Then Qis called an algebro-geometricAKNSpotential(or simplyalgebro-geometric) ifQis a stationary solution of some AKNS equation.

The goal of this section is to prove the following theorem.

Theorem 3. Let Q=

0 −iq ip 0

and assume either thatp, q are rational functions bounded at infinity or else that p, q are meromorphicω-periodic functions bounded at the ends of the period strip. ThenQ is an algebro-geometric AKNS potential if and only if for allz∈Call solutions of the equationJy+Qy=zyare meromorphic with respect to the independent variable.

Before we begin the proof of this result let us recall the following two results which were proven by Gesztesy and myself in [9]. The first one (Theorem4below) asks that p and q are meromorphic and provides one direction in the proof of Theorem 3. The second one (Theorem5 below) is the analogue of Theorem3 for the case of elliptic coefficients and is stated here for comparison purposes.

Theorem 4. Let Q=

0 −iq ip 0

wherep, q are meromorphic functions. If Q is an algebro-geometric AKNS potential then for all z ∈C all solutions of the equation Jy+Qy=zyare meromorphic with respect to the independent variable.

Theorem 5. Let Q=

0 −iq ip 0

with p, q elliptic functions with a common period lattice. ThenQ is an elliptic algebro-geometric AKNS potential if and only if for allz∈Call solutions of the equationJy+Qy=zy are meromorphic with respect to the independent variable.

Now we are ready to prove Theorem3:

Proof of Theorem 3. We only need to prove that Q is algebro-geometric if all solutions ofLy =zyare meromorphic since the converse follows from Theorem4.

SupposeQis periodic. The desired conclusion follows from Theorem2once we have checked its hypotheses. But Conditions 1 and 2 are satisfied by our assumptions while Conditions3and4 hold automatically when the eigenvalues ofQ0(=J) are distinct. For convenience, however, let us mention that the eigenvalues of

B(λ) =

iλ −iq^∗(∞) ip^∗(∞) −iλ

are ±

q^∗(∞)p^∗(∞)−λ² and that these are distinct for all but two values of λ.

The eigenvectors may be chosen as v₁= 1

2λ

iλ+z₁(λ) ip^∗(∞)

and v₂= 1 2λ

iq^∗(∞) iλ+z₁(λ)

wherez₁(λ) is the branch of

q^∗(∞)p^∗(∞)−λ² which is asymptotically equal to iλ.The proof for the rational case is virtually the same.

The solutions of Jy +Qy = zy are analytic at every point which is neither a pole of pnor of q. Since it is a matter of routine calculations to check whether a solution of Jy +Qy = zy is meromorphic at a pole of p or q and since there are only finitely many poles ofQmodulo periodicity, Theorem3 provides an easy method which allows one to determine whether a rational function Qbounded at infinity or a meromorphic simply periodic function Qbounded at the ends of the period strip is a stationary solution of an equation in the AKNS hierarchy.

5. The lemmas

Lemma 1. Let M be a field of meromorphic functions on C and consider the differential expression L = _n

j=0QjD^n−j where Q0 ∈ C^m×m is invertible and Q_j∈M^m×m forj= 1, . . . , n. Suppose that there exist differential expressions

L_∞=ⁿ

j=0

Q_∞,j D^n−j and U = g j=0

U_j(τ(x))D^g−j with the following properties:

1. Q_∞,0,. . .,Q_∞,n are in C^m×mandQ_∞,0=Q₀. 2. τ is a meromorphic function onC.

3. There is a set Λ⊂Cwith at least g+n+ 1distinct elements such that, for each λ∈Λ, the matrixB(λ) =_n

j=0λ^n−jQ∞,j has m linearly independent eigenvectorsv₁(λ),. . .,v_m(λ)∈C^mrespectively associated with the (possibly degenerate)eigenvaluesz₁(λ),. . . ,z_m(λ).

4. U₀, . . . , U_g∈M^m×mandU₀ is invertible.

5. U(vj(λ) exp(λx))is a solution ofLy=zj(λ)y forj= 1, . . . , m.

Finally, define the algebra

C={C∈C^m×m: [Q_∞,0, C] =· · ·= [Q_∞,n, C] = 0}.

Then there exists a nonempty set F ⊂C[u] with the following property: for each polynomialf ∈F and each polynomialh∈ C[u]there exists a differential expression P with coefficients in M^m×m such that [P, L] = 0. In fact, P is given by P U = Uh(D)f(L_∞).

Proof. Consider the differential expressionV =LU−UL∞ and fixλ∈Λ. Since L_∞(v_j(λ) exp(λx)) =z_jv_j(λ) exp(λx)

we obtain

V(vj(λ) exp(λx)) = (L−zj)U(vj(λ) exp(λx)) = 0.

V is a differential expression of order g+n at most, i.e.,V =_g+n

k=0Vk(x)D^k for suitable matricesVk. Hence

0 = exp(−λx)V(v_j(λ) exp(λx)) = _g+n

k=0

V_k(x)λ^k

v_j(λ).

For fixed xandλwe now have anm×m matrixV(λ, x) =_g+n

k=0V_k(x)λ^k whose kernel contains all eigenvectors ofB(λ) and is thereforem-dimensional. This means that V(λ, x) = 0. Since this is the case for at leastg+n+ 1 different values ofλ we conclude thatV0=· · ·=Vg+n= 0 and hence that

LU =UL_∞.

SinceU₀is invertibleUy= 0 hasmglinearly independent solutions. Let{y₁, . . . , y_mg} be a basis of ker(U). With each element y of this basis we may associate a differential expression H with coefficients in C^m×m in the following way. Since y ∈ker(U), so isL_∞y and, in fact,L^j_∞y for everyj∈N. Since ker(U) is finite- dimensional there exists ak∈Nand complex numbersα₀, . . . , α_k such thatα₀= 0 and

k j=0

α_k−jL^j_∞y = 0.

Then defineH =_k

j=0αk−jL^j_∞. Since the expressionsH commute among them- selves we obtain that

ker(U)⊂ker _mg

=1

H

. Hence the set

F ={f ∈C[u] : ker(U)⊂ker(f(L_∞))}

is not empty.

Note that [L_∞, D] = 0 and [L_∞, C] = 0 ifC∈ C. For anyh∈ C[u] and anyf ∈F let P_∞ =h(D)f(L_∞). Then [P_∞, L_∞] = 0 and ker(U) ⊂ker(P_∞) ⊂ker(UP_∞).

Corollary 1 in Appendix B shows that there is an expression P such that P U = UP_∞. Hence [P, L]U =P LU−LP U =UP_∞L_∞−UL_∞P_∞=U[P_∞, L_∞] = 0 and thus, recalling thatU₀ is invertible, [P, L] = 0.

Lemma 2. Let

B(λ) = n j=0

λ^n−jBj

where B₀, . . . , B_n ∈C^m×m and where B₀ is invertible. Suppose the characteristic polynomial of B has the prime factorization _ν

j=1f_j(λ, z)^mj. If weight nr+s is assigned to the monomialλ^sz^r, then the weight of the heaviest monomial infj is a multiple ofn, saynk_j and the coefficients ofz^kj andλ^nkj inf_j are nonzero.

LetΛbe the set of all complex numbersλsatisfying the following two conditions:

1. (f1. . . fν)(λ,·)has k1+· · ·+kν distinct roots.

2. If zj,k is a root offj(λ,·), thenλis a simple root of(f1. . . fν)(·, zj,k).

Then the complement ofΛ is finite.

Moreover, there is an integerhand there are complex numbers ρj,k,r such that, for sufficiently largeλ, the roots of fj(λ,·),j= 1, . . . , ν, are given by

z_j,k=λⁿ

ρ_j,k,0+^∞

r=1

ρ_j,k,rλ^−r/h

, k= 1, . . . , k_j where the numbersρ_j,k,0 are different from zero.

Proof. First we agree, as usual, that the weight of a polynomial is equal to the weight of its heaviest monomial. It is then easy to see that the characteristic polynomial f(λ, z) = det(B(λ)−z) has weight mn. Suppose f = g₁g₂ and let f =_mn

j=0α_jw_j where w_j is a polynomial all of whose terms have weightj. Doing the same with g1 and g2 one can show that any factor off has a weight which is a multiple of n, saykn, and that the coefficients of z^k and λ^kn in that factor are nonzero. In particular then, this is true for the prime factors.

Therefore fj(λ,·) has kj distinct roots for all but finitely many values of λ.

Moreover, by Bezout’s theorem, the curves defined by fj and f intersect only in finitely many points ifjis different from. Hence the first condition is satisfied for all but finitely many values ofλ.

The discriminant of (f₁. . . f_ν)(·, z) is a polynomial inz. Hence there are at most finitely many values of z for which (f₁. . . f_ν)(·, z) has multiple roots. For each of these exceptional z-values there are only finitely many of the multiple roots.

Hence there are only finitely many values of λ such that there is a z for which (f₁. . . f_ν)(·, z) has a multiple root.

The last statement follows from standard considerations of the behavior of alge- braic functions near a point. In particular, the power nonλis determined by an inspection of the Newton polygon associated withfj. Lemma 3. Let

B(λ) =ⁿ

j=0

λ^n−jB_j

whereB₀, . . . , B_n∈C^m×m and whereB₀ is invertible. Define

A(z) =







0 1m 0 · · · 0

... ...

0 · · · 1_m

B₀⁻¹(z−Bn) −B₀⁻¹Bn−1 −B⁻¹₀ Bn−2 · · · −B₀⁻¹B1





,

a matrix whose n² entries arem×mblocks.

The vector v∈C^m is an eigenvector of B(λ) associated with the eigenvaluez if and only if

u=





 λvv λⁿ⁻¹... v







is an eigenvector of A(z) associated with the eigenvalue λ. In particular, z has geometric multiplicity k as an eigenvalue of B(λ) if and only if λ has geometric

multiplicity kas an eigenvalue of A(z). Also,

det(A(z)−λ) = (−1)^nmdet(B₀⁻¹) det(B(λ)−z).

(3)

IfB is diagonalizable (as an element ofA^m×m), thenA(z)is diagonalizable for all but finitely many values ofz.

Moreover, ifzj,kis a zero offj(λ,·), then there are complex numbersσj,k, ,rand an integer hsuch that the eigenvaluesµj,k,1,. . .,µj,k,mn ofA(zj,k)are given by

µj,k, =λ

σj,k, ,0+ ∞ r=1

σj,k, ,rλ^−r/h

, = 1, . . . , mn

where the numbersσj,k, ,0 are different from zero.

Proof. ThatB(λ)v=zvif and only ifA(z)u=λufollows immediately from direct computation. The validity of (3) is proven by blockwise Gaussian elimination.

Assume now thatB is diagonalizable and letT ∈A^m×mbe an invertible matrix whose columns are eigenvectors ofB. The determinant ofT is an algebraic function in λwhich is zero or infinity only for finitely many distinct values of λand B(λ) is diagonalizable for allλ but these. From Lemma2 we know also that there are only finitely many values of λ for which _ν

j=1f_j(·, z) has repeated zeros. To all these exceptional values of λcorrespond finitely many eigenvalues z ofB(λ). We assume now thatz is a complex number distinct from all those values. Ifµis now an eigenvalue ofA(z) then it is a zero off_j(·, z) for somej but not a zero off (·, z), if = j. Hence its algebraic multiplicity is m_j. Additionally, z is an eigenvalue of geometric multiplicity mj of B(µ), since B(µ) is diagonalizable. The previous argument shows thatµhas geometric multiplicitymj as eigenvalue ofA(z). Since this is true for any eigenvalue of A(z), the matrix A(z) must be diagonalizable.

The last statement follows again from standard considerations of the behavior of algebraic functions near a point, using thatzj,kis an algebraic function ofλ(whose behavior near infinity is of the form given in Lemma2) and thatµj,k,r are algebraic

functions ofzj,k.

Lemma 4. LetRbe an integral domain,Qits fraction field,gan element ofR[z], andK a field extension of Qin which g splits into linear factors. SupposeA is a matrix inR[z]^j×k. Then there exist kvectorsv1, . . . , vk∈R[z]^k with the following property: ifz0∈K is any of the roots ofgand if the dimension ofker(A(z0))isµ, thenv1(z0),. . . ,vµ(z0) are linearly independent solutions ofA(z0)x= 0.

Proof. Suppose g has the prime factorization g^m₁¹. . . g_ν^mν. If g(z0) = 0 then precisely one of the prime factors of g, say g , satisfies g (z0) = 0. Note that F =Q[z]/g is a field and that we may viewAas an element ofF^j×k. SinceF is isomorphic to a subfield ofK anyK^k-solution ofA(z0)x= 0 is a scalar multiple of a representative of an F^k-solution of Ax = 0 (evaluated atz0) and vice versa.

Therefore there is a basis{x ,1(z0), . . . , x ,µ(z0)}of ker(A(z0)) where thex ,r are inQ[z]^k. By choosing appropriate multiples inRwe may even assume that thex _,r are inR[z]^k. Notice that ifz₀is another root ofg then{x ,1(z₀), . . . , x ,µ(z₀)}is a basis of ker(A(z₀)). We define alsox ,r = 0 forr=µ + 1, . . . , k.

Forr= 1, . . . , k we now let

vr= ν =1





 ν

=1

=

g





x ,r.

This proves the lemma once we recall that g (z0) = 0 =g (z0) implies that =

.

Lemma 5. SupposeA∈C^mn×mnandT ∈A^mn×mn have the following properties:

1. The first (n−1)m rows ofAare zero.

2. T is invertible at and near infinity and its columnsT1:mn,j have the form

T1:mn,j =





 v_j µ_jv_j

...

µⁿ⁻¹_j vj





,

where theµj are complex-valued algebraic functions ofλwith the asymptotic behavior µj(λ) = λ(σj+o(1)) as λ tends to infinity and where the vj are C^m-valued algebraic functions of λwhich are holomorphic at infinity.

Then(T⁻¹AT)(λ)is bounded asλtends to infinity.

Proof. The first (n−1)mrows ofAT are zero. Consequently we need to consider only the lastmcolumns ofT⁻¹. LetB_n,. . . ,B₁denote them×mmatrices which occupy the lastmrows ofA(with decreasing index as one moves from left to right) and letτ^∗ denote the row-vector in the lastmcolumns of rowin T⁻¹ (note that τ ∈A^m). Then

(T⁻¹AT) ,k= n j=1

µ^n−j_k τ^∗Bjvk.

We will show below thatτ has the asymptotic behaviorτ =λ¹⁻ⁿ(τ_0, +o(1)) with τ_0, ∈C^mas λtends to infinity. Hence

(T⁻¹AT) ,k= n j=1

λ^1−j(σ_k^n−jτ_0, ^∗ Bjvk(∞) +o(1)) =σ_kⁿ⁻¹τ_0, ^∗ B1vk(∞) +o(1) asλtends to infinity and this will prove the claim.

The minor ofT which arises when one deletes rowrand columnssofT will be denoted byMs,r. We have then that

(T⁻¹)_r,s= (−1)^r+s

det(T) det(M_s,r).

The k-th entry in row mα+β, where β ∈ {1, . . . , m} and α ∈ {0, . . . , n−1}, equalsλ^αtimes a function which is bounded asλtends to infinity. Hence det(T) = λ^N(t0+o(1)) for some nonzero complex number t0 and for N = mn(n−1)/2.

By the same argument we have that det(M_mα+β,r) =λ^N(m_mα+β,r+o(1)) where N=N−αandm_mα+β,r ∈C. Hence

(T⁻¹)_r,s= (−1)^r+sλ^−αm_mα+β,r+o(1)

t₀ .