New York Journal of Mathematics

New York J. Math. 8 (2002) 9–30.

On Commuting Matrix Differential Operators

Rudi Weikard

Abstract. If the differential expressions P and L are 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 matrixvalued 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 a P such that [P, L] = 0 while P and L are not both polynomials of any other differential expression. The result is applied to the AKNS hierarchy where L = JD + Q is a first order expression whose coefficients J and Q are 2 × 2 matrices. It is therefore an elementary exercise to determine whether a given matrix Q with 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.

9^{ISSN 1076-9803/02}

1. Introduction

Consider the differential expression

L = Q_{0} \frac{d^{n}}{dx^{n}} + ⋯ + Q_{n}.

When does a differential expression P exist which commutes with L? This question has drawn attention for well over one hundred years and its relationship with completely 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.

If L = d^{2}/dx^{2} + q the problem is related to the Korteweg-de Vries (KdV) hierarchy which, according to Lax [17], can be represented as the hierarchy of equations q_{t} = [P_{2n+1}, L] for n = 0, 1, …, where P_{2n+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 case n = 1). Relying on a classical result of Picard [21], Gesztesy and myself discovered in [8] that, when q is an elliptic function, the existence of an expression P_{2n+1} which commutes with L is equivalent to the property that for all z ∈ ℂ all solutions of the equation Ly = zy are meromorphic 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 Q_{0}, …, Q_{n} of L are m × m matrices 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 expression A such that both P and L are in ℂ[A].

I will give sufficient conditions for the coefficients Q_{j} which guarantee the existence of a P such that (P, L) is a nontrivially commuting pair when mn is larger than one.^{2} Theorem 1 covers the rational case while Theorem 2 covers the periodic case. These results are then applied to the AKNS hierarchy to obtain a characterization of all rational and simply periodic algebro-geometric AKNS potentials (see Theorem 3).

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

1The definition is motivated by the following observation. The expressions P and L commute if they are both polynomials of another differential expression A, i.e., if P, L ∈ ℂ[A]. Note that this does not happen in the case discussed above, i.e., when L = d^{2}/dx^{2} + q and P is of odd order, unless q is constant.

2When m = n = 1 and [P, L] = 0 then P is necessarily a polynomial of L.

The original theorems cover the scalar case. The generalizations, which are quoted in Appendix A, 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: Sections 2 and 3 contain the statements and proofs of Theorems 1 and 2, respectively. Section 4 contains a short description of the AKNS hierarchy as well as Theorem 3 and its proof. The proofs of Theorems 1 and 2 rely 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 theorems 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. If 𝔽 is a field we denote by 𝔽[x] the ring of polynomials with coefficients in 𝔽 and by 𝔽(x) the associated quotient field. The ring of j ×k matrices with entries in 𝔽 is denoted by 𝔽^{j×k}. The letter 𝔸 represents the field of algebraic functions in one variable over the complex numbers. The symbol 1 denotes an identity matrix. Occasionally it is useful to indicate its dimension by a subscript as in 1_{k}. Similarly 0 and 0_{j×k} denote zero matrices. Polynomials are to be regarded as polynomials over ℂ unless the contrary is explicitly stated.

2. The rational case

Theorem 1. Let L be the differential expression given by

Ly = Q_{0}y^{(n)} + Q_{1}y^{(n−1)} + ⋯ + Q_{n}y.

Suppose that the following conditions are satisfied:

1. Q_{0}, …, Q_{n} ∈ ℂ(x)^{m×m} are bounded at infinity.

2. Q_{0} is constant and invertible.

3. The matrix

B(λ) = λ^{n}Q_{0} + λ^{n−1}Q_{1}(∞) + ⋯ + Q_{n}(∞)

is diagonalizable (as an element of 𝔸^{m×m}).

4. There are linearly independent eigenvectors v_{1}, …, v_{m} ∈ 𝔸^{m} of B such that lim_{λ→∞} v_{j}(λ), j = 1, …, m, exist and are linearly independent eigenvectors of Q_{0}. In particular Q_{0} is diagonalizable.

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

Note that Conditions 3 and 4 are automatically satisfied if all eigenvalues of Q_{0} are algebraically simple.

Proof. Without loss of generality we will assume that Q_{0} 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 a P which commutes nontrivially with L.

Let 𝕄 = ℂ(x). For j = 0, …, n let Q_{∞,j} = Q_{j}(∞) and let the function τ be the identity. The eigenvectors of B, which are linearly independent as elements of 𝔸^{m}, become linearly dependent (as elements of ℂ^{m}) for at most finitely many values of λ, since the determinant of the matrix whose columns are these eigenvectors is an algebraic function. Conditions 1–3 of Lemma 1 are then satisfied. Next we have to construct U such that Conditions 4 and 5 are also satisfied.

Let the characteristic polynomial of B be given by

det(B(λ) − z) = ∏_{j=1}^{ν} f_{j}(λ, z)^{m_{j}}

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. (f_{1} … f_{ν})(λ, ·) has k_{1} + ⋯ + k_{ν} distinct roots.

3. if z_{j,k}(λ) is a root of f_{j}(λ, ·), then λ is a simple root of (f_{1} … f_{ν})(·, z_{j,k}(λ)). Until further notice we will think of this value of λ as fixed and, accordingly, we will typically suppress the dependence on λ of 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 of A(z, ∞) is a constant multiple of ∏_{j=1}^{ν} f_{j}(λ, z)^{m_{j}} and hence λ is an eigenvalue of A(z_{j,k}, ∞) of algebraic multiplicity m_{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 of m_{j} linearly independent solutions

ψ_{j,k,ℓ}(x) = R_{j,k,ℓ}(x) exp(λx), ℓ = 1, …, m_{j}

of ψ′ = A(z_{j,k}, ·)ψ where the components of R_{j,k,ℓ} are rational functions. The common denominator q of these components is a polynomial in x whose coefficients are independent of λ and z_{j,k} since the poles of the solutions of Ly = zy may occur only at points where one of the coefficient matrices Q_{j} has a pole. Moreover, q may be chosen such that the entries of qQ_{j} are polynomials for all j ∈ {1, …, n}.

The R_{j,k,ℓ}, ℓ = 1, …, m_{j}, may have poles at infinity whose order can be determined from asymptotic considerations. We denote the largest order of these poles, i.e., the largest degree of the numerators of the components of the R_{j,k,ℓ} by s and perform the substitution

ψ(x) = \frac{exp(λx)}{q(x)} ∑_{j=0}^{s} α_{j}x^{s−j}.

This turns the equation ψ′ = A(z, ·)ψ into the equivalent equation

∑_{ℓ=0}^{s+s′} x^{s+s′−ℓ} ∑_{j+k=ℓ}{(s − j)q_{k−1} − Γ_{k}(λ, z)}α_{j} = 0 (1)

where s′ = deg(q) and where the Γ_{k} and q_{k} are defined respectively by

q(x)A(z, x) + (q′(x) − λq(x)) = ∑_{k=0}^{s′} Γ_{k}(λ, z)x^{s′−k} and q(x) = ∑_{k=0}^{s′} q_{k}x^{s′−k}

(quantities whose index is out of range are set equal to zero). Equation (1) represents a system of (s + s′ + 1)mn linear homogeneous equations for the (s + 1)mn unknown components of the coefficients α_{j} and is thus equivalent to the equation Ã(λ, z)β = 0 where Ã is an appropriate (s+s′ +1)mn×(s+1)mn matrix and β is a vector with (s+1)mn components comprising all components of all the α_{j}. Lemma 4 applies to the equation Ã(λ, z)β = 0 with R = ℂ[λ] and g(λ, z) = det(B(λ) − z). We therefore conclude that there are polynomials β_{1}, …, β_{(s+1)mn} in ℂ[λ, z]^{(s+1)mn} (some of which might be zero) such that

β_{1}(λ, z_{j,k}), …, β_{m_{j}}(λ, z_{j,k})

are linearly independent solutions of Ã(λ, z_{j,k})β = 0 for k = 1, …, k_{j} and j = 1, …, ν. Hence

ψ_{j,k,ℓ}(x) = \frac{exp(λx)}{q(x)}(x^{s}1_{mn}, …, x^{0}1_{mn})β_{ℓ}(λ, z_{j,k}).

Using next that f_{j}(λ, z_{j,k}) = 0 and the fact that z^{k_{j}} has a constant nonvanishing coefficient in f_{j}(λ, ·) we obtain that ψ_{j,k,ℓ} can be expressed as

ψ_{j,k,ℓ}(x) = \frac{exp(λx)}{q(x)}(x^{s}1_{mn}, …, x^{0}1_{mn}) ∑_{r=0}^{k_{j}−1} β̃_{ℓ,j,r}(λ)z_{j,k}^{r}

where the β̃_{ℓ,j,r} are elements of ℂ[λ]^{(s+1)mn}. (They are independent of the subscript k.) The first m components of each ψ_{j,k,ℓ} form a solution y_{j,k,ℓ} of Ly = z_{j,k}y. One obtains

y_{j,k,ℓ}(x) = exp(λx) ∑_{r=0}^{k_{j}−1} γ_{ℓ,j,r}(λ, x)z_{j,k}^{r},

where γ_{ℓ,j,r}(λ, ·) ∈ ℂ(x)^{m} and γ_{ℓ,j,r}(·, x) ∈ ℂ[λ]^{m}.

Now define

S_{j} = (γ_{1,j,0}, …, γ_{1,j,k_{j}−1}, γ_{2,j,0}, …, γ_{m_{j},j,k_{j}−1}),

V_{j} = (\begin{matrix}1 & ⋯ & 1 \\ ⋮ & & ⋮ \\ z_{j,1}^{k_{j}−1} & ⋯ & z_{j,k_{j}}^{k_{j}−1} \end{matrix}) ,

Z_{j} = ⊕_{r=1}^{m_{j}}V_{j}, and Y_{j}(λ, x) = S_{j}(λ, x)Z_{j} exp(λx).

The matrix Y_{j} is a m × m_{j}k_{j} matrix. The m_{j} columns whose index is equal to k modulo k_{j} are the linearly independent solutions of Ly = z_{j,k}y whose asymptotic behavior is given by exp(λx). Finally we define the m × m matrices

S(λ, x) = (S_{1}(λ, x), …, S_{ν}(λ, x)), Z = ⊕_{j=1}^{n}Z_{j}, and
Y (λ, x) = (Y_{1}(λ, x), …, Y_{ν}(λ, x)) = S(λ, x)Z exp(λ, x).

We now study the asymptotic behavior of Y (λ, x) as λ tends to infinity. By Lemma 3 the matrix A(z_{j,k}, ∞) is diagonalizable and there is a positive integer h such that the eigenvalues of A(z_{j,k}, ∞) are given by

μ_{j,k,ℓ} = λ (σ_{j,k,ℓ,0} + ∑_{r=1}^{∞} σ_{j,k,ℓ,r}λ^{−r/h}) , ℓ = 1, …, mn

where the numbers σ_{j,k,ℓ,0} are different from zero. Define the diagonal matrices M_{r} = diag(σ_{j,k,1,r}, …, σ_{j,k,mn,r}) and order the eigenvalues in such a way that, for r = 0, …, h − 1,

M_{r} = (\begin{matrix}σ_{j,k,1,r}1_{p_{r}} & 0 \\ 0 & Σ_{j,k,r} \end{matrix}) ,

where p_{0} ≥ p_{1} ≥ ⋯ ≥ 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 σ_{j,k,1,0} = 1, σ_{j,k,1,1} = ⋯ = σ_{j,k,1,h−1} = 0, and p_{h−1} ≥ m_{j}. There are p_{0} eigenvalues which are asymptotically equal to λ and there are p_{h−1} eigenvalues which differ from λ by a function which stays bounded as λ tends to infinity.

To each eigenvalue μ_{j,k,ℓ} we have an eigenvector u_{j,k,ℓ} of the form

u_{j,k,ℓ} = ( v_{j,k,ℓ} ; μ_{j,k,ℓ}v_{j,k,ℓ} ; ⋮ ; μ_{j,k,ℓ}^{n−1}v_{j,k,ℓ}) ,

where v_{j,k,ℓ} is an appropriate eigenvector of B(μ_{j,k,ℓ}) associated with the eigenvalue z_{j,k}, and can, by assumption, be chosen to be holomorphic at infinity. Define T_{j,k} to be (mn) × (mn) matrix whose columns are the vectors u_{j,k,1}, …, u_{j,k,mn}. Then T_{j,k}(λ) is invertible at and near infinity. Let

Ă_{j,k}(λ, x) = λ^{−1}T_{j,k}^{−1}A(z_{j,k}, x)T_{j,k} = λ^{−1} diag(μ_{j,k,1}, …, μ_{j,k,mn}) + λ^{−1}X_{j,k}(λ, x)

where, according to Lemma 5, X_{j,k}(λ, x) is bounded as λ tends to infinity. Furthermore, X_{j,k}(λ, x) tends to zero as x tends to infinity. Hence, given a δ > 0, there is an x_{0}(δ) and a number r(δ) such that ∥X_{j,k}(λ, x)∥ < δ whenever |x−x_{0}(δ)| ≤ r(δ). The matrix Ă_{j,k} satisfies now the assumptions of Lemma 6 with ρ = λ^{−1/h}, Ω = {x : |x − x_{0}(δ)| < r(δ)}, and S = {ρ : 0 < |ρ| < ρ_{0}} for some suitable constant ρ_{0}. The matrix Γ is the upper left p_{h−1} × p_{h−1} block of M_{h} and hence diagonal. The matrix Δ(x) is the upper left p_{h−1} × p_{h−1} block of X_{j,k}(∞, x). Hence Lemma 6 guarantees the existence of p_{h−1} linearly independent solutions for λy′ = Ă_{j,k}y whose asymptotic behavior is given by

(2) (\begin{matrix}exp(Γ(x − x_{0}))(1_{p_{h−1}} + Υ(x)) \\ 0_{(mn−p_{h−1})×p_{h−1}} \end{matrix}) exp(λ, x).

Moreover, given any ε > 0 there is a δ > 0 such that ∥Υ(x)∥ < ε for all x ∈ {x : |x − x_{0}(δ)| < r(δ)}. Since the first m_{j} entries in the diagonal of Γ are zero we obtain that the asymptotic behavior of the first m_{j} columns of matrix (2) is given by

E_{j,k}(x) = (\begin{matrix}1_{m_{j}} + Υ_{1,1}(x)) \\ exp(Γ_{2,2}(x − x_{0}))Υ_{2,1}(x) \\ 0_{(mn−p_{h−1})×m_{j}} \end{matrix}) exp(λ, x)

where Υ_{1,1} and Υ_{2,1} are the upper left m_{j} × m_{j} block and the lower left (p_{h−1} − m_{j}) × m_{j} block of Υ, respectively, and where Γ_{2,2} is the lower right (p_{h−1} − m_{j}) × (p_{h−1} − m_{j}) block of Γ.

We have now arrived at the following result: the m_{j} columns of Y_{j} whose index is equal to k modulo k_{j} have asymptotic behavior whose leading order is given by the first m rows of T_{j,k}E_{j,k}(x)C where C is an appropriate constant and invertible m_{j} × m_{j} matrix. By choosing the eigenvectors u_{j,k,1}, …, u_{j,k,m_{j}} (which are all associated with the eigenvalue μ_{j,k,1} = λ) appropriately we may assume that C = 1. Hence, up to terms of a negligible size, the linearly independent eigenvectors of Q_{0} are the columns of Y exp(−λx) = SZ when λ and x are large. This implies that Y is invertible.

Similarly, considering the differential expression L_{∞} defined by

(L_{∞}y)(x) = Q_{0}y^{(n)}(x) + Q_{1}(∞)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)

where Z is as before. The m_{j} columns of Y_{∞,j} whose index is equal to k modulo k_{j} are those solutions of L_{∞}y = z_{j,k}y 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_{∞}(v exp(λx)) = (B(λ)v) exp(λx), the columns of S_{∞}Z are eigenvectors of B(λ), which, to leading order as λ tends to infinity, are eigenvectors of Q_{0}.

Let d ∈ ℂ[λ] be such that dS(·, x)S_{∞}(·)^{−1} becomes a polynomial (at least d(λ) = det(S_{∞}(λ)) will do). Then we may define matrices U_{j} ∈ ℂ(x)^{m×m} by the equation

∑_{j=0}^{g} λ^{g−j}U_{j}(x) = d(λ)S(λ, x)S_{∞}(λ)^{−1}

and a differential expression

U = ∑_{j=0}^{g} U_{j}(x)D^{g−j}.

Then, obviously,

U(S_{∞}(λ)Z exp(λx)) = d(λ)S(λ, x)Z exp(λx) = d(λ)Y (λ, x).

Since Y Y_{∞}^{−1} is close to the identity when λ and x are sufficiently large we obtain that U_{0} is invertible and hence that Conditions 4 and 5 of Lemma 1 are satisfied.

Applying Lemma 1 gives now the existence of a nonempty set F of polynomials such that the differential expression P defined by P U = UDf(L_{∞}) commutes with L when f ∈ F. Assume that P and L commute trivially. Then, by the first statement of Lemma 7, Q_{0} is a multiple of the identity and P and L are polynomials of a unique first order differential expression G = D + G_{1} where G_{1} ∈ ℂ(x)^{m×m} is bounded at infinity and where G_{1}(∞) 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 G rather than L we know that we also have a differential expression Ũ and a nonempty set F̃ of polynomials such that, when f ∈ F̃, the differential expression P defined by P U = UCf(D + G_{1}(∞)) commutes with G and hence with L for all matrices C ∈ ℂ^{m×m}. The second statement of Lemma 7 shows that P is not a polynomial of G, if C is not a multiple of the identity. Hence, in this case, (P, L) is a nontrivially commuting pair. □

3. The simply periodic case

If f is an ω-periodic function we will use f^{∗} to denote the one-valued function given by f^{∗}(t) = f(\frac{ω}{2πi} log(t)). Conversely, if a function f^{∗} is given f(x) will refer to f^{∗}(exp(2πix/ω)). We say that a periodic function f is bounded at the ends of the period strip if f^{∗} 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 function f is a meromorphic periodic function bounded at the ends of the period strips if and only if f^{∗} 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 period ω will be denoted by ℙ_{ω}.

Theorem 2. Let L be the differential expression given by

Ly = Q_{0}y^{(n)} + Q_{1}y^{(n−1)} + ⋯ + Q_{n}y.

Suppose that the following conditions are satisfied:

1. Q_{0}, …, Q_{n} ∈ ℙ_{ω}^{m×m} are bounded at the ends of the period strip.

2. Q_{0} is constant and invertible.

3. The matrix

B(λ) = λ^{n}Q_{0} + λ^{n−1}Q_{1}^{∗}(∞) + ⋯ + Q_{n}^{∗}(∞)

is diagonalizable (as an element of 𝔸^{m×m}).

4. There are linearly independent eigenvectors v_{1}, …, v_{m} ∈ 𝔸^{m} of B such that lim_{λ→∞} v_{j}(λ), j = 1, …, m, exist and are linearly independent eigenvectors of Q_{0}. In particular Q_{0} is diagonalizable.

If mn > 1 and if, for all z ∈ ℂ, all solutions of Ly = zy are meromorphic, then there exists a differential expression P with coefficients in ℙ_{ω}^{m×m} 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 that ω = 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 expression U: The role of Halphen’s theorem (or, more precisely, Theorem 2.4 of [11]) is now played by Theorem 1 in [25] (quoted in Appendix A), which is a variant Floquet’s theorem. We have therefore the existence of m_{j} linearly independent functions

ψ_{j,k,ℓ}(x) = R_{j,k,ℓ}^{∗}(e^{ix}) exp(λx), ℓ = 1, …, m_{j}

where the components of R_{j,k,ℓ}^{∗} are rational functions. The substitution

y(x) = \frac{exp(λx)}{q(e^{ix})} ∑_{j=0}^{s} α_{j}e^{ix(s−j)}

turns the equation y′ = A(z_{j,k}, ·)y into a system of linear algebraic equation with m_{j} linearly independent solutions. This way one shows as before that

ψ_{j,k,ℓ}(x) = \frac{exp(λx)}{q(e^{ix})}(e^{six}1_{mn}, …, e^{ix}1_{mn}, 1_{mn}) ∑_{r=0}^{k_{j}−1} β̃_{ℓ,j,r}(λ)z_{j,k}^{r}

where the β̃_{ℓ,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 first m components of all the resulting vectors provides once more an m × m matrices S, Z, and Y = SZ exp(λx). Again the entries of S are polynomials with respect to λ but now they are rational functions with respect to e^{ix}. By considering the constant coefficient expression

L_{∞} = Q_{0} \frac{d^{n}}{dx^{n}} + ⋯ + Q_{n}^{∗}(∞)

one obtains also matrices S_{∞} and Y_{∞} = S_{∞}Z exp(λx) and U is defined as before through a multiple of S(λ, x)S_{∞}(λ)^{−1}. The investigation of the asymptotic behavior of Y and Y_{∞} as λ tends to infinity, which leads to proving the invertibility of U_{0}, is unchanged as it did not use the special structure of the Q_{j}, except that one should choose exp(ix_{0}) large rather than x_{0} large.

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

4. Application to the AKNS system

Let L = Jd/dx + Q(x), where

J = (\begin{matrix}i & 0 \\ 0 & −i \end{matrix}) and Q(x) = (\begin{matrix}0 & −iq(x) \\ ip(x) & 0 \end{matrix}) .

Note that J^{2} = −1_{2} and that JQ + 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 happen P_{n+1} has to be very special. It can be recursively computed in the following way: Let

P_{n+1} = ∑_{ℓ=0}^{n+1}(k_{n+1−ℓ}(x) + v_{n+1−ℓ}(x)J + W_{n+1−ℓ}(x))L^{ℓ},

where the k_{j} and v_{j} are scalar-valued and where the W_{j} are 2 × 2 matrices with vanishing diagonal elements. Requiring that [P_{n+1}, L] is a multiplication operator yields k′_{j} = 0 for j = 0, …, n + 1 and the recursion relations

W_{0} = 0

v′_{j}1_{2} = W_{j}Q + QW_{j}, W_{j+1} = \frac{1}{2}J(W′_{j} − 2v_{j}Q_{j}), j = 0, …, n + 1.

This gives finally

[P_{n+1}, L] = 2v_{n+1}JQ − JW′_{n+1}.

The first few AKNS equations are

Q_{t} = −c_{0}Q′ + 2c_{1}JQ,

Q_{t} = −\frac{c_{0}}{2} J(Q′′ − 2Q^{3}) − c_{1}Q′ + 2c_{2}JQ,

Q_{t} = \frac{c_{0}}{4} (Q′′′ − 6Q^{2}Q′) − c_{1}J(Q′′ − 2Q^{3}) − c_{2}Q′ + 2c_{3}JQ.

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

Definition 2. Suppose p and q are meromorphic functions. Then Q is called an algebro-geometric AKNS potential (or simply algebro-geometric) if Q is a stationary solution of some AKNS equation.

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

Theorem 3. Let Q = ( \begin{matrix}0 & −iq \\ ip & 0 \end{matrix} ) and assume either that p, q are rational functions bounded at infinity or else that p, q are meromorphic ω-periodic functions bounded at the ends of the period strip. Then Q is an algebro-geometric AKNS potential if and only if for all z ∈ ℂ all solutions of the equation Jy′+Qy = zy are 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 (Theorem 4 below) asks that p and q are meromorphic and provides one direction in the proof of Theorem 3. The second one (Theorem 5 below) is the analogue of Theorem 3 for the case of elliptic coefficients and is stated here for comparison purposes.

Theorem 4. Let Q = ( \begin{matrix}0 & −iq \\ ip & 0 \end{matrix} ) where p, q are meromorphic functions. If Q is an algebro-geometric AKNS potential then for all z ∈ ℂ all solutions of the equation Jy′ + Qy = zy are meromorphic with respect to the independent variable.

Theorem 5. Let Q = ( \begin{matrix}0 & −iq \\ ip & 0 \end{matrix} ) with p, q elliptic functions with a common period lattice. Then Q is an elliptic algebro-geometric AKNS potential if and only if for all z ∈ ℂ all solutions of the equation Jy′ + Qy = zy are meromorphic with respect to the independent variable.

Now we are ready to prove Theorem 3:

Proof of Theorem 3. We only need to prove that Q is algebro-geometric if all solutions of Ly = zy are meromorphic since the converse follows from Theorem 4. Suppose Q is periodic. The desired conclusion follows from Theorem 2 once we have checked its hypotheses. But Conditions 1 and 2 are satisfied by our assumptions while Conditions 3 and 4 hold automatically when the eigenvalues of Q_{0} (= J) are distinct. For convenience, however, let us mention that the eigenvalues of

B(λ) = (\begin{matrix}iλ & −iq^{∗}(∞) \\ ip^{∗}(∞) & −iλ \end{matrix})

are ±\sqrt{q^{∗}(∞)p^{∗}(∞) − λ^{2}} and that these are distinct for all but two values of λ. The eigenvectors may be chosen as

v_{1} = \frac{1}{2λ} (\begin{matrix}iλ + z_{1}(λ) \\ ip^{∗}(∞) \end{matrix}) and v_{2} = \frac{1}{2λ} (\begin{matrix}iq^{∗}(∞) \\ iλ + z_{1}(λ) \end{matrix})

where z_{1}(λ) is the branch of \sqrt{q^{∗}(∞)p^{∗}(∞) − λ^{2}} 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 p nor 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 of Q modulo periodicity, Theorem 3 provides an easy method which allows one to determine whether a rational function Q bounded at infinity or a meromorphic simply periodic function Q bounded 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 = ∑_{j=0}^{n} Q_{j}D^{n−j} where Q_{0} ∈ ℂ^{m×m} is invertible and Q_{j} ∈ 𝕄^{m×m} for j = 1, …, n. Suppose that there exist differential expressions

L_{∞} = ∑_{j=0}^{n} Q_{∞,j} D^{n−j} and U = ∑_{j=0}^{g} U_{j}(τ(x))D^{g−j}

with the following properties:

1. Q_{∞,0}, …, Q_{∞,n} are in ℂ^{m×m} and Q_{∞,0} = Q_{0}.

2. τ is a meromorphic function on ℂ.

3. There is a set Λ ⊂ ℂ with at least g + n + 1 distinct elements such that, for each λ ∈ Λ, the matrix B(λ) = ∑_{j=0}^{n} λ^{n−j}Q_{∞,j} has m linearly independent eigenvectors v_{1}(λ), …, v_{m}(λ) ∈ ℂ^{m} respectively associated with the (possibly degenerate) eigenvalues z_{1}(λ), …, z_{m}(λ).

4. U_{0}, …, U_{g} ∈ 𝕄^{m×m} and U_{0} is invertible.

5. U(v_{j}(λ) exp(λx)) is a solution of Ly = z_{j}(λ)y for j = 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 polynomial f ∈ F and each polynomial h ∈ 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 expression V = LU − UL_{∞} and fix λ ∈ Λ. Since

L_{∞}(v_{j}(λ) exp(λx)) = z_{j}v_{j}(λ) exp(λx)

we obtain

V (v_{j}(λ) exp(λx)) = (L − z_{j})U(v_{j}(λ) exp(λx)) = 0.

V is a differential expression of order g + n at most, i.e., V = ∑_{k=0}^{g+n} V_{k}(x)D^{k} for suitable matrices V_{k}. Hence

0 = exp(−λx)V (v_{j}(λ) exp(λx)) = (∑_{k=0}^{g+n} V_{k}(x)λ^{k}) v_{j}(λ).

For fixed x and λ we now have an m × m matrix Ṽ(λ, x) = ∑_{k=0}^{g+n} V_{k}(x)λ^{k} whose kernel contains all eigenvectors of B(λ) and is therefore m-dimensional. This means that Ṽ(λ, x) = 0. Since this is the case for at least g + n + 1 different values of λ we conclude that V_{0} = ⋯ = V_{g+n} = 0 and hence that

LU = UL_{∞}.

Since U_{0} is invertible Uy = 0 has mg linearly independent solutions. Let {y_{1}, …, y_{mg}} be a basis of ker(U). With each element y_{ℓ} of this basis we may associate a differential expression H_{ℓ} with coefficients in ℂ^{m×m} in the following way. Since y_{ℓ} ∈ ker(U), so is L_{∞}y_{ℓ} and, in fact, L_{∞}^{j}y_{ℓ} for every j ∈ ℕ. Since ker(U) is finitedimensional there exists a k ∈ ℕ and complex numbers α_{0}, …, α_{k} such that α_{0} ≠ 0 and

∑_{j=0}^{k} α_{k−j}L_{∞}^{j}y_{ℓ} = 0.

Then define H_{ℓ} = ∑_{j=0}^{k} α_{k−j}L_{∞}^{j}. Since the expressions H_{ℓ} commute among themselves we obtain that

ker(U) ⊂ ker (∏_{ℓ=1}^{mg} H_{ℓ}) .

Hence the set

F = {f ∈ ℂ[u] : ker(U) ⊂ ker(f(L_{∞}))}

is not empty.

Note that [L_{∞}, D] = 0 and [L_{∞}, C] = 0 if C ∈ C. For any h ∈ C[u] and any f ∈ 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 that U_{0} is invertible, [P, L] = 0. □

Lemma 2. Let

B(λ) = ∑_{j=0}^{n} λ^{n−j}B_{j}

where B_{0}, …, B_{n} ∈ ℂ^{m×m} and where B_{0} is invertible. Suppose the characteristic polynomial of B has the prime factorization ∏_{j=1}^{ν} f_{j}(λ, z)^{m_{j}}. If weight nr + s is assigned to the monomial λ^{s}z^{r}, then the weight of the heaviest monomial in f_{j} is a multiple of n, say nk_{j} and the coefficients of z^{k_{j}} and λ^{nk_{j}} in f_{j} are nonzero.

Let Λ be the set of all complex numbers λ satisfying the following two conditions:

1. (f_{1} … f_{ν})(λ, ·) has k_{1} + ⋯ + k_{ν} distinct roots.

2. If z_{j,k} is a root of f_{j}(λ, ·), then λ is a simple root of (f_{1} … f_{ν})(·, z_{j,k}).

Then the complement of Λ is finite.

Moreover, there is an integer h and there are complex numbers ρ_{j,k,r} such that, for sufficiently large λ, the roots of f_{j}(λ, ·), j = 1, …, ν, are given by

z_{j,k} = λ^{n} (ρ_{j,k,0} + ∑_{r=1}^{∞} ρ_{j,k,r}λ^{−r/h}) , k = 1, …, k_{j}

where the numbers ρ_{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 = g_{1}g_{2} and let f = ∑_{j=0}^{mn} α_{j}w_{j} where w_{j} is a polynomial all of whose terms have weight j. Doing the same with g_{1} and g_{2} one can show that any factor of f has a weight which is a multiple of n, say kn, 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 f_{j}(λ, ·) has k_{j} distinct roots for all but finitely many values of λ. Moreover, by Bezout’s theorem, the curves defined by f_{j} and f_{ℓ} intersect only in finitely many points if j is different from ℓ. Hence the first condition is satisfied for all but finitely many values of λ.

The discriminant of (f_{1} … f_{ν})(·, z) is a polynomial in z. Hence there are at most finitely many values of z for which (f_{1} … 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 (f_{1} … f_{ν})(·, z) has a multiple root.

The last statement follows from standard considerations of the behavior of algebraic functions near a point. In particular, the power n on λ is determined by an inspection of the Newton polygon associated with f_{j}. □

Lemma 3. Let

B(λ) = ∑_{j=0}^{n} λ^{n−j}B_{j}

where B_{0}, …, B_{n} ∈ ℂ^{m×m} and where B_{0} is invertible. Define

A(z) = (\begin{matrix}0 & 1_{m} & 0 & ⋯ & 0 \\ ⋮ & & & & ⋮ \\ 0 & & & ⋯ & 1_{m} \\ B_{0}^{−1}(z − B_{n}) & −B_{0}^{−1}B_{n−1} & −B_{0}^{−1}B_{n−2} & ⋯ & −B_{0}^{−1}B_{1} \end{matrix}) ,

a matrix whose n^{2} entries are m × m blocks.

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

u = ( v ; λv ; ⋮ ; λ^{n−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 k as an eigenvalue of A(z). Also,

(3) det(A(z) − λ) = (−1)^{nm} det(B_{0}^{−1}) det(B(λ) − z).

If B is diagonalizable (as an element of 𝔸^{m×m}), then A(z) is diagonalizable for all but finitely many values of z.

Moreover, if z_{j,k} is a zero of f_{j}(λ, ·), then there are complex numbers σ_{j,k,ℓ,r} and an integer h such that the eigenvalues μ_{j,k,1}, …, μ_{j,k,mn} of A(z_{j,k}) are given by

μ_{j,k,ℓ} = λ (σ_{j,k,ℓ,0} + ∑_{r=1}^{∞} σ_{j,k,ℓ,r}λ^{−r/h}) , ℓ = 1, …, mn

where the numbers σ_{j,k,ℓ,0} are different from zero.

Proof. That B(λ)v = zv if and only if A(z)u = λu follows immediately from direct computation. The validity of (3) is proven by blockwise Gaussian elimination.

Assume now that B is diagonalizable and let T ∈ 𝔸^{m×m} be an invertible matrix whose columns are eigenvectors of B. The determinant of T is an algebraic function in λ which is zero or infinity only for finitely many distinct values of λ and B(λ) is diagonalizable for all λ but these. From Lemma 2 we know also that there are only finitely many values of λ for which ∏_{j=1}^{ν} f_{j}(·, z) has repeated zeros. To all these exceptional values of λ correspond finitely many eigenvalues z of B(λ). We assume now that z is a complex number distinct from all those values. If μ is now an eigenvalue of A(z) then it is a zero of f_{j}(·, z) for some j but not a zero of f_{ℓ}(·, z), if ℓ ≠ j. Hence its algebraic multiplicity is m_{j}. Additionally, z is an eigenvalue of geometric multiplicity m_{j} of B(μ), since B(μ) is diagonalizable. The previous argument shows that μ has geometric multiplicity m_{j} as eigenvalue of A(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 that z_{j,k} is an algebraic function of λ (whose behavior near infinity is of the form given in Lemma 2) and that μ_{j,k,r} are algebraic functions of z_{j,k}. □

Lemma 4. Let R be an integral domain, Q its fraction field, g an element of R[z], and K a field extension of Q in which g splits into linear factors. Suppose A is a matrix in R[z]^{j×k}. Then there exist k vectors v_{1}, …, v_{k} ∈ R[z]^{k} with the following property: if z_{0} ∈ K is any of the roots of g and if the dimension of ker(A(z_{0})) is μ, then v_{1}(z_{0}), …, v_{μ}(z_{0}) are linearly independent solutions of A(z_{0})x = 0.

Proof. Suppose g has the prime factorization g_{1}^{m_{1}} … g_{ν}^{m_{ν}}. If g(z_{0}) = 0 then precisely one of the prime factors of g, say g_{ℓ}, satisfies g_{ℓ}(z_{0}) = 0. Note that F_{ℓ} = Q[z]/⟨g_{ℓ}⟩ is a field and that we may view A as an element of F_{ℓ}^{j×k}. Since F_{ℓ} is isomorphic to a subfield of K any K^{k}-solution of A(z_{0})x = 0 is a scalar multiple of a representative of an F_{ℓ}^{k}-solution of Ax = 0 (evaluated at z_{0}) and vice versa. Therefore there is a basis {x_{ℓ,1}(z_{0}), …, x_{ℓ,μ_{ℓ}}(z_{0})} of ker(A(z_{0})) where the x_{ℓ,r} are in Q[z]^{k}. By choosing appropriate multiples in R we may even assume that the x_{ℓ,r} are in R[z]^{k}. Notice that if z′_{0} is another root of g_{ℓ} then {x_{ℓ,1}(z′_{0}), …, x_{ℓ,μ_{ℓ}}(z′_{0})} is a basis of ker(A(z′_{0})). We define also x_{ℓ,r} = 0 for r = μ_{ℓ} + 1, …, k.

For r = 1, …, k we now let

v_{r} = ∑_{ℓ=1}^{ν} ( ∏_{\begin{matrix}ℓ′=1 \\ ℓ′≠ℓ \end{matrix}}^{ν} g_{ℓ′} ) x_{ℓ,r}.

This proves the lemma once we recall that g_{ℓ}(z_{0}) = 0 = g_{ℓ′}(z_{0}) implies that ℓ = ℓ′. □

Lemma 5. Suppose A ∈ ℂ^{mn×mn} and T ∈ 𝔸^{mn×mn} have the following properties: 1. The first (n − 1)m rows of A are zero.

2. T is invertible at and near infinity and its columns T_{1:mn,j} have the form

T_{1:mn,j} = ( v_{j} ; μ_{j}v_{j} ; ⋮ ; μ_{j}^{n−1}v_{j}) ,

where the μ_{j} are complex-valued algebraic functions of λ with the asymptotic behavior μ_{j}(λ) = λ(σ_{j} + o(1)) as λ tends to infinity and where the v_{j} are ℂ^{m}-valued algebraic functions of λ which are holomorphic at infinity.

Then (T^{−1}AT)(λ) is bounded as λ tends to infinity.

Proof. The first (n − 1)m rows of AT are zero. Consequently we need to consider only the last m columns of T^{−1}. Let B_{n}, …, B_{1} denote the m × m matrices which occupy the last m rows of A (with decreasing index as one moves from left to right) and let τ_{ℓ}^{∗} denote the row-vector in the last m columns of row ℓ in T^{−1} (note that τ_{ℓ} ∈ 𝔸^{m}). Then

(T^{−1}AT)_{ℓ,k} = ∑_{j=1}^{n} μ_{k}^{n−j}τ_{ℓ}^{∗}B_{j}v_{k}.

We will show below that τ_{ℓ} has the asymptotic behavior τ_{ℓ} = λ^{1−n}(τ_{0,ℓ} +o(1)) with τ_{0,ℓ} ∈ ℂ^{m} as λ tends to infinity. Hence

(T^{−1}AT)_{ℓ,k} = ∑_{j=1}^{n} λ^{1−j}(σ_{k}^{n−j}τ_{0,ℓ}^{∗}B_{j}v_{k}(∞) + o(1)) = σ_{k}^{n−1}τ_{0,ℓ}^{∗}B_{1}v_{k}(∞) + o(1)

as λ tends to infinity and this will prove the claim.

The minor of T which arises when one deletes row r and columns s of T will be denoted by M_{s,r}. We have then that

(T^{−1})_{r,s} = \frac{(−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}, equals λ^{α} times a function which is bounded as λ tends to infinity. Hence det(T) = λ^{N}(t_{0} + o(1)) for some nonzero complex number t_{0} 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 − α and m_{mα+β,r} ∈ ℂ. Hence

(T^{−1})_{r,s} = (−1)^{r+s}λ^{−α} \frac{m_{mα+β,r} + o(1)}{t_{0}}.

For the first part of the proof we need only the case α = n − 1. □

Lemma 6. Let Ω ⊂ ℂ be an open simply connected set containing x_{0} and S ⊂ ℂ a sector centered at zero. Suppose that A : S × Ω → ℂ^{n×n} is holomorphic on S × Ω and admits a uniform asymptotic expansion

A(ρ, x) ∼ ∑_{r=0}^{∞} A_{r}(x)ρ^{r}

as ρ tends to zero. Suppose that, for r = 0, …, h − 1, the matrices A_{r} are constant and have the block-diagonal form

A_{r} = (\begin{matrix}σ_{r}1_{p_{r}} & 0 \\ 0 & Σ_{r} \end{matrix})

where p_{0} ≥ p_{1} ≥ ⋯ ≥ p_{h−1} = p and where σ_{r} is not an eigenvalue of Σ_{r}. Denote the upper left p × p block of A_{h} by A_{h;1,1} and assume that A_{h;1,1}(x) = Γ + Δ(x) where Γ ∈ ℂ^{p×p} and Δ : Ω → ℂ^{p×p}. Let α = ∑_{r=0}^{h−1} σ_{r}ρ^{r−h}.

Then there exists a subsector S′ of S and an n×p matrix Y (ρ, x) whose columns are linearly independent solutions of ρ^{h}y′ = Ay and for which

R(ρ, x) = Y (ρ, x) exp(−αx)

has in S′ an asymptotic expansion of the form

R(ρ, x) ∼ ∑_{j=0}^{∞} R_{j}(x)ρ^{j}

as ρ tends to zero. Moreover, for every positive ε there exists a positive δ such that ∥Δ(x)∥ < δ for all x ∈ Ω implies

R_{0}(x) = (\begin{matrix}exp(Γ(x − x_{0}))(1_{p} + Υ(x)) \\ 0_{(n−p)×p} \end{matrix})

with ∥Υ(x)∥ < ε for all x ∈ Ω.

Proof. The key to the proof of this lemma is Theorem 26.2 in Wasow [23] which we have (essentially) quoted in Appendix C and which implies immediately Corollary 2. A repeated application of this corollary shows that there are p linearly independent solutions y_{j} of ρ^{h}y′ = Ay of the form

y_{j} = P_{0}Q_{0} … P_{h−1}Q_{h−1}w_{j} exp(αx), j = 1, …, p

where the P_{k} and Q_{k} are matrices and where the w_{j} are vectors whose properties are described presently. Let p_{−1} = n. Then P_{k} is an p_{k−1} × p_{k−1} matrix which is asymptotically equal to 1_{p_{k−1}}. The matrix Q_{k} is a constant p_{k−1} × p_{k} matrix whose upper block is equal to 1_{p_{k}} and whose lower block is a zero matrix. Finally, the w_{j} are linearly independent solutions of the p × p-system w′ = B(ρ, x)w where

B(ρ, x) = ρ^{−h}Q^{∗} (A(ρ, x) − ∑_{r=0}^{h−1} A_{r}(x)ρ^{r}) Q.

Note that B(ρ, x) has the asymptotic behavior

B(ρ, x) ∼ ∑_{r=0}^{∞} B_{r}(x)ρ^{r}

as ρ tends to zero where B_{0}(x) = A_{h;1,1}(x).

The equation w′ = B(ρ, x)w has a fundamental matrix W whose asymptotic behavior is given by

W (ρ, x) ∼ ∑_{r=0}^{∞} W_{r}(x)ρ^{r}

where

W_{0}(x) = exp (∫_{x_{0}}^{x} A_{h;1,1}(t)dt) = exp (Γ(x − x_{0}) + ∫_{x_{0}}^{x} Δ(t)dt)

is an invertible matrix. Since ∥ exp(T_{1} + T_{2}) − exp(T_{1})∥ ≤ ∥T_{2}∥ exp(∥T_{1}∥ + ∥T_{2}∥) we have that

W_{0}(x) = exp(Γ(x − x_{0}))(1_{p} + Υ(x))

where the norm of Υ becomes small if the norm of Δ becomes small. The fact that the matrices P_{k} are asymptotically equal to identity matrices and that the upper blocks of the Q_{k} are equal to identity matrices gives now the desired conclusion. □

Lemma 7. Suppose that 𝕄, L, L_{∞}, U, C, and F are as in Lemma 1. Given an expression P_{∞} let P be defined by P U = UP_{∞}. Then the following two statements hold:

1. Let P_{∞} = Df(L_{∞}), where f is a monic polynomial in F. If (P, L) is a trivially commuting pair, then Q_{0} and Q_{∞,1} are multiples of the identity. Moreover, there exists a first order differential expression G = D + Q_{0}^{−1}(Q_{1} − η_{1})/n (where η_{1} is a suitable constant) such that both P and L are polynomials of G.

2. Let P_{∞} = Cf(L_{∞}), where C ∈ C and where f ∈ F is monic. If both P and L are polynomials of an expression D + G_{1} then C is a multiple of the identity matrix.

Proof. Assume that

L = ∑_{j=0}^{n} Q_{j}D^{n−j} = ∑_{j=0}^{n′} η_{j}G^{n′−j} and P = ∑_{j=0}^{r} P_{j}D^{r−j} = ∑_{j=0}^{r′} γ_{j}G^{r′−j},

where G is a differential expression of order k and the coefficients η_{j} and γ_{j} are complex numbers.

To prove the first statement assume also that P_{∞} = Df(L_{∞}) where f ∈ F has degree s. Since the order of L is equal to n = kn′ and the order of P is equal to r = sn + 1 = kr′ we have necessarily k = 1, n′ = n, and r′ = r = sn + 1. Therefore we assume now that G = G_{0}D + G_{1}.

Note that LU = UL_{∞}, P U = UP_{∞}, and Q_{∞,0} = Q_{0} imply that

U_{0}Q_{0} = Q_{0}U_{0}, U_{0}P_{∞,0} = P_{0}U_{0}, (4)

U_{1}Q_{0} + U_{0}Q_{∞,1} = Q_{0}U_{1} + Q_{1}U_{0} + nQ_{0}U′_{0}, (5)

and that

U_{1}P_{0} + U_{0}P_{∞,1} = P_{0}U_{1} + P_{1}U_{0} + rP_{0}U′_{0}, (6)

where P_{∞,j} is the coefficient of D^{r−j} in P_{∞}. Since P_{∞,0} = Q_{0}^{s} we find firstly that P_{0} = U_{0}Q_{0}^{s}U_{0}^{−1} = Q_{0}^{s}. Next, since Q_{0} = η_{0}G_{0}^{n} and P_{0} = γ_{0}G_{0}^{sn+1}, we have that G_{0} = η_{0}^{s}γ_{0}^{−1}1. Hence G_{0}, Q_{0}, and P_{0} are all multiples of the identity matrix. Therefore we can (and will) assume from now that G_{0} = 1 by changing the coefficients η_{j} and γ_{j} appropriately. In particular, Q_{0} = η_{0}1.

We find next that

Q_{1} = (nη_{0}G_{1} + η_{1}1), P_{1} = (rγ_{0}G_{1} + γ_{1}1),

and that

P_{∞,1} = (sQ_{∞,1} + κ)Q_{0}^{s−1}

where κ is the coefficient of u^{s−1} in f(u) if n = 1 and κ = 0 if n > 1. Inserting these expressions into (5) and (6) and eliminating the terms with U′_{0} gives

Q_{∞,1} = [nκ + η_{0}^{−s}(rη_{1}γ_{0} − nγ_{1}η_{0})]1.

We also obtain that G_{1} = Q_{0}^{−1}(Q_{1} − η_{1})/n. This proves the first statement of the lemma.

To prove the second statement, let G = D + G_{1}. This implies, as before, that Q_{0} = η_{0}1 and P_{∞,0} = P_{0} = γ_{0}1. On the other hand, since P_{∞} = Cf(L_{∞}), we have that P_{∞,0} = Cη_{0}^{s}. Thus C is a multiple of the identity. □

Appendix A. The theorems of Halphen and Floquet

The proofs of Theorems 1 and 2 rely on results of Halphen [12] and Floquet [6], [7], or rather on generalizations to systems of their results. These generalizations were proven in [11] and [25], respectively, and are repeated here for the convenience of the reader.

Theorem 6. Let A ∈ ℂ(x)^{n×n} with entries bounded at infinity and suppose that the first-order system y′ = Ay has a meromorphic fundamental system of solutions. Then y′ = Ay has a fundamental matrix of the type

Y (x) = R(x) exp(diag(λ_{1}x, …, λ_{n}x)),

where λ_{1}, …, λ_{n} are the eigenvalues of A(∞) and R ∈ ℂ(x)^{n×n}.

Theorem 7. Suppose that A is an n × n-matrix whose entries are meromorphic, ω-periodic functions which are bounded at the ends of the period strip. If the firstorder system y′ = Ay has only meromorphic solutions, then there exists a constant n × n-matrix J in Jordan normal form and an n × n-matrix R^{∗} whose entries are rational functions over ℂ such that the following statements hold:

1. The eigenvalues of A^{∗}(0) and J are the same modulo iℤ if multiplicities are properly taken into account. More precisely, suppose that there are nonnegative integers ν_{1}, …, ν_{r−1} such that λ, λ + iν_{1}, …, λ + iν_{r−1} are all the eigenvalues of A^{∗}(0) which are equal to λ modulo iℤ. Then λ is an eigenvalue of J with algebraic multiplicity r.

2. The equation y′ = Ay has a fundamental matrix Y given by

Y (x) = R^{∗}(e^{2πix/ω}) exp(Jx).

In particular every entry of Y has the form f(e^{2πix/ω}, x)e^{λx}, where λ + iν is an eigenvalue of A^{∗}(0) for some nonnegative integer ν and where f is rational function in its first argument and a polynomial in its second argument.

Appendix B. Higher order systems of differential equations

In this section we recall two basic facts about systems of linear differential equations of order higher than one.

Consider the system

T y = T_{0}(x)y^{(n)} + T_{1}(x)y^{(n−1)} + ⋯ + T_{n}(x)y = 0 (7)

where the T_{j} are m × m matrices whose entries are continuous functions on some real interval or complex domain Ω and where T_{0}(x) is invertible for every x ∈ Ω. Using the analogue of the standard transformation which turns a higher order scalar equation into a first order system, one finds that the system (7) is equivalent to the first order system u′ = Au where A is the mn × mn matrix

(\begin{matrix}0 & 1_{m} & 0 & ⋯ & 0 \\ ⋮ & & & & ⋮ \\ 0 & & & ⋯ & 1_{m} \\ −T_{0}^{−1}T_{n} & −T_{0}^{−1}T_{n−1} & −T_{0}^{−1}T_{n−2} & ⋯ & −T_{0}^{−1}T_{1} \end{matrix})

in which all entries represent m × m blocks. From this it follows immediately that a fundamental system of solutions of T y = 0 has mn elements.

The other property we need is about the existence of a factorization of an n-th order expression into n first order factors.

Theorem 8. Let T be the differential expression defined in (7). Suppose that F_{1}, …, F_{n−1} are m × m matrices whose entries are continuous functions on Ω and which are invertible for every x ∈ Ω. Define F_{n} = T_{0}F_{1}^{−1} … F_{n−1}^{−1}. Then there exist m × m matrices Φ_{1}, …, Φ_{n} such that

T = (F_{n}D − Φ_{n}) … (F_{1}D − Φ_{1}).

Proof. Denote the elements of a fundamental system of solutions by y_{1}, …, y_{mn} and define, for j = 1, …, n, the m × m matrices

Y_{j} = (y_{m(j−1)+1}, …, y_{mj}).

Next define W_{1} = Y_{1} and Φ_{1} = F_{1}Y′_{1}Y_{1}^{−1} and suppose we have determined matrices Φ_{1}, …, Φ_{j−1}. We will show below that

W_{j} = (F_{j−1}D − Φ_{j−1}) … (F_{1}D − Φ_{1})Y_{j}

is invertible so that we can define

Φ_{j} = F_{j}W′_{j}W_{j}^{−1}.

Now let

S = (F_{n}D − Φ_{n}) … (F_{1}D − Φ_{1}).

Then S(Y_{j}) = (F_{n}D − Φ_{n}) … (F_{j}D − Φ_{j})W_{j} = 0, i.e., S and T have the same solutions. S and T are therefore equivalent to the same first order system. Since they have the same leading coefficient we finally obtain S = T.

We complete the proof by showing that the matrices W_{j} are invertible, i.e., that their columns W_{j,1}, …, W_{j,m} are linearly independent. This is true for j = 1 since the columns of W_{1} are the solutions y_{1}, …, y_{m} which are linearly independent. Assume that W_{1}, …, W_{j−1} are invertible and that

0 = ∑_{k=1}^{m} α_{k}W_{j,k}.

Then

0 = (F_{j−1}D − Φ_{j−1}) … (F_{1}D − Φ_{1}) ∑_{k=1}^{m} α_{k}y_{m(j−1)+k}.

Since the space of solutions of (F_{j−1}D − Φ_{j−1}) … (F_{1}D − Φ_{1})y = 0 is spanned by y_{1}, …, y_{m(j−1)} we obtain that

∑_{k=1}^{m} α_{k}y_{m(j−1)+k} = ∑_{ℓ=1}^{m(j−1)} β_{ℓ}y_{ℓ}.

But since y_{1}, …, y_{mj} are linearly independent it follows that all α_{1} = ⋯ = α_{n} = 0 (and β_{1} = ⋯ = β_{m(j−1)} = 0). Hence the columns of W_{j} are linearly independent and W_{j} is invertible. □

Corollary 1. Let S and T be differential expressions with matrix coefficients and invertible leading coefficients. If ker S ⊂ ker T then there exists a differential expression R such that RS = T.

Appendix C. Wasow’s theorem

For the reader’s convenience we provide here a slightly adapted version of Theorem 26.2 in Wasow [23]. The adaptation makes use of formulas (25.19) and (25.20) in [23].

Theorem 9. Let Ω ⊂ ℂ be an open simply connected set containing the point x_{0} and let S be a sector {ρ : 0 < |ρ| < ρ_{0}, α_{0} < arg(ρ) < β_{0}}. Suppose that A : S × Ω → ℂ^{n×n} is holomorphic and admits a uniform asymptotic expansion

A(ρ, x) ∼ ∑_{r=0}^{∞} A_{r}(x)ρ^{r}

on S ×Ω. Furthermore suppose that A_{0} is diagonal, i.e., A_{0} = diag(λ_{1}, …, λ_{n}) and that the sets {λ_{1}(x_{0}), …, λ_{p}(x_{0})} and {λ_{p+1}(x_{0}), …, λ_{n}(x_{0})} are disjoint. Then there exists a subsector S^{∗} of S and a subregion Ω^{∗} of Ω and ℂ^{n×n}-valued functions P and B with the following properties:

1. P and B are holomorphic in S^{∗} × Ω^{∗}.

2. P − 1 and B have the block forms

P (ρ, x) − 1 = (\begin{matrix}0 & P_{1,2}(ρ, x) \\ P_{2,1}(ρ, x) & 0 \end{matrix})

and

B(ρ, x) = (\begin{matrix}B_{1,1}(ρ, x) & 0 \\ 0 & B_{2,2}(ρ, x) \end{matrix}) ,

and the blocks have asymptotic expansion P_{j,k}(ρ, x) ∼ ∑_{r=1}^{∞} P_{r;j,k}(x)ρ^{r} and B_{j,j}(ρ, x) ∼ ∑_{r=0}^{∞} B_{r;j,j}(x)ρ^{r}, as ρ tends to zero.

3. B_{0} = A_{0} and A_{0}P_{1} − P_{1}A_{0} = B_{1} − A_{1}.

On Commuting Matrix Differential Operators 29

4. the transformation y = P z takes the differential equation ρ^{h}y′ = Ay into ρ^{h}z′ = By.

Corollary 2. If λ_{1}(x) = ⋯ = λ_{p}(x) = σ for all x ∈ Ω and

B̃(ρ, x) = \frac{1}{ρ}(B_{1,1} − B_{0;1,1}(x)) ∼ ∑_{r=1}^{∞} B_{r;j,j}(x)ρ^{r−1},

then the equation ρ^{h}y′ = Ay has p linearly independent solutions of the form y(x) = P Qw(x) exp(σxρ^{−h}), where

Q = (\begin{matrix}1_{p×p} \\ 0_{(n−p)×p} \end{matrix})

and w is a solution of the p × p system

ρ^{h−1}w′ = B̃(ρ, x)w.

References

1. J.L. Burchnall and T.W. Chaundy, Commutative ordinary differential operators, Proc. London Math. Soc. Ser. 2 21 (1923), 420–440.

2. J.L. Burchnall and T.W. Chaundy, Commutative ordinary differential operators, Proc. Roy. Soc. London A 118 (1928), 557–583.

3. I. V. Cherednik, Regularity of “finite-zone” solutions of integrable matrix differential equations, Soviet Phys. Dokl. 27 (1982), 716–718, MR 84f:35123, Zbl 0552.35017.

4. L. A. Dickey, On the τ-function of matrix hierarchies of integrable equations, J. Math. Phys. 32 (1991), 2996–3002, MR 93d:58068, Zbl 0737.35097.

5. B. A. Dubrovin, Completely integrable Hamiltonian systems associated with matrix operators and abelian varieties, Funct. Anal. Appl. 11 (1977), 265–277, Zbl 0413.58012.

6. G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques, C. R. Acad. Sci. Paris 91 (1880), 880–882.

7. G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques, Ann. Sci. École Norm. Sup. 12 (1883), 47–88.

8. F. Gesztesy and R. Weikard, Picard potentials and Hill’s equation on a torus, Acta Math. 176 (1996), 73–107, MR 97f:14046.

9. F. Gesztesy and R. Weikard, A characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy, Acta Math. 181 (1998), 63–108, MR 99k:14052, Zbl 0955.34073.

10. F. Gesztesy and R. Weikard, Elliptic algebro-geometric solutions of the KdV and AKNS hierarchies—an analytic approach, Bull. Amer. Math. Soc. (N.S.) 35 (1998), 271–317, MR 99i:58075, Zbl 0909.34073.

11. F. Gesztesy, K. Unterkofler and R. Weikard, On a theorem of Halphen and its application to integrable systems, J. Math. Anal. Appl. 251 (2000), 504–526, MR 2001i:37108, Zbl 0966.34078.

12. G.-H. Halphen, Sur une nouvelle classe d’équations différentielles linéaires intégrables, C. R. Acad. Sci. Paris 101 (1885), 1238–1240.

13. A. R. Its and V. B. Matveev, Schrödinger operators with the finite-band spectrum and the N-soliton solutions of the Korteweg-de Vries equation, Theoret. and Math. Phys. 23 (1975), 343–355, MR 57 #18570.

14. I. M. Krichever, Algebraic curves and commuting matricial differential operators, Funct. Anal. Appl. 10 (1976), 144–146, Zbl 0347.35077.

15. I. M. Krichever, Integration of nonlinear equations by the methods of algebraic geometry, Funct. Anal. Appl. 11 (1977), 12–26, Zbl 0368.35022.

16. I. M. Krichever, Methods of algebraic geometry in the theory of nonlinear equations, Russ. Math. Surv. 32 (1977), 185–213. See also: Integrable Systems, Selected Papers, Lond. Math. Soc. Lect. Note Ser. 60, Lond. Math. Soc, 1981, 141–169, Zbl 0461.35075.

17. P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Commun. Math. Phys. 21 (1968), 467–490, MR 38 #3620, Zbl 0162.41103.

30 Rudi Weikard

18. A. I. Markushevich, Theory of Functions in a Complex Variable (three volumes in one), Chelsea 1965.

19. P. van Moerbeke, Integrable foundations of string theory, Lectures on integrable systems (Sophia-Antipolis, 1991), 163–267, World Sci. Publishing, River Edge, NJ, 1994.

20. D. Mumford, An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg deVries equation and related nonlinear equation, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), 115–153, Tokyo, 1978, MR 83j:14041, Zbl 0423.14007.

21. E. Picard, Sur une classe d’équations différentielles linéaires, C. R. Acad. Sci. Paris 90 (1880), 128–131.

22. A. Treibich, Matrix elliptic solitons, Duke Math. J. 90 (1997), 523–547, MR 98m:14030, Zbl 0909.35116.

23. W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Interscience 1965, MR 34 #3041.

24. R. Weikard, On commuting differential operators, Electron. J. Differential Equations 2000 (2000), No. 19, 1–11, MR 2001a:34146, Zbl 0953.34073.

25. R. Weikard, Floquet theory for linear differential equations with meromorphic solutions, E. J. Qualitative Theory of Diff. Equ., No. 8. (2000), 1–6, MR 2001j:34113, Zbl 0969.34073.

Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170, USA

rudi@math.uab.edu http://www.math.uab.edu/rudi/

This paper is available via http://nyjm.albany.edu:8000/j/2002/8-2.html.