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A CHARACTERIZATION OF ALL ELLIPTIC SOLUTIONS OF THE AKNS HIERARCHY

F. GESZTESY1 AND R. WEIKARD2

Abstract. An explicit characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy is presented. More precisely, we show that a pair of elliptic functions (p; q) is an algebro-geometric AKNS potential, that is, a solution of some equation of the stationary AKNS hierarchy, if and only if the associated linear i 0 di erential system J + Q = E , where J = 0 ?i , Q = ip0x) ?iq(x) , ( 0 has a fundamental system of solutions which are meromorphic with respect to the independent variable for in nitely many and hence for all values of the spectral parameter E 2 C . Our approach is based on (an extension of) a classical theorem of Picard, which guarantees the existence of solutions which are elliptic of the second kind for nthorder ordinary di erential equations with elliptic coe cients associated with a common period lattice. The fundamental link between Picard's theorem and elliptic algebro-geometric solutions of completely integrable hierarchies of nonlinear evolution equations has recently been established in connection with the KdV hierarchy. The current investigation appears to be the rst of its kind associated with matrixvalued Lax pairs. As by-products we o er a detailed Floquet analysis of Dirac-type di erential expressions with periodic coe cients speci cally emphasizing algebrogeometric coe cients and a constructive reduction of singular hyperelliptic curves and their Baker-Akhiezer functions to the nonsingular case.

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1. Introduction Before describing our approach in some detail, we shall give a brief account of the history of the problem of characterizing elliptic algebro-geometric solutions of completely integrable systems. This theme dates back to a 1940 paper of Ince 50] who studied what is presently called the Lame{Ince potential q(x) = ?n(n + 1)}(x + !3); n 2 N ; x 2 R (1.1)

Date : May 14. Based upon work supported by the US National Science Foundation under Grants No. DMS9623121 and DMS-9401816. 1991 Mathematics Subject Classi cation. Primary 35Q55, 34L40; Secondary 58F07. Key words and phrases. Elliptic algebro-geometric solutions, AKNS hierarchy, Floquet theory, Dirac-type operators.

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in connection with the second-order ordinary di erential equation

y00(E; x) + q(x)y(E; x) = Ey(E; x); E 2 C :

(1.2)

Here }(x) = }(x; !1; !3) denotes the elliptic Weierstrass function with fundamental periods 2!1 and 2!3 (Im(!3=!1) 6= 0). In the special case where !1 is real and !3 is purely imaginary, the potential q(x) in (1.1) is real-valued and Ince's striking result 50], in modern spectral theoretic terminology, yields that the spectrum of the unique self-adjoint operator associated with the di erential expression L2 = d2=dx2 + q(x) in L2 (R ) exhibits nitely many bands (respectively gaps), that is, (L2) = (?1; E2n ]

n m=1

E2m?1 ; E2m?2 ] ; E2n < E2n?1 < : : : < E0 :

(1.3)

What we call the Lame{Ince potential has, in fact, a long history and many investigations of it precede Ince's work 50]. Without attempting to be complete we refer the interested reader, for instance, to 2], 3], Sect. 59, 6], Ch. IX, 9], Sect. 3.6.4, 18], Sects. 135{138, 19], 20], 22], 39], 48], p. 494{498, 49], p. 118{122, 266{418, 475{478, 51], p. 378{380, 54], 57], p. 265{275, 71], 73], 98], 100], 102], Ch. XXIII as pertinent publications before and after Ince's fundamental paper. Following the traditional terminology, any real-valued potential q that gives rise to a spectrum of the type (1.3) is called an algebro-geometric KdV potential. The proper extension of this notion to general complex-valued meromorphic potentials q then proceeds via the KdV hierarchy of nonlinear evolution equations obtained from appropriate Lax pairs (P2n+1(t); L2 (t)), with L2 (t) = d2=dx2 + q(x; t), P2n+1(t) a di erential expression of order 2n + 1, whose coe cients are certain di erential polynomials in q(x; t) (i.e., polynomials in q and its x-derivatives), and t 2 R an additional deformation parameter. Varying n 2 N f0g, the collection of all Lax equations d L = P ; L ] ; that is, q = P ; L ] (1.4) 2n+1 2 t 2n+1 2 dt 2 then de nes the celebrated KdV hierarchy. In particular, q(x; t) is called an algebrogeometric solution of (one of) the nth equation in (1.4) if it satis es for some (and 0 hence for all) xed t0 2 R one of the higher-order stationary KdV equations in (1.4) associated with some n1 n0. Therefore, without loss of generality, one can focus on characterizing stationary elliptic algebro-geometric solutions of the KdV hierarchy (and similarly in connection with other hierarchies of soliton equations). The stationary KdV hierarchy, characterized by qt = 0 or P2n+1 ; L2] = 0, is intimately connected with the question of commutativity of ordinary di erential expressions. In particular, if P2n+1; L] = 0, a celebrated theorem of Burchnall and Chaundy

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16], 17] implies that P2n+1 and L2 satisfy an algebraic relationship of the form

P22n+1

=

2n Y

and hence de ne a (possibly singular) hyperelliptic curve (branched at in nity)

m=0

(L2 ? Em ) for some fEmg2n=0 m

C

(1.5) (1.6)

w2 =

2n Y

It is the curve (1.6), which signi es that q in L2 = d2=dx2 + q(x) represents an algebro-geometric KdV potential. While these considerations pertain to general solutions of the stationary KdV hierarchy, we now concentrate on the additional restriction that q be an elliptic function (i.e., meromorphic and doubly periodic) and hence return to the history of elliptic algebro-geometric potentials q for L2 = d2 =dx2 + q(x), or, equivalently, elliptic solutions of the stationary KdV hierarchy. Ince's remarkable algebro-geometric result (1.3) remained the only explicit elliptic algebro-geometric example until the KdV ow 3 1 qt = 4 qxxx + 2 qqx with the initial condition q(x; 0) = ?6}(x) was explicitly integrated by Dubrovin and Novikov 26] in 1975 (see also 29]{ 31], 53]), and found to be of the type

m=0

(E ? Em ):

q(x; t) = ?2

3 X

for appropriate fxj (t)g1 j 3. Given these results it was natural to ask for a systematic account of all elliptic solutions of the KdV hierarchy, a problem posed, for instance, in 70], p. 152. In 1977, Airault, McKean and Moser, in their seminal paper 1], presented the rst systematic study of the isospectral torus IR(q0) of real-valued smooth potentials q0 (x) of the type

j =1

}(x ? xj (t))

(1.7)

q0 (x) = ?2

M X j =1

}(x ? xj )

(1.8)

with an algebro-geometric spectrum of the form (1.3). In particular, the potential (1.8) turned out to be intimately connected with completely integrable many-body systems of the Calogero-Moser-type 19], 67] (see also 20], 22]). This connection with integrable particle systems was subsequently exploited by Krichever 58] in his fundamental construction of elliptic algebro-geometric solutions of the KadomtsevPetviashvili equation. The next breakthrough occurred in 1988 when Verdier 99] published new explicit examples of elliptic algebro-geometric potentials. Verdier's examples spurred a urry of activities and inspired Belokolos and Enol'skii 11], Smirnov

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84], and subsequently Taimanov 90] and Kostov and Enol'skii 56] to nd further such examples by combining the reduction process of Abelian integrals to elliptic integrals (see 7], 8], 9], Ch. 7, 10]) with the aforementioned techniques of Krichever 58], 59]. This development nally culminated in a series of recent results of Treibich and Verdier 95], 96], 97], where it was shown that a general complex-valued potential of the form

q(x) = ?

4 X

(!2 = !1 + !3; !4 = 0) is an algebro-geometric potential if and only if dj =2 are triangular numbers, that is, if and only if dj = gj (gj + 1) for some gj 2 Z; 1 j 4: (1.10) We shall refer to potentials of the form (1.9), (1.10) as Treibich-Verdier potentials. The methods of Treibich and Verdier are based on hyperelliptic tangent covers of the torus C = ( being the period lattice generated by 2!1 and 2!3). The state of the art of elliptic algebro-geometric solutions up to 1993 was recently reviewed in issues 1 and 2 of volume 36 of Acta Applicandae Math., see, for instance, 12], 33], 60], 86], 91], 94] and also in 13], 24], 25], 32], 47], 81], 87], 93]. In addition to these investigations on elliptic solutions of the KdV hierarchy, the study of other soliton hierarchies, such as the modi ed KdV hierarchy, nonlinear Schrodinger hierarchy, and Boussinesq hierarchy has also begun. We refer, for instance, to 21], 28], 39], 40], 63], 64], 66], 79], 80], 82], 83], 85], 88]. Despite these (basically algebro-geometric) approaches described thus far, an e cient characterization of all elliptic solutions of the KdV hierarchy remained elusive until recently. The nal breakthrough in this characterization problem in 43], 44] became possible due to the application of the most powerful analytic tool in this context, a theorem of Picard. This result of Picard (cf. Theorem 6.1) is concerned with the existence of solutions which are elliptic of the second kind of nth -order ordinary di erential equations with elliptic coe cients. The main hypothesis in Picard's theorem for a second-order di erential equation of the form y00 (x) + q(x)y(x) = Ey(x); E 2 C ; (1.11) with an elliptic potential q, relevant in connection with the KdV hierarchy (cf. the second-order di erential expression L2 in (1.4)), assumes the existence of a fundamental system of solutions meromorphic in x. Hence we call any elliptic function q which has this property for all values of the spectral parameter a Picard-KdV potential. The characterization of all elliptic algebro-geometric solutions of the stationary KdV hierarchy, then reads as follows: Theorem 1.1. ( 43], 44]) q is an elliptic algebro-geometric potential if and only if it is a Picard-KdV potential.

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j =1

dj }(x ? !j )

(1.9)

In particular, Theorem 1.1 sheds new light on Picard's theorem since it identi es the elliptic coe cients q for which there exists a meromorphic fundamental system of solutions of (1.11) precisely as the elliptic algebro-geometric solutions of the stationary KdV hierarchy. Moreover, we stress its straightforward applicability based on an elementary Frobenius-type analysis which decides whether or not (1.11) has a meromorphic fundamental system for each E 2 C . Related results and further background information on our approach can be found in 38], 39]{ 41], 42], 45]. After this somewhat detailed description of the history of the problem under consideration, we now turn to the content of the present paper. The principal objective in this paper is to prove an analogous characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy and hence to extend the preceding formalism to matrix-valued di erential expressions. More precisely, replace the scalar second-order di erential equation (1.2) by the 2 2 rst-order system J 0(E; x) + Q(x) (E; x) = E (E; x); E 2 C ; (1.12) where (E; x) = ( 1 (E; x); 2(E; x))t (\t" abbreviating transpose) and i 0 J = 0 ?i ; (1.13) 0 Q(x) = Q1;1 (x) Q1;2(x) = ip(x) ?iq(x) : (1.14) Q2;1 (x) Q2;2(x) 0 Similarly, replace the scalar KdV di erential expression L2 by the 2 2 matrix-valued di erential expression L(t) = Jd=dx + Q(x; t), t a deformation parameter. The AKNS hierarchy of nonlinear evolution equations is then constructed via appropriate Lax pairs (Pn+1(t); L(t)), where Pn+1(t) is a 2 2 matrix-valued di erential expression of order n + 1 (cf. Section 2 for an explicit construction of Pn+1). In analogy to the KdV hierarchy, varying n 2 N f0g, the collection of all Lax equations d L = P ; L] ; (1.15) n+1 dt then de nes the AKNS hierarchy of nonlinear evolution equations for (p(x; t); q(x; t)). Algebro-geometric AKNS solutions are now introduced as in the KdV context and stationary AKNS solutions, characterized by pt = 0; qt = 0 or Pn+1; L] = 0, again yield an algebraic relationship between Pn+1 and L of the type

2 Pn+1 = 2Y n+1

and hence a (possibly singular) hyperelliptic curve (not branched at in nity)

m=0

+1 (L ? Em) for some fEm g2n=0 m 2Y n+1

C

(1.16) (1.17)

w2

=

m=0

(E ? Em):

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In order to characterize all elliptic solutions of the AKNS hierarchy we follow our strategy in the KdV context and consider 2 2 rst-order systems of the form J 0(x) + Q(x) (x) = E (x); E 2 C ; (1.18) with Q an elliptic 2 2 matrix of the form (1.14). Again we single out those elliptic Q such that (1.18) has a fundamental system of solutions meromorphic in x for all values of the spectral parameter E 2 C and call such Q Picard-AKNS potentials. Our principal new result in this paper, a characterization of all elliptic algebro-geometric solutions of the stationary AKNS hierarchy, then simply reads as follows: Theorem 1.2. An elliptic potential Q is an algebro-geometric AKNS potential if and only if it is a Picard-AKNS potential (i.e., if and only if for in nitely many and hence for all E 2 C , (1.18) has a fundamental system of solutions meromorphic with respect to x). The proof of Theorem 1.2 in Section 6 (Theorem 6.4) relies on three main ingredients: A purely Floquet theoretic part to be discussed in detail in Sections 4 and 5, the fact that meromorphic algebro-geometric AKNS potentials are Picard potentials using gauge transformations in Section 3, and an elliptic function part described in Section 6. The corresponding Floquet theoretic part is summarized in Theorems 4.7, 4.8, 5.1{5.4. In particular, Theorems 4.7 and 4.8 illustrate the great variety of possible values of algebraic multiplicities of (anti)periodic and Dirichlet eigenvalues in the general case where L is non-self-adjoint. Theorem 5.1 on the other hand reconstructs the (possibly singular) hyperelliptic curve (1.17) associated with the 2 2 periodic matrix Q (not necessarily elliptic), which gives rise to two linearly independent Floquet solutions of J 0 + Q = E for all but nitely many values of E 2 C . Our use of gauge transformations in Section 3, in principle, suggests a constructive method to relate -functions associated with a singular curve Kn and -functions of ^^ the associated desingularized curve Kn, which appears to be of independent interest. The elliptic function portion in Section 6 consists of several items. First of all we describe a matrix generalization of Picard's (scalar) result in Theorem 6.1. In Theorem 6.3 we prove the key result that all 4!j -periodic eigenvalues associated with Q lie in certain strips Sj = fE 2 C j j Im(j!j j?1!j E )j Cj g; j = 1; 3 (1.19) for suitable constants Cj > 0. Then S1 and S3 do not intersect outside a su ciently large disk centered at the origin. A combination of this fact and Picard's Theorem 6.1 then yields a proof of Theorem 1.2 (see the proof of Theorem 6.4). We close Section 6 with a series of remarks that put Theorem 1.2 into proper perspective: Among a variety of points, we stress, in particular, its straightforward applicability based on an elementary Frobenius-type analysis, its property of complementing Picard's original result, and its connection with the Weierstrass theory of

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reduction of Abelian to elliptic integrals. Finally, Section 7 rounds o our presentation with a few explicit examples. The result embodied by Theorems 1.1 and 1.2 in the context of the KdV and AKNS hierarchies, uncovers a new general principle in connection with elliptic algebrogeometric solutions of completely integrable systems: The existence of such solutions appears to be in a one-to-one correspondence with the existence of a meromorphic (with respect to the independent variable) fundamental system of solutions for the underlying linear Lax di erential expression (for all values of the corresponding spectral parameter E 2 C ). Even though the current AKNS case is technically much more involved than the KdV case in 44] (and despite the large number of references at the end) we have made every e ort to keep this presentation self-contained. 2. The AKNS Hierarchy, Recursion Relations, and Hyperelliptic

Curves

Q ( Q () 0 Q(x) = Q1;1 (x) Q1;2(x) = ip(x) ?iq(x) : (2.3) x) 2;2 x 0 2;1 In order to explicitly construct higher-order matrix-valued di erential expressions Pn+1, n 2 N 0 (=N f0g) commuting with L, which will be used to de ne the stationary AKNS hierarchy, one can proceed as follows (see 37] for more details). De ne functions f`, g`, and h` by the following recurrence relations, f?1 = 0; g0 = 1; h?1 = 0; i i f`+1 = 2 f`;x ? iqg`+1; g`+1;x = pf` + qh`; h`+1 = ? 2 h`;x + ipg`+1 (2.4) for ` = ?1; 0; 1; :::. The functions f`, g`, and h` are polynomials in the variables p; q; px; qx; ::: and c1 ; c2; ::: where the cj denote integration constants. Assigning weight k +1 to p(k) and q(k) and weight k to ck one nds that f`, g`+1, and h` are homogeneous of weight ` + 1.

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In this section we brie y review the construction of the AKNS hierarchy using a recursive approach. This method was originally introduced by Al'ber 4] in connection with the Korteweg-de Vries hierarchy. The present case of the AKNS hierarchy was rst systematically developed in 37]. Suppose that q = iQ1;2 ; p = ?iQ2;1 2 C 1(R ) (or meromorphic on C ) and consider the Dirac-type matrix-valued di erential expression d i 0 d 0 L = J dx + Q(x) = 0 ?i dx + ip(x) ?iq(x) ; (2.1) 0 where we abbreviate i 0 J = 0 ?i ; (2.2)

Explicitly, one computes, f0 = ?iq; 1 f1 = 2 qx + c1(?iq); i 1 i f2 = 4 qxx ? 2 pq2 + c1 ( 2 qx ) + c2(?iq); g0 = 1; g1 = c1; g2 = 1 pq + c2 ; 2 i g3 = ? 4 (px q ? pqx) + c1( 1 pq) + c3; 2 h0 = ip; 1 h1 = 2 px + c1(ip); i i 1 h2 = ? 4 pxx + 2 p2q + c1 ( 2 px ) + c2 (ip); etc: Next one de nes the matrix-valued di erential expression Pn+1 by

(2.5)

Pn+1 = ?

where One veri es that

n+1 X `=0

(gn?`+1J + iAn?`)L`;

(2.6) (2.7)

0 A` = h ?0f` ; ` = ?1; 0; 1; : : : : `

gn?`+1J + iAn?`; L] = 2iAn?`L ? 2iAn?`+1; (2.8) where ; ] denotes the commutator. This implies Pn+1; L] = 2iAn+1: (2.9) The pair (Pn+1; L) represents a Lax pair for the AKNS hierarchy. Introducing a deformation parameter t into (p; q), that is, (p(x); q(x)) ! (p(x; t); q(x; t)), the AKNS hierarchy (cf., e.g., 68], Chs. 3, 5 and the references therein) is de ned as the collection of evolution equations (varying n 2 N 0 ) d L(t) ? P (t); L(t)] = 0 (2.10) n+1 dt

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or equivalently, by that is, by

(x; t) AKNSn(p; q) = pt(x; t) ? 2hn+1(x; t) = 0; qt(x; t) ? 2fn+1

(2.11)

2iEH ? 2pG AKNSn(p; q) = pqt + ii((Hn;x ? 2iEF n ? 2qG n+1)) = 0: (2.12) ? Fn;x + t n n+1 Explicitly, one obtains for the rst few equations in (2.11) ? 2ic p AKNS0 (p; q) = pt ? px + 2ic 1q = 0 qt ? q x 1 i p ? ip2 q ? c p ? 2ic p (2.13) AKNS1 (p; q) = pt + 2i qxx + ipq2 ? c1q x + 2ic 2q = 0 qt ? 2 xx 1 x 2 ? 3 pp + ( 2i xx 2 +1 2 AKNS2 (p; q) = qpt+ 14qpxxx? 32pqqxq+ cc1? ipq ? ip q2))? cc pqx ? 22ic3 pq = 0 1 ( 2 xx + ipq ? 2 x + ic3 t 4 xxx 2 x etc: The stationary AKNS hierarchy is then de ned by the vanishing of the commutator of Pn+1 and L, that is, by Pn+1; L] = 0; n 2 N 0 ; (2.14) or equivalently, by fn+1 = hn+1 = 0; n 2 N 0 : (2.15) Next, we introduce Fn, Gn+1, and Hn which are polynomials with respect to E 2 C ,

Fn(E; x) = Gn+1(E; x) = Hn(E; x) =

n X

`=0 n+1 X n X `=0 `=0

fn?`(x)E ` ; gn+1?`(x)E ` ; hn?`(x)E ` ;

(2.17) (2.18) (2.19) (2.20) (2.16)

and note that (2.15) becomes Fn;x(E; x) = ?2iEFn(E; x) + 2q(x)Gn+1(E; x); Gn+1;x(E; x) = p(x)Fn(E; x) + q(x)Hn(E; x); Hn;x(E; x) = 2iEHn(E; x) + 2p(x)Gn+1(E; x): These equations show that G2 +1 ? FnHn is independent of x. Hence n R2n+2 (E ) = Gn+1(E; x)2 ? Fn(E; x)Hn(E; x)

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is a monic polynomial in E of degree 2n + 2. One can use (2.17){(2.20) to derive di erential equations for Fn and Hn separately by eliminating Gn+1. One obtains 2 2 2 q(2FnFn;xx ? Fn;x + 4(E 2 ? pq)Fn ) ? qx (2FnFn;x + 4iEFn ) = ?4q3 R2n+2 (E ); (2.21) 2 + 4(E 2 ? pq )H 2 ) ? p (2H H ? 4iEH 2 ) = ?4p3 R p(2HnHn;xx ? Hn;x x n n;x 2n+2 (E ): n n (2.22) Next, assuming Pn+1; L] = 0, one infers

2 Pn+1 =

n+1 X

Hence,

`;m=0

(gn?`+1J + iAn?`)(gn?m+1J + iAn?m )L`+m:

(2.23)

2 Pn+1 = ?Gn+1 (L; x)2 + Fn(L; x)Hn(L; x) = ?R2n+2 (L); (2.24) that is, whenever Pn+1 and L commute they necessarily satisfy an algebraic relationship. In particular, they de ne a (possibly singular) hyperelliptic curve Kn of (arithmetic) genus n of the type

Kn :

w2

= R2n+2 (E ); R2n+2 (E ) =

2Y n+1

The functions f`, g`, and h`, and hence the matrices A` and the di erential expressions P` de ned above, depend on the choice of the integration constants c1 ; c2; :::; c` (cf. (2.5)). In the following we make this dependence explicit and write f`(c1; : : : ; c`), g`(c1; : : : ; c`), h`(c1; : : : ; c`), A`(c1; :::; c`), P`(c1; :::; c`), etc. In particular, we denote homogeneous quantities, where c` = 0; ` 2 N by f^` = f`(0; : : : ; 0), g` = g`(0; : : : ; 0), ^ ^ ^ ^ h` = h`(0; : : : ; 0), A` = A`(0; : : : ; 0), P` = P`(0; :::; 0), etc. In addition, we note that

m=0

+1 (E ? Em ) for some fEmg2n=0 m

C:

(2.25)

f`(c1; :::; c`) =

and

` X k=0

c`?k f^k ; g`(c1; :::; c`) =

` X k=0

c`?k gk ; h` (c1; :::; c`) = ^

` X

^ c`?k hk ; k=0 (2.26) (2.27) (2.28)

A`(c1; :::; c`) =

de ning c0 = 1. In particular, then

` X k=0

^ c`?k Ak ; ^ cr?`P`:

Pr (c1 ; :::; cr ) =

10

r X `=0

Next suppose that Pn+1 is any 2 2 matrix-valued di erential expression such that Pn+1; L] represents multiplication by a matrix whose diagonal entries are zero. This implies that the leading coe cient of Pn+1 is a constant diagonal matrix. Since any constant diagonal matrix can be written as a linear combination of J and I (the identity matrix in C 2 ), we infer the existence of complex numbers n+1 and n+1 such that ^ S1 = Pn+1 ? n+1Pn+1 ? n+1Ln+1 (2.29) is a di erential expression of order at most n whenever Pn+1 is of order n + 1. Note that ^ ^ S1 ; L] = Pn+1; L] ? n+1 Pn+1; L] = Pn+1; L] ? 2i n+1An+1 (2.30) represents multiplication with zero diagonal elements. An induction argument then shows that there exists Sn+1 such that

Sn+1 = Pn+1 ?

n+1 X `=1

^ ( `P` + `L`) and Sn+1; L] = Pn+1; L] ? 2i

n+1 X `=1

^ ` A` :

(2.31)

Since the right-hand side of the last equation is multiplication with a zero diagonal, Sn+1 is a constant diagonal matrix, that is, there exist complex numbers 0 and 0 such that Sn+1 = 0J + 0I . Hence

Pn+1 =

n+1 X `=0

^ ( `P` + `L`) and Pn+1; L] = 2i

n+1 X `=0

^ ` A` :

(2.32)

Consequently, if all ` = 0, then Pn+1 is a polynomial of L, and Pn+1 and L commute irrespective of p and q. If, however, r 6= 0 and ` = 0 for ` > r, then

Pn+1 = r Pr (

r X `

In this case Pn+1 and L commute if and only if

`=0 r

n+1 X r?1 0 ` ; :::; ) + `L : r r `=0 r?1 ; :::; 0 ) = 0; r r

(2.33)

^ A` = Ar (

(2.34)

that is, if and only if (p; q) is a solution of some equation of the stationary AKNS P +1 hierarchy. In this case Pn+1 ? n=0 `L` and L satisfy an algebraic relationship of ` the type (2.24). Suppose on the other hand that Pn+1 is a matrix-valued di erential expression such that (Pn+1 ? Kr (L))2 = ?R2n+2 (L) for some polynomials Kr and R2n+2 . Then L commutes with (Pn+1 ? Kr (L))2 and this enforces that L also commutes with Pn+1 ? Kr (L) and hence with Pn+1. Thus we proved the following theorem.

11

Theorem 2.1. Let L be de ned as in (2.1). If Pn+1 is a matrix-valued di erential

Theorem 2.1 represents a matrix-valued generalization of a celebrated result due to Burchnall and Chaundy 16], 17] in the special case of scalar di erential expressions. By the arguments presented thus far in this section it becomes natural to make the following de nition. We denote by M2 (C ) the set of all 2 2 matrices over C .

expression of order n +1 which commutes with L, whose leading coe cient is di erent from a constant multiple of J n+1 , then there exist polynomials Kr and R2n+2 of degree r n + 1 and 2n + 2, respectively, such that (Pn+1 ? Kr (L))2 = ?R2n+2 (L).

0 De nition 2.2. A function Q : R ! M2 (C ) of the type Q = ip ?0iq is called an algebro-geometric AKNS potential if (p; q) is a stationary solution of some equation of the AKNS hierarchy (2.15). By a slight abuse of notation we will also call (p; q) an algebro-geometric AKNS potential in this case. The following theorem gives a su cient condition for Q to be algebro-geometric.

Theorem 2.3. Assume that Fn(E; x) = Pn=0 fn?`(x)E ` with f0(x) = ?iq(x) is a ` polynomial of degree n in E , whose coe cients are twice continuously di erentiable complex-valued functions on (a; b) for some ?1 a < b 1. Moreover, suppose that q has (at most) nitely many zeros on each compact interval on R with 1 the

only possible accumulation points. If

1 nq(x)?2F (E; x)F (E; x) ? F (E; x)2 + 4(E 2 ? p(x)q(x))F (E; x)2 n n;xx n;x n 4q(x)3 o ? ? qx(x) 2Fn(E; x)Fn;x(E; x) + 4iEFn(E; x)2 ; (2.35)

is independent of x, then p, q and all coe cients f` of Fn are in C 1((a; b)). Next, de ne

geometric AKNS potential.

g0(x) = 1; (2.36) g`+1(x) = q(1 ) (? 1 f`;x(x) ? if`+1 (x)); x 2 h`(x) = q(1 ) (?g`+1;x(x) ? ip(x)f` (x)) (2.37) x for ` = 0; : : : ; n, where fn+1 = 0. Then the di erential expression Pn+1 de ned by (2.6) commutes with L in (2.1). In particular, if (a; b) = R then (p; q) is an algebro12

Proof. The expression (2.35) is a monic polynomial of degree 2n + 2 with constant coe cients. We denote it by

R2n+2 (E ) =

2n+2 X

We now compare coe cients in (2.35) and (2.38) starting with the largest powers. First of all this yields qx = i 1 q ? 2f1; (2.39) which shows that q 2 C 3 ((a; b)), and secondly that 2 p = (?4 2q2 ? 3qx + 2qqxx ? 8iqf2 ? 4f12 + 8qxf1)=(4q3); (2.40) which shows that p 2 C 1 ((a; b)). Comparing the coe cients of E 2n?` allows one to express 4q2f`;xx as a polynomial in p, q, qx, the coe cients f`, their rst derivatives, and the second derivatives of f0 ,...,f`?1 . Therefore, one may show recursively that f`;xx 2 C 1 ((a; b)) for any ` 2 f1; :::; ng. Equations (2.39) and (2.40) then show that q 2 C 4((a; b)) and p 2 C 2((a; b)). Thus it follows that the f`;xx are in C 2((a; b)). An induction argument now completes the proof of the rst part of the theorem. Next, introducing

m=0

2n+2?m E

m;

0

= 1:

(2.38)

Gn+1(E; x) =

n+1 X `=0

gn+1?`

(x)E `;

Hn(E; x) =

n X `=0

hn?`(x)E `;

(2.41)

one nds that Fn, Gn+1, and Hn satisfy equations (2.17) and (2.18). Equating (2.35) with R2n+2 yields R2n+2 = G2 +1 ? FnHn. Hence G2 +1 ? FnHn does not depend on n n x and therefore di erentiating with respect to x results in 2Gn+1Gn+1;x ? FnHn;x ? Fn;xHn = 0 which shows that equation (2.19) also holds. This in turn proves that f`, g`, and h` satisfy the recurrence relations given above with fn+1 = hn+1 = 0. The commutativity of Pn+1 and L now follows as before. The same proof yields the following result. Corollary 2.4. Assume that Fn(E; x) = Pn=0 fn?`(x)E `, f0 (x) = ?iq(x) is a poly` nomial of degree n in E whose coe cients are meromorphic in x. If 1 nq(x)?2F (E; x)F (E; x) ? F (E; x)2 + 4(E 2 ? p(x)q(x))F (E; x)2 n n;xx n;x n 4q(x)3 o ? ? qx(x) 2Fn(E; x)Fn;x(E; x) + 4iEFn(E; x)2 ; (2.42)

is independent of x, then (p; q) is a meromorphic algebro-geometric AKNS potential. Finally, we mention an interesting scale invariance of the AKNS equations (2.11).

13

AKNSn(p; q) = 0 (2.43) for some n 2 N 0 . Consider the scale transformation (p(x; t); q(x; t)) ! (p(x; t); q(x; t)) = (Ap(x; t); A?1 q(x; t)); A 2 C nf0g: (2.44) Then AKNSn (p; q) = 0: (2.45) We omit the straightforward proof which can be found, for instance, in 37]. In the particular case of the nonlinear Schrodinger (NS) hierarchy, where (2.46) p(x; t) = q(x; t); (2.44) further restricts A to be unimodular, that is, jAj = 1: (2.47) Note that the KdV hierarchy as well as the modi ed Korteweg-de Vries (mKdV) hierarchy are contained in the AKNS hierarchy. In fact, setting all integration constants c2`+1 equal to zero the nth KdV equation is obtained from the (2n)th AKNS system by the constraint p(x; t) = 1 (2.48) while the nth mKdV equation is obtained from the (2n)th AKNS system by the constraint p(x; t) = q(x; t): (2.49)

Lemma 2.5. Suppose (p; q) satis es one of the AKNS equations (2.11),

3. Gauge Transformations for the Stationary AKNS Hierarchy This section is devoted to a study of meromorphic properties of solutions (E; x) of L = E with respect to x 2 C under the assumption that Q is a meromorphic algebro-geometric AKNS potential associated with a (possibly singular) hyperelliptic curve Kn. Meromorphic properties of (E; x) will enter at a crucial stage in the proof of our main characterization result, Theorem 6.4. In the following we denote the order of a meromorphic function f at the point x 2 C by ordx(f ). Proposition 3.1. If Q is a meromorphic algebro-geometric AKNS potential then ordx(p) + ordx(q) ?2 for every x 2 C .

14

Proof. Assume the contrary (i.e., ordx(p) + ordx(q) ?1) and choose E such that R2n+2 (E ) = 0, where R2n+2 is the polynomial de ning the hyperelliptic curve associated with Q. De ne Fn as in (2.16) and denote its order at x by r. Then ordx(?4pqFn(E; )2) is strictly smaller than 2r ? 2 while the order of any other term on the left-hand side of (2.21) is at least 2r ? 2. This is impossible since the right and thus the left-hand side of (2.21) vanishes identically. Therefore, to discuss algebro-geometric AKNS potentials we only need to consider the case where p and q have Laurent expansions of the form

p(x) =

1 X

j =0

pj (x ? x0

)j?1+m;

q(x) =

1 X

j =0

qj (x ? x0 )j?1?m;

(3.1)

with m an integer and at least one of the numbers p0 and q0 di erent from zero. If B = diag(?m; 0), the change of variables y = xB w transforms the di erential P equation Jy0 + Qy = Ey into the equation w0 = (R=x + S + 1 Aj+1xj )w, where j =0 m 0 0 R = p q00 ; S = ?iE 1 ?1 ; Aj = p q0j : (3.2) 0 0 j We now make the ansatz

w(x) =

where T and the relation

j

1 X

j =0

j (x ? x0 )

j +T ;

(3.3)

are suitable constant matrices. This ansatz yields the recurrence

R 0 ? 0 T = 0;

j +1 ? j +1 (T

R

+ j + 1) = ?S j ?

j X l=0

(3.4)

Al+1

j ?l

= Bj ;

(3.5)

where the last equality de nes Bj . In the following we denote the lth column of j ( and Bj by !j(l) and bjl) , respectively. Proposition 3.2. Suppose Q is a meromorphic potential of L = E and x0 is a pole of Q where p and q have Laurent expansions given by (3.1). The equation L = E has a fundamental system of solutions which are meromorphic in a vicinity of x0 if and only if (i) the eigenvalues and m ? (where, without loss of generality, > m ? ) of R are distinct integers and (ii) b2 ?m?1 is in the range of R ? . P Proof. A fundamental matrix of w0 = (R=x + S + 1 Aj+1 xj )w may be written as j =0 (3.3) where T is P1 in Jordan normal form. If all solutions of L = E and hence of w0 = (R=x + S + j=0 Aj+1xj )w are meromorphic near x0 , then T must be a diagonal

15

matrix with integer eigenvalues. Equation (3.4) then shows that the eigenvalues of T are the eigenvalues of R and that R is diagonalizable. But since at least one of p0 and q0 is di erent from zero, R is not diagonalizable if is a double eigenvalue of R, a case which is therefore precluded. This proves (i). Since T is a diagonal matrix, equation (3.5) implies (R + ? m ? j ? 1)!j(2) = b(2) (3.6) +1 j for j = 0; 1; :::. Statement (ii) is just the special case where j = 2 ? m ? 1. Conversely, assume that (i) and (ii) are satis ed. If the recurrence P relations (3.4), (3.5) are satis ed, that is, if w is a formal solution of w0 = (R=x + S + 1 Aj+1xj )w j =0 then it is also an actual solution near x0 (see, e.g., Coddington and Levinson 23], Sect. 4.3). Since R has distinct eigenvalues it has linearly independent eigenvectors. Using these as the columns of 0 and de ning T = ?1 R 0 yields (3.4). Since T is 0 a diagonal matrix, (3.5) is equivalent to the system b(1) = (R ? ? j ? 1)!j(1) ; (3.7) j +1 b(2) = (R + ? m ? j ? 1)!j(2) : (3.8) j +1 Next, we note that R ? ? j ? 1 is invertible for all j 2 N 0 . However, R + ? m ? j ? 1 is only invertible if j 6= 2 ? m ? 1. Hence a solution of the proposed form exists if and only if b(2)?m?1 is in the range of R ? , which is guaranteed by hypothesis 2 (ii). Note that B0 = ?EJ 0 ? A1 0 (3.9) is a rst-order polynomial in E . As long as R has distinct eigenvalues and j 2 ? m ? 1, we may compute j recursively from (3.7) and (3.8) and Bj from the equality on the right in (3.5). By induction one can show that j is a polynomial of degree j and that Bj is a polynomial of degree j + 1 in E . This leads to the following result. Theorem 3.3. Suppose Q is a meromorphic potential of L = E . The equation L = E has a fundamental system of solutions which are meromorphic with respect to the independent variable for all values of the spectral parameter E 2 C whenever this is true for a su ciently large nite number of distinct values of E . Proof. By hypothesis, Q has countably many poles. Let x0 be any one of them. Near x0 the functions p and q have the Laurent expansions (3.1). The associated matrix R has eigenvalues and m ? , which are independent of E . The vector v = (q0; ? )t spans R ? , and the determinant of the matrix whose columns are v and b2 ?m?1 is a polynomial in E of degree 2 ? m. Our hypotheses and Proposition 3.2 imply that this determinant has more than 2 ? m zeros and hence is identically equal to zero. This shows that b2 ?m?1 is a multiple of v for every value of E . Applying Proposition

16

3.2 once more then shows that all solutions of L = E are meromorphic near x0 for all E 2 C . Since x0 was arbitrary, this concludes the proof. Next, let fE0; :::; E2n+1 g be a set of not necessarily distinct complex numbers. We recall (cf. (2.25)),

Kn = fP

= (E; V ) j V 2

= R2n+2 (E ) =

2Y n+1

We introduce the meromorphic function ( ; x) on Kn by +1 (3.11) (P; x) = V +FG(nE; (E; x) ; P = (E; V ) 2 Kn: x) n We remark that can be extended to a meromorphic function on the compacti cation (projective closure) of the a ne curve Kn. This compacti cation is obtained by joining two points to Kn. Next we de ne Z x dx0 ?iE + q(x0 ) (P; x0)] ; (3.12) 1 (P; x; x0 ) = exp (3.13) 2 (P; x; x0 ) = (P; x) 1 (P; x; x0 ); where the simple Jordan arc from x0 to x in (3.12) avoids poles of q and . One veri es with the help of (2.17){(2.20), that 2 (3.14) x (P; x) = p(x) ? q (x) (P; x) + 2iE (P; x): From this and (2.6) we nd L (P; x; x0 ) = E (P; x; x0); Pn+1 (P; x; x0 ) = iV (P; x; x0); (3.15) where (P; x; x ) (P; x; x0) = 1 (P; x; x0) : (3.16)

2 0

m=0

(E ? Em )g:

(3.10)

x0

One observes that the two branches (E; x; x0 ) = ( ;1(E; x; x0 ); ;2(E; x; x0))t of (P; x; x0) represent a fundamental system of solutions of Ly = Ey for all E 2 +1 C n ffEm g2n=0 f j (x0 )gn=1 g, since j m +( W ( ?(E; x; x0 ); +(E; x; x0 )) = F2VE;E )) : (3.17) n ( x0 Here W (f; g) denotes the determinant of the two columns f and g and V+( ) (resp. V?( )) denotes the branch of V ( ) on the upper (resp. lower) sheet of Kn (we follow the notation established in 37]). In the special case where Kn is nonsingular, that is, Em 6= Em for m 6= m0, the explicit representation of (P; x; x0) in terms of the Riemann theta function associated with Kn immediately proves that (E; x; x0 ) are meromorphic with respect to x 2 C

0

17

+1 for all E 2 C n ffEm g2n=0 f j (x0 )gn=1g. A detailed account of this theta function m j representation can be found, for instance, in Theorem 3.5 of 37]. In the following we demonstrate how to use gauge transformations to reduce the case of singular curves Kn to nonsingular ones. Let (p; q) be meromorphic on C , the precise conditions on (p; q) being immaterial (at least, temporarily) for introducing gauge transformations below. De ne L and Q as in (2.1), (2.3) and consider the formal rst-order di erential system L = E . Introducing, (x A(E; x) = ?iEx) ?qiE) ; (3.18) p( ? L = E is equivalent to x(E; x) + A(E; x) (E; x) = 0. Next we consider the gauge transformation, ~ (E; x) = ?(E; x) (E; x); (3.19) ~(x ~ A(E; x) = ?iEx) ?qiE) = ?(E; x)A(E; x)?(E; x)?1 ? ?x(E; x)?(E; x)?1; p( ? ~ (3.20) implying ~ x(E; x) + A(E; x) ~ (E; x) = 0; that is, L ~ (E; x) = E ~ (E; x); ~ ~ (3.21) ~ with L de ned as in (4.1), (4.2) replacing (p; q) by (~; q). In the following we make p~ the explicit choice (cf., e.g., 55]), i iq ~ 2 q(x) (0) ~ ~ ?(E; x) = E ? E ? (0) (E; x) (E; x) 2 ?(x) ; E 2 C n fE g (3.22) i i ~ 2 2 ~ for some xed E 2 C and (0) (E; x) = (0) (E; x)= (0) (E; x); ~ (3.23) 2 ~ 1 ~ (0) ~ (0) ~ ~ ~ where (0) (E; x) = ( 1 (E; x); 2 (E; x))t is any solution of L = E . Using (3.14), equation (3.20) becomes ~ p(x) = (0) (E; x); ~ (3.24) ~ ~ q(x) = ?qx(x) ? 2iEq(x) + q(x)2 (0) (E; x): ~ (3.25) Moreover, one computes for ~ = ( ~1 ; ~2 )t in terms of = ( 1 ; 2 )t, ~1 (E; x) = (E ? E ) 1 (E; x) + i q(x)( 2 (E; x) ? (0) (E; x) 1 (E; x)); ~ ~ (3.26) 2 ~2 (E; x) = ? i ( 2(E; x) ? (0) (E; x) 1 (E; x)): ~ (3.27) 2

18

In addition, we note that and therefore,

i ~ det(?(E; x)) = ? 2 (E ? E )

(3.28)

i ~ (3.29) W ( ~ 1(E ); ~ 2(E )) = ? 2 (E ? E )W ( 1(E ); 2(E )); where j (E; x), j = 1; 2 are two linearly independent solutions of L = E . Our rst result proves that gauge transformations as de ned in this section leave the class of meromorphic algebro-geometric potentials of the AKNS hierarchy invariant. Theorem 3.4. Suppose (p; q) is a meromorphic algebro-geometric AKNS potential. ~ ~ Fix E 2 C and de ne (~; q) as in (3.24), (3.25), with (0) (E; x) de ned as in (3.23). p~ ~ Suppose (0) (E; x) is meromorphic in x. Then (~; q) is a meromorphic algebrop~

geometric AKNS potential. Proof. The upper right entry G1;2 (E; x; x0 ) of the Green's matrix of L is given by 0 0 ) = i +;1 (E; x; x0 ) ?;1 (E; x ; x0 ) ; x x0 : G1;2(E; x; x W ( (E; ; x ); (E; ; x )) (3.30) ? 0 + 0 Combining (3.11){(3.13) and (3.14), its diagonal (where x = x0 ) equals n (3.31) G1;2(E; x; x) = iFV(E; x) : 2 +(E ) ~ The corresponding diagonal of the upper right entry G1;2 (E; x; x) of the Green's ~ matrix of L is computed to be ~; ~ ~ ( ~ G1;2 (E; x; x) = i +~1(E; x) ?;1 (E; x) = iFn+1~ E; x) ; (3.32) W ( ?(E; ); ~ +(E; )) 2(E ? E )V+(E ) where ~ ~ i ~ Fn+1(E; x) =2iFn(E; x)f(E ? E ) + 2 q(x) +(E; x) ? (0) (E; x)]g ~ ~ f(E ? E ) + 2i q(x) ?(E; x) ? (0) (E; x)]g; (3.33) using (3.17), (3.26), (3.29), and Fn (E; x) : (3.34) +;1 (E; x; x0 ) ?;1 (E; x; x0 ) = Fn(E; x0) By (3.11), +(E; x) + ?(E; x) = 2Gn+1(E; x)=Fn(E; x) and +(E; x) ?(E; x) = ~ Hn(E; x)=Fn(E; x). From this, (2.5), and (2.16), it follows that Fn+1 ( ; x) is a polynomial of degree n + 1 with leading coe cient ~ ~ iqx (x) ? 2Eq(x) ? iq(x)2 (0) (E; x) = ?iq(x): ~ (3.35)

19

~ ~ Finally, using L ~ = E ~ , that is, (3.21), one veri es that G1;2(E; x; x) satis es the di erential equation, ? ? ~1 ~1 q 2G1;2G1;2;xx ? G2;2;x + 4(E 2 ? pq)G2;2 ? qx 2G1;2G1;2;x + 4iE G2;2 = q3 : (3.36) ~ ~ ~ ~~ ~ 1 ~ ~ ~ ~ ~ Hence Fn+1(E; x) satis es the hypotheses of Corollary 2.4 (with n replaced by n + 1 and (p; q) replaced by (~; q)) and therefore, (~; q) is a meromorphic algebro-geometric p~ p~ AKNS potential. ~ We note here that (3.36) implies also that Fn+1 satis es ~2 ~2 q(2Fn+1Fn+1;xx ? Fn+1;x + 4(E 2 ? pq)Fn+1) ? qx(2Fn+1Fn+1;x + 4iE Fn+1) ~ ~ ~ ~~ ~ 2 ~ ~ ~ ~ = ? 4~3(E ? E )2 R2n+2(E ); q (3.37) that is, (~; q) is associated with the curve p~ ~ ~ Kn+1 = f(E; V ) j V 2 = (E ? E )2R2n+2 (E )g: (3.38) Corollary 3.5. Suppose (p; q) is a meromorphic algebro-geometric AKNS potential associated with the hyperelliptic curve

Kn

= f(E; V ) j V 2

= R2n+2 (E ) =

2Y n+1

~ which has a singular point at (E; 0), that is, R2n+2 has a zero of order r 2 at the ~ point E . Choose ~ (0) (E; x) = Gn+1 (E; x) ~ (3.40) ~ Fn(E; x) ~ (cf. (3.11)) and de ne (~; q) as in (3.24), (3.25). Then (0) (E; ) is meromorphic p~ and the meromorphic algebro-geometric AKNS potential (~; q) is associated with the p~ hyperelliptic curve ~~ ~ ~ Kn = f(E; V ) j V 2 = R2n?2s+4(E ) = (E ? E )2?2sR2n+2 (E )g (3.41) ~~ for some 2 s (r=2) + 1. In particular, Kn and Kn have the same structure near ~ any point E 6= E . ~ Proof. Since V 2 = R2n+2 (E ) we infer that V+ (E ) has at least a simple zero at E . Hence ~ +1 ~ ~ (E; x) ? (0) (E; x) = F V+(E ) + Gn+1(E; x)Fn (E; x) ? Gn~ (E; x)Fn(E; x) Fn (E; x)Fn(E; x) n (E; x) (3.42)

20

m=0

(E ? Em )g;

(3.39)

~ ~ also have at least a simple zero at E . From (3.33) one infers that Fn+1 (E; x) has a ~ , that is, zero of order at least 2 at E ~ ~ ~ Fn+1 (E; x) = (E ? E )sFn+1?s(E; x); s 2: (3.43) ~~ De ne n = n +1 ? s. Then Fn still satis es the hypothesis of Corollary 2.4. Moreover, ~ ~ ~ inserting (3.43) into (3.37) shows that (E ?E )2s must be a factor of (E ?E )2R2n+2 (E ). Thus, 2s r + 2 and hence ~~2 ~~2 q(2FnFn;xx ? Fn;x + 4(E 2 ? pq)Fn ) ? qx(2FnFn;x + 4iE Fn ) = ?4~3R2~ +2(E ); ~ ~~ ~~ ~~ ~~2 ~ ~~ ~~ q ~n (3.44) where ~n ~ R2~+2(E ) = (E ? E )2?2s R2n+2(E ) (3.45) is a polynomial in E of degree 0 < 2n ? 2s + 4 < 2n + 2. This proves (3.41). ~ In view of our principal result, Theorem 6.4, our choice of (0) (E; x) led to a curve ~ ~~ Kn which is less singular at E than Kn, without changing the structure of the curve ~~ elsewhere. By iterating the procedure from Kn to Kn one ends up with a curve which ~ 0). Repeating this procedure for each singular point of Kn is nonsingular at (E; ^^ then results in a nonsingular curve Kn and a corresponding Baker-Akhiezer function ^ (P; x; x0) which is meromorphic with respect to x 2 C (this can be seen by using ^^ their standard theta function representation, cf., e.g., 37]). Suppose that Kn was obtained from Kn by applying the gauge transformation ?(E; x) = ?N (E; x):::?1 (E; x); (3.46) where each of the ?j is of the type (3.22). Then the branches of (P; x) = ?(E; x)?1 ^ (P; x; x0 ) = ?1(E; x)?1:::?N (E; x)?1 ^ (P; x; x0) (3.47) are linearly independent solutions of L = E for all E 2 C nfE0 ; :::; E2n+1 ; 1(x0 ); :::; n (x0 )g. These branches are meromorphic with respect to x since i ( ? 2i (3.48) ?j (E; x)?1 = 2i ~ ? i (0) (E; x) E ? E ??i 2 qxx) (0) (E; x) ~ ~ 2 q( ) ~ E?E 2 maps meromorphic functions to meromorphic functions in view of the fact that q ~ ~ ~ and (0) (E; ) = Gn+1(E; )=Fn(E; ) are meromorphic. Combining these ndings and Theorem 3.3 we thus proved the principal result of this section. Theorem 3.6. Suppose (p; q) is a meromorphic algebro-geometric AKNS potential. Then the solutions of L = E are meromorphic with respect to the independent variable for all values of the spectral parameter E 2 C .

21

Miura transformation and algebro-geometric methods to prove results of the type stated in Corollary 3.5. An alternative approach in the KdV context has recently been found by Ohmiya 72]. The present technique to combine gauge transformations, the polynomial recursion approach to integrable hierarchies based on hyperelliptic curves (such as the KdV, AKNS, and Toda hierarchies), and the fundamental meromorphic function ( ; x) on Kn (cf. (3.11)), yields a relatively straightforward and uni ed treatment, further details of which will appear elsewhere. To the best of our knowledge this is the rst such approach for the AKNS hierarchy. A systematic study of the construction used in Theorem 3.6 yields explicit connections between the -function associated with the possibly singular curve Kn and the ^^ Riemann theta function of the nonsingular curve Kn. This seems to be of independent interest and will be pursued elsewhere. 4. Floquet Theory Throughout this section we will assume the validity of the following basic hypothesis. Hypothesis 4.1. Suppose that p; q 2 L1 (R ) are complex-valued periodic functions loc of period > 0 and that L is a 2 2 matrix-valued di erential expression of the form d L = J dx + Q; (4.1) where i 0 0 J = 0 ?i ; Q = ip ?0iq : (4.2) We note that where I is the 2 2 identity matrix in C 2 . Given Hypothesis 4.1, we uniquely associate the following densely and maximally de ned closed linear operator H in L2 (R)2 with the matrix-valued di erential expresssion L, Hy = Ly; D(H ) = fy 2 L2 (R )2 j y 2 ACloc(R )2 ; Ly 2 L2 (R )2 g: (4.4) One easily veri es that L is unitarily equivalent to 0 ?1 d + 1 (p + q) i(p ? q) ; (4.5) 1 0 dx 2 i(p ? q) ?(p + q) a form widely used in the literature. We consider the di erential equation Ly = Ey where L satis es Hypothesis 4.1 and where E is a complex spectral parameter. De ne 0(E; x; x0 ; Y0) = eE(x?x0)J Y0 for

22

Remark 3.7. In the case of the KdV hierarchy, Ehlers and Knorrer 27] used the

?J 2 = I and JQ + QJ = 0;

(4.3)

Y0 2 M2 (C ). The matrix function (E; ; x0; Y0) is the unique solution of the integral equation Y (x) = 0(E; x; x0 ; Y0) +

Z x

if and only if it satis es the initial value problem JY 0 + QY = EY; Y (x0 ) = Y0: (4.7) Since @ 0 (E; x; x ; Y ) = EJ eE(x?x0)J Y = E eE(x?x0)J JY ; (4.8) 0 0 0 0 @x0 di erentiating (4.6) with respect to x0 yields @ (E; x; x ; Y ) =eE(x?x0)J (EJ ? JQ(x ))Y 0 0 0 0 @x0 Z x (4.9) + eE(x?x )J JQ(x0 ) @ (E; x0; x0 ; Y0; )dx0; @x0 x0 that is, @ (E; x; x ; Y ) = (E; x; x ; (E + Q(x ))JY ); (4.10) 0 0 0 0 0 @x0 taking advantage of the fact that (4.6) has unique solutions. In contrast to the Sturm-Liouville case, the Volterra integral equation (4.6) is not suitable to determine the asymptotic behavior of solutions as E tends to in nity. The following treatment circumvents this di culty and closely follows the outline in 62], Section 1.4. Suppose L sati es Hypothesis 4.1, p; q 2 C n(R ), and then de ne recursively, a1(x) = iQ(x);

0

x0

eE(x?x )J JQ(x0 )Y (x0 )dx0

0

(4.6)

bk (x) = ?i Q(t)ak (t)dt; 0 ak+1(x) = ?ak;x(x) + iQ(x)bk (x); k = 1; : : : ; n: Next let A : R2 ! M 2 (C ) be the unique solution of the integral equation A(x; y) = an+1 (x ? y) +

Introducing

Z y

Z x

(4.11)

0

Q(x ? y0)

Z x?y

0

y?y

0

Q(x0 )A(x0; y ? y0)dx0 dy0:

Z x

(4.12) (4.13)

an(E; x) = ^

Z x

0

A(x; y)e?2iEy dy;

^n (E; x) = ?i b

23

0

Q(y)^n(E; y)dy a

and

u1(E; x) = I + u2(E; x) =

we infer that

n X k=1

n X k=1

bk (x)(2iE )?k + ^n (E; x)(2iE )?n; b

(4.14) (4.15)

ak (x)(2iE )?k + an(E; x)(2iE )?n; ^

Y1(E; x) = eiExf(I + iJ )u1(E; x) + (I ? iJ )u2(E; x)g; (4.16) Y2(E; x) = e?iExf(I ? iJ )u1(?E; x) ? (I + iJ )u2(?E; x)g (4.17) satisfy the di erential equation JY 0 + QY = EY: (4.18) Since jA(x; y)j is bounded on compact subsets of R 2 one obtains the estimates jeiExan(E; x)j; jeiEx^n (E; x)j CR2 ejx Im(E)j ^ b (4.19) for a suitable constant C > 0 as long as jxj is bounded by some R > 0. ^ The matrix Y (E; x; x0 ) = (Y1(E; x ? x0) + Y2(E; x ? x0))=2 is also a solution of ^ JY 0 + QY = EY and satis es Y (E; x0 ; x0) = I + Q(x0 )=(2E ). Therefore, at least for su ciently large jE j, the matrix function ^ ^ (E; ; x0; I ) = Y (E; ; x0)Y (E; x0 ; x0)?1 (4.20) is the unique solution of the initial value problem JY 0 + QY = EY , Y (x0 ) = I . Hence, if p; q 2 C 2(R ), one obtains the asymptotic expansion ?iE ?iE 1 0 (E; x0 + ; x0 ; I ) = e 0 eiE + 2iE 2p(x e sin(E ) 2q(x0) sin(E ) ? eiE 0) + O(ej Im(E)j E ?2); (4.21) where

=

Z x0 +

From this result we infer in particular that the entries of ( ; x0 + ; x0 ; I ), which are entire functions, have order one whenever q(x0 ) and p(x0) are nonzero. Denote by T the operator de ned by Ty = y( + ) on the set of C 2 -valued functions on R and suppose L satis es Hypothesis 4.1. Then T and L commute and this implies that T (E ), the restriction of T to the (two-dimensional) space V (E ) of solutions of Ly = Ey, maps V (E ) into itself. Choosing as a basis of V (E ) the columns of (E; ; x0; I ), the operator T (E ) is represented by the matrix (E; x0 + ; x0 ; I ). In

24

x0

p(t)q(t)dt:

(4.22)

particular, det(T (E )) = det( (E; x0 + ; x0 ; I )) = 1. Therefore, the eigenvalues (E ) of T (E ), the so called Floquet multipliers, are determined as solutions of 2 ? tr(T (E )) + 1 = 0: (4.23) These eigenvalues are degenerate if and only if 2(E ) = 1 which happens if and only if the equation Ly = Ey has a solution of period 2 . Hence we now study asymptotic properties of the spectrum of the densely de ned closed realization H2 ;x0 of L in L2 ( x0 ; x0 + 2 ])2 given by H2 ;x0 y = Ly; D(H2 ;x0 ) = fy 2 L2 ( x0 ; x0 + 2 ])2 j y 2 AC ( x0 ; x0 + 2 ])2 ; y(x0 + 2 ) = y(x0); Ly 2 L2 ( x0 ; x0 + 2 ])2g: (4.24) Its eigenvalues, which are called the (semi-)periodic eigenvalues of L, and their multiplicities are given, respectively, as the zeros and their multiplicities of the function tr(T (E ))2 ? 4. The asymptotic behavior of these eigenvalues is described in the following result. Theorem 4.2. Suppose that p; q 2 C 2(R ). Then the eigenvalues Ej , j 2 Z of H2 ;x0 are x0 -independent and satisfy the asymptotic behavior E2j ; E2j?1 = j + O( j1j ) (4.25) j as jj j tends to in nity, where all eigenvalues are repeated according to their algebraic multiplicities. In particular, all eigenvalues of H2 ;x0 are contained in a strip = fE 2 C j j Im(E )j C g (4.26) for some constant C > 0. Proof. Denoting A(E; x) = (E + Q(x))J (cf.(3.18)), equation (4.10) implies @ (E; x; x0 ; I )=@x = ?A(E; x) (E; x; x0 ; I ); (4.27) @ (E; x; x0 ; I )=@x0 = (E; x; x0 ; I )A(E; x) (4.28) and hence @ tr(T (E ))=@x0 = 0: (4.29) Thus the eigenvalues of H2 ;x0 are independent of x0. According to (4.21), tr(T (E )) is asymptotically given by tr(T (E )) = 2 cos(E ) + sin(E )E ?1 + O(ej Im(E)j E ?2): (4.30) Rouche's theorem then implies that two eigenvalues E lie in a circle centered at j =a with radius of order 1=jj j. To prove that the eigenvalues may be labeled in the manner indicated, one again uses Rouche's theorem with a circle of su ciently large radius centered at the origin of the E -plane in order to compare the number of zeros of tr(T (E ))2 ?4 and 4 cos(E )2 ? 4 in the interior of this circle.

25

The conditional stability set S (L) of L in (4.1) is de ned to be the set of all spectral parameters E such that Ly = Ey has at least one bounded nonzero solution. This happens if and only if the Floquet multipliers (E ) of Ly = Ey have absolute value one. Hence S (L) = fE 2 C j ? 2 tr(T (E )) 2g: (4.31) It is possible to prove that the spectrum of H coincides with the conditional stability set S (L) of L, but since we do not need this fact we omit a proof. In the following we record a few properties of S (L) to be used in Sections 5 and 6. Theorem 4.3. Assume that p; q 2 C 2(R ). Then the conditional stability set S (L) consists of a countable number of regular analytic arcs, the so called spectral bands. At most two spectral bands extend to in nity and at most nitely many spectral bands are closed arcs. The point E is a band edge, that is, an endpoint of a spectral band, if and only if tr(T (E ))2 ? 4 has a zero of odd order. Proof. The fact that S (L) is a set of regular analytic arcs whose endpoints are odd order zeros of (tr(T (E )))2 ? 4 and hence countable in number, follows in standard manner from the fact that tr(T (E )) is entire with respect to E . (For additional details on this problem, see, for instance, the rst part of the proof of Theorem 4.2 in 101].) From the asymptotic expansion (4.21) one infers that tr(T (E )) is approximately equal to 2 cos(E ) for jE j su ciently large. This implies that the Floquet multipliers are in a neighborhood of e iE . If E0 2 S (L) and jE0j is su ciently large, then it is close to a real number. Now let E = jE0jeit , where t 2 (? =2; 3 =2]. Whenever this circle intersects S (L) then t is close to 0 or . When t is close to 0, the Floquet multiplier which is near eiE moves radially inside the unit circle while the one close to e?iE leaves the unit disk at the same time. Since this can happen at most once, there is at most one intersection of the circle of radius jE0j with S (L) in the right half-plane for jE j su ciently large. Another such intersection may take place in the left half-plane. Hence at most two arcs extend to in nity and there are no closed arcs outside a su ciently large disk centered at the origin. Since there are only countably many endpoints of spectral arcs, and since outside a large disk there can be no closed spectral arcs, and at most two arcs extend to in nity, the conditional stability set consists of at most countably many arcs. Subsequently we need to refer to components of vectors in C 2 . If y 2 C 2 , we will denote the rst and second components of y by y1 and y2, respectively, that is, y = (y1; y2)t , where the superscript \t" denotes the transpose of a vector in C 2 . The boundary value problem Ly = zy, y1(x0 ) = y1(x0 + ) = 0 in close analogy to the scalar Sturm-Liouville case, will be called the Dirichlet problem for the interval x0 ; x0 + ] and its eigenvalues will therefore be called Dirichlet eigenvalues (associated with the interval x0; x0 + ]). In the corresponding operator theoretic formulation

26

one introduces the following closed realization HD;x0 of L in L2 ( x0 ; x0 + ])2 , HD;x0 y = Ly; D(HD;x0 ) = fy 2 L2 ( x0 ; x0 + ])2 j y 2 AC ( x0 ; x0 + ])2 ; y1(x0) = y1(x0 + ) = 0; Ly 2 L2 ( x0 ; x0 + ])2 g: (4.32) The eigenvalues of HD;x0 and their algebraic multiplicities are given as the zeros and their multiplicities of the function g(E; x0) = (1; 0) (E; x0 + ; x0 ; I )(0; 1)t; (4.33) that is, the entry in the upper right corner of (E; x0 + ; x0 ; I ). Theorem 4.4. Suppose p; q 2 C 2(R ). If q(x0) 6= 0 then there are countably many Dirichlet eigenvalues j (x0 ), j 2 Z, associated with the interval x0 ; x0 + ]. These eigenvalues have the asymptotic behavior j + O( 1 ) (4.34) j (x0 ) = jj j as jj j tends to in nity, where all eigenvalues are repeated according to their algebraic multiplicities. Proof. From the asymptotic expansion (4.21) we obtain that ( g(E; x0) = ?iqEx0 ) sin(E ) + O(ej Im(E)j E ?2 ): (4.35) Rouche's theorem implies that one eigenvalue E lies in a circle centered at j = with radius of order 1=jj j and that the eigenvalues may be labeled in the manner indicated (cf. the proof of Theorem 4.2). We now turn to the x-dependence of the function g(E; x). Theorem 4.5. Assume that p; q 2 C 1(R ). Then the function g(E; ) satis es the di erential equation q(x)(2g(E; x)gxx(E; x) ? gx(E; x)2 + 4(E 2 ? p(x)q(x))g(E; x)2) ? qx(x)(2g(E; x)gx(E; x) + 4iEg(E; x)2) = ?q(x)3 (tr(T (E ))2 ? 4): (4.36) Proof. Since g(E; x) = (1; 0) (E; x; x + ; I )(0; 1)t we obtain from (4.27) and (4.28), gx(E; x) =(1; 0)( (E; x; x + ; I )A(E; x) ? A(E; x) (E; x; x + ; I ))(0; 1)t; (4.37) gxx(E; x) =(1; 0)( (E; x; x + ; I )A(E; x)2 ? 2A(E; x) (E; x; x + ; I )A(E; x) + A(E; x)2 (E; x; x + ; I ) + (E; x; x + ; I )Ax(E; x) ? Ax(E; x) (E; x; x + ; I ))(0; 1)t; (4.38) where we used periodicity of A, that is, A(E; x + ) = A(E; x). This yields the desired result upon observing that tr( (z; x + ; x; I )) = tr(T (E )) is independent of x.

27

De nition 4.6. The algebraic multiplictiy of E as a Dirichlet eigenvalue (x) of

HD;x is denoted by (E; x). The quantities (4.39) i (E ) = minf (E; x) j x 2 R g; and (4.40) m (E; x) = (E; x) ? i (E ) will be called the immovable part and the movable part of the algebraic multiplicP ity (E; x), respectively. The sum E2C m(E; x) is called the number of movable Dirichlet eigenvalues. If q(x) 6= 0 the function g( ; x) is an entire function with order of growth equal to one. The Hadamard factorization theorem then implies g(E; x) = FD (E; x)D(E ); (4.41) where Y FD (E; x) = gD (x)ehD (x)E E m (0;x) (1 ? (E= )) m(z;x)e m (z;x)E ; (4.42) D(E ) = ed0 E E i (E)

Y

2C nf0g

(1 ? (E= )) i(E)e i(E)E ;

2C nf0g

(4.43)

for suitable numbers gD (x) and hD (x) and d0. De ne U (E ) = (tr(T (E ))2 ? 4)=D(E )2: (4.44) Then Theorem 4.2 shows that ? q(x)3 U (E ) =q(x)(2FD (E; x)FD;xx(E; x) ? FD;x(E; x)2 + 4(E 2 ? q(x)p(x))FD (E; x)2 ) ? qx(x)(2FD (E; x)FD;x(E; x0 ) + 4iEFD (E; x)2): (4.45) As a function of E the left-hand side of this equation is entire (see Proposition 5.2 in 101] for an argument in a similar case). Introducing s(E ) = ordE (tr(T (E ))2 ? 4) we obtain the following important result. Theorem 4.7. Under the hypotheses of Theorem 4.5, s(E ) ? 2 i(E ) 0 for every E 2 C. We now de ne the sets E1 = fE 2 C j s(E ) > 0; i(E ) = 0g and E2 = fE 2 C j s(E ) ? 2 i (E ) > 0g. Of course, E1 is a subset of E2 which, in turn, is a subset of the set of zeros of tr(T (E ))2 ? 4 and hence isolated and countable.

28

Theorem 4.8. Assume Hypothesis 4.1 and that Ly = Ey has degenerate Floquet multipliers (equal to 1) but two linearly independent Floquet solutions. Then E is an immovable Dirichlet eigenvalue, that is, i(E ) > 0. Moreover, E1 is contained

in the set of all those values of E such that Ly = Ey does not have two linearly independent Floquet solutions. Proof. If Ly = Ey has degenerate Floquet multipliers (E ) but two linearly independent Floquet solutions then every solution of Ly = Ey is Floquet with multiplier (E ). This is true, in particular, for the unique solution y of the initial value problem Ly = Ey, y(x0) = (0; 1)t. Hence y(x0 + ) = (0; )t and y is a Dirichlet eigenfunction regardless of x0 , that is, i(E ) > 0. If E 2 E1 then s(E ) > 0 and Ly = Ey has degenerate Floquet multipliers. Since (E ) = 0, there cannot be two linearly independent Floquet solutions. i Spectral theory for nonself-adjoint periodic Dirac operators has very recently drawn considerable attention in the literature and we refer the reader to 46] and 92].

5. Floquet Theory and Algebro-Geometric Potentials In this section we will obtain necessary and su cient conditions in terms of Floquet theory for a function Q : R ! M2 (C ) which is periodic with period > 0 and which has zero diagonal entries to be algebro-geometric (cf. De nition 2.2). Throughout this section we assume the validity of Hypothesis 4.1. We begin with su cient conditions on Q and recall the de nition of U (E ) in (4.44). Theorem 5.1. Suppose that p; q 2 C 2(R ) are periodic with period > 0. If U (E ) is a polynomial of degree 2n + 2 then the following statements hold. (i) deg(U ) is even, that is, n is an integer. (ii)The number of movable Dirichlet eigenvalues (counting algebraic multiplicities) equals n. (iii) S (L) consists of nitely many regular analytic arcs. (iv) p; q 2 C 1(R ). (v) There exists a 2 2 matrix-valued di erential expression Pn+1 of order n + 1 with leading coe cient J n+2 which commutes with L and satis es Y 2 (5.1) Pn+1 = (L ? E )s(E)?2 i (E) :

Proof. The asymptotic behavior of Dirichlet and periodic eigenvalues (Theorems 4.2 and 4.4) shows that s(E ) 2 and (E; x) 1 when jE j is suitably large. Since (U (E ) is a polynomial, s(E ) > 0 implies that s(E ) = 2 i(E ) = 2. If (x) is a Dirichlet eigenvalue outside a su ciently large disk, then it must be close to m = for some integer m and hence close to a point E where s(E ) = 2 i(E ) = 2. Since there is only one Dirichlet eigenvalue in this vicinity we conclude that (x) = E is independent of x. Hence, outside a su ciently large disk, there is no movable

29

E 2F2

Dirichlet eigenvalue, that is, FD ( ; x) is a polynomial. Denote its degree, the number of movable Dirichlet eigenvalues, by n. By (4.45) U (E ) is a polynomial of degree ~ 2~ + 2. Hence n = n and this proves parts (i) and (ii) of the theorem. n ~ Since asymptotically s(E ) = 2, we infer that s(E ) = 1 or s(E ) 3 occurs at only nitely many points E . Hence, by Theorem 4.3, there are only nitely many band edges, that is, S (L) consists of nitely many arcs, which is part (iii) of the theorem. Let (x) be the leading coe cient of FD ( ; x). From equation (4.45) we infer that ? (x)2 =q(x)2 is the leading coe cient of U (E ) and hence (x) = ciq(x) for a suitable constant c. Therefore, F ( ; x) = FD ( ; x)=c is a polynomial of degree n with leading coe cient iq(x) satisfying the hypotheses of Theorem 2.3. This proves that p; q 2 C 1(R ) and that there exists a 2 2 matrix-valued di erential expression Pn+1 of order n + 1 and leading coe cient J n+2 which commutes with L. The di erential 2 expressions Pn+1 and L satisfy Pn+1 = R2n+2 (L), where

R2n+2(E ) = U (E )=(4c2 ) =

Y

2F2

(E ? )s( )?2 i ( ) ;

(5.2)

concluding parts (iv) and (v) of the theorem. Theorem 5.2. Suppose that p; q 2 C 2 (R ) are periodic of period > 0 and that the di erential equation Ly = Ey has two linearly independent Floquet solutions for all but nitely many values of E . Then U (E ) is a polynomial. Proof. Assume that U (E ) in (4.44) is not a polynomial. At any point outside a large disk where s(E ) > 0 we have two linearly independent Floquet solutions and hence, by Theorem 4.8, i(E ) 1. On the other hand, we infer from Theorem 4.2 that s(E ) 2 and hence s(E ) ? 2 i(E ) = 0. Therefore, s(E ) ? 2 i(E ) > 0 happens only at nitely many points and this contradiction proves that U (E ) is a polynomial. Theorem 5.3. Suppose that p; q 2 C 2 (R ) are periodic of period > 0 and that the associated Dirichlet problem has n movable eigenvalues for some n 2 N . Then U (E ) is a polynomial of degree 2n + 2. Proof. If there are n movable Dirichlet eigenvalues, that is, if deg(FD ( ; x)) = n then (4.45) shows that U (E ) = (tr(T (E ))2 ? 4)=D(E )2 is a polynomial of degree 2n +2. Next we prove that U (E ) being a polynomial, or the number of movable Dirichlet eigenvalues being nite, is also a necessary condition for Q to be algebro-geometric. Theorem 5.4. Suppose L satis es Hypothesis 4.1. Assume there exists a 2 2 matrix-valued di erential expression Pn+1 of order n + 1 with leading coe cient J n+2 which commutes with L but that there is no such di erential expression of smaller order commuting with L. Then U (E ) is a polynomial of degree 2n + 2.

30

^ Proof. Without loss of generality we may assume that Pn+1 = Pc1;:::;cn+1 for suitable constants cj . According to the results in Section 2, the polynomial

Fn(E; x) =

n X `=0

fn?`(c1 ; :::; cn?`)(x)E `

(5.3)

satis es the hypotheses of Theorem 2.3. Hence the coe cients f` and the functions p and q are in C 1(R). Also the f`, and hence Pn+1, are periodic with period . Next, let (x0) be a movable Dirichlet eigenvalue. Since (x) is a continuous function of x 2 R and since it is not constant, there exists an x0 2 R such that s( (x0)) = 0, that is, (x0) is neither a periodic nor a semi-periodic eigenvalue. Suppose that for this choice of x0 the eigenvalue := (x0 ) has algebraic multiplicity k. Let V = ker((HD;x0 ? )k ) be the algebraic eigenspace of . Then V has a basis fy1; :::; yk g such that (HD;x0 ? )yj = yj?1 for j = 1; :::; k, agreeing that y0 = 0. Moreover, we introduce Vm := span fy1; :::; ymg and V0 = f0g. First we show by induction that there exists a number such that (T ? )y; (Pn+1 ? )y 2 Vm?1 , whenever y 2 Vm. Let m = 1. Then (HD;x0 ? )y = 0 implies y = y1 for some constant and hence y is a Floquet solution with multiplier = y1;2(x0 + ), that is, (T ? )y = 0. (We de ne, in obvious notation, yj;k , k = 1; 2 to be the k-th component of yj , 1 j m.) Since Pn+1 commutes with both L and T , we nd that Pn+1y is also a Floquet solution with multiplier . Since s( ) = 0, the geometric eigenspace of is one-dimensional and hence Pn+1y = y for a suitable constant . P Now assume that the statement is true for 1 m < k. Let y = m+1 j yj 2 Vm+1 . j =1 Note that (T ? )y satis es Dirichlet boundary conditions. Hence (HD;x0 ? )(T ? )y = (T ? )

m+1 X j =1 j (HD;x0 ?

)yj =

m+1 X j =1

j (T

? )yj?1

(5.4)

Pm? is an element of Vm?1 , say equal to v = j=11 j yj . The nonhomogeneous equation (HD;x0 ? )w = v has the general solution

w=

m?1 X j =1

j yj +1 +

y1 ;

(5.5)

where is an arbitary constant. Since w is in Vm , the particular solution (T ? )y of (HD;x0 ? )w = v is in Vm too. Since Q is in nitely often di erentiable, so are the functions y1; :::; yk . Hence L can be applied to (Pn+1 ? )y and one obtains (L ? )(Pn+1 ? )y =

m+1 X j =1 j (Pn+1 ?

31

)yj?1 =

m?1 X j =1

j yj

(5.6)

for suitable constants j . Thus there are numbers (Pn+1 ? )y =

m?1 X j =1

1

and

2

such that (5.7)

^ j yj +1 + 1 y + 2 y1 ;

where y is the solution of Ly = y with y(x0 ) = (1; 0)t. Note that (Pn+1y ? y)1 (x0 ) = ^ ^ (Pn+1y)1(x0 ) = 1. Let w = (T ? )y and v = (Pn+1 ? )w. Then w 2 Vm and v 2 Vm?1 . Hence, (Pn+1y ? y)1(x0 + a) = (T (Pn+1 ? )y)1(x0 ) = ((Pn+1 ? )Ty)1(x0) = (Pn+1( y + w))1(x0 ) = (Pn+1y)1(x0 ) + (Pn+1w)1(x0 ) = 1 : (5.8) On the other hand, (Pn+1y ? y)1(x0 + a) = 1 = since y1(x0 + a) = 1= . Thus ^ 0 = 1( ? 1= ) which implies 1 = 0 and (Pn+1 ? )y 2 Vm. Hence we have shown that T and Pn+1 map V into itself. In particular, (Pn+1y)1(x0 ) = 0 for every y 2 V . Next observe that the functions y1; :::; yk de ned above satisfy (L ? )j ym = ym?j , agreeing that ym = 0 whenever m 0. Consequently, j j X j X j r r (L ? )j ?r y = jy = (5.9) L m m r ym+r?j : r r=0 r=0 Moreover,

Pn+1ym = ?

=?

n+1 X

j r ign+1?j ym+r?j;1 ? ifn?j ym+r?j;2 : (5.10) ?ign+1?j ym+r?j;2 + ihn?j ym+r?j;1 j =0 r=0 r Since (Pn+1yj )1(x0 ) = yj;1(x0 ) = 0, evaluating the rst component of (5.10) at x0 yields n+1 n+1 X X 0 = (Pn+1ym)1 (x0) = i ym?`;2(x0 ) 1 j:::(j ? ` + 1) j?`fn?j (x0 ) `! j=` `=0 n X 1 ` = i ym?`;2(x0 ) `! @ F`n ( ; x0): (5.11) @E `=0 Letting m run from 1 through k shows that is a zero of Fn( ; x0) of order at least k. Therefore, there can be at most n movable Dirichlet eigenvalues counting multiplicities. However, if there were less than n movable Dirichlet eigenvalues then, by Theorems 5.1 and 5.3, there would exist a di erential expression of order less than

32

j =0 n+1 X X j

(gn+1?j J + iAn?j ) Lj ym

n + 1 which commutes with L without being a polynomial of L. Hence there are precisely n movable Dirichlet eigenvalues and deg (U ) = 2n + 2. 6. A Characterization of Elliptic Algebro-Geometric AKNS

Picard's theorem yields su cient conditions for a linear (scalar) nth -order di erential equation, whose coe cients are elliptic functions with a common period lattice spanned by 2!1 and 2!3, to have a fundamental system of solutions which are elliptic of the second kind. We start by generalizing Picard's theorem to rst-order systems. Let Tj , j = 1; 3, be the operators de ned by Tj y = y( + 2!j ). In analogy to the scalar case we call y elliptic of the second kind if it is meromorphic and y( + 2!j ) = j y( ) for some j 2 C n f0g; j = 1; 3: (6.1) Theorem 6.1. Suppose that the entries of A : C ! Mn(C 1 ) are elliptic functions with common fundamental periods 2!1 and 2!3 . Assume that the rst-order di erential system 0 = A has a meromorphic fundamental system of solutions. Then there exists at least one solution 1 which is elliptic of the second kind. If in addition, the restriction of either T1 or T3 to the (n-dimensional) space W of solutions of 0 = A has distinct eigenvalues, then there exists a fundamental system of solutions of 0 = A which are elliptic of the second kind. Proof. T1 is a linear operator mapping W into itself and thus has an eigenvalue 1 and an associated eigenfunction u1, that is, 0 = A has a solution u1 satisfying u1(x + 2!1) = 1u1(x). Now consider the functions u1(x); u2(x) = u1(x + 2!3); :::; um(x) = u1(x + 2(m ? 1)!3); (6.2) where m 2 f1; :::; ng is chosen such that the functions in (6.2) are linearly independent but including u1(x +2m!3 ) would render a linearly dependent set of functions. Then, um(x + 2!3) = b1 u1(x) + :::: + bm um(x): (6.3) ~ Next, denote the restriction of T3 to the span V of fu1; :::; umg by T3. It follows ~ ~ from (6.3) that the range of T3 is again V . Let 3 be an eigenvalue of T3 and v the associated eigenvector, that is, v is a meromorphic solution of the di erential equation 0 = A satisfying v (x + 2!3 ) = 3 v (x). But v also satis es v (x + 2!1 ) = 1 v (x) since every element of V has this property. Hence v is elliptic of the second kind. The numbers 1 and 3 are the Floquet multipliers corresponding to the periods 2!1 and 2!3, respectively. The process described above can be performed for each multiplier corresponding to the period 2!1. Moreover, the roles of 2!1 and 2!3 may of course be interchanged. The last statement of the theorem follows then from the observation that solutions associated with di erent multipliers are linearly independent.

33

Potentials

What we call Picard's theorem following the usual convention in 3], p. 182{185, 18], p. 338{343, 48], p. 536{539, 57], p. 181{189, appears, however, to have a longer history. In fact, Picard's investigations 76]{ 78] in the scalar nth -order case were inspired by earlier work of Hermite in the special case of Lame's equation 49], p. 118{122, 266{418, 475{478 (see also 9], Sect. 3.6.4 and 102], p. 570{ 576). Further contributions were made by Mittag-Le er 65], and Floquet 34]{ 36]. Detailed accounts on Picard's di erential equation can be found in 48], p. 532{574, 57], p. 198{288. For a recent extension of Theorem 6.1 see 38]. Picard's Theorem 6.1 motivates the following de nition. De nition 6.2. A 2 2 matrix Q whose diagonal entries are zero and whose o diagonal entries are elliptic functions with a common period lattice is called a PicardAKNS potential if and only if the di erential equation J 0 + Q = E has a meromorphic fundamental system of solutions (with respect to the independent variable) for in nitely many values of the spectral parameter E 2 C . Recall from Theorem 3.3 that J 0 + Q = E has a meromorphic fundamental system of solutions for all values of E if this is true for a su ciently large nite number of values of E . In the following assume, without loss of generality, that Re(!1) > 0, Re(!3) 0, Im(!3 =!1) > 0. The fundamental period parallelogram then consists of the points E = 2!1s + 2!3t, where 0 s; t < 1. We introduce 2 (0; ) by ! ei = !3 !1 (6.4) 1 !3 and for j = 1; 3, Qj ( ) = tj Q(tj + x0 ); (6.5) where tj = !j =j!j j. Subsequently, the point x0 will be chosen in such a way that no pole of Qj , j = 1; 3 lies on the real axis. (This is equivalent to the requirement that no pole of Q lies on the line through the points x0 and x0 + 2!1 nor on the line through x0 and x0 + 2!3. Since Q has only nitely many poles in the fundamental period parallelogram this can always be achieved.) For such a choice of x0 we infer that the entries of Qj ( ) are real-analytic and periodic of period j = 2j!j j whenever is restricted to the real axis. Using the variable transformation x = tj + x0 , (x) = ( ) one concludes that is a solution of J 0 (x) + Q(x) (x) = E (x) (6.6) if and only if is a solution of J 0 ( ) + Qj ( ) ( ) = ( ); (6.7) where = tj E . Theorem 4.2 is now applicable and yields the following result.

34

Theorem 6.3. Let j = 1 or 3. Then all 4!j -periodic (i.e., all 2!j -periodic and all

2!j -semi-periodic) eigenvalues associated with Q lie in the strip Sj given by Sj = fE 2 C j j Im(tj E )j Cj g (6.8) for suitable constants Cj > 0. The angle between the axes of the strips S1 and S3 equals 2 (0; ). Theorem 6.3 applies to any elliptic potential Q whether or not it is algebrogeometric. Next we present our principal result, a characterization of all elliptic algebro-geometric potentials of the AKNS hierarchy. Given the preparations in Sections 3{5, the proof of our principal result, Theorem 6.4 below, will be fairly short. Theorem 6.4. Q is an elliptic algebro-geometric AKNS potential if and only if it is a Picard-AKNS potential. Proof. The fact that any elliptic algebro-geometric AKNS potential is a Picard potential is a special case of Theorem 3.12. Conversely, assume that Q is a Picard-AKNS potential. Choose R > 0 large enough such that the exterior of the closed disk D(0; R) of radius R centered at the origin contains no intersection of S1 and S3 (de ned in (6.8)), that is, (C nD(0; R)) \ (S1 \ S3) = ;: (6.9) Let j; ( ) be the Floquet multipliers of Qj , that is, the solutions of 2 ? tr(T ) + 1 = 0: (6.10) j j j Then (6.9) implies that for E 2 C nD(0; R) at most one of the eigenvalues 1(t1 E ) and 3 (t3E ) can be degenerate. In particular, at least one of the operators T1 and T3 has distinct eigenvalues. Since by hypothesis Q is Picard, Picard's Theorem 6.1 applies with A = ?J (Q ? E ) and guarantees the existence of two linearly independent solutions 1 (E; x) and 2 (E; x) of J 0 + Q = E which are elliptic of the second kind. Then j;k ( ) = k (tj + x0 ), k = 1; 2 are linearly independent Floquet solutions associated with Qj . Therefore the points for which J 0 + Qj = has only one Floquet solution are necessarily contained in D(0; R) and hence nite in number. This is true for both j = 1 and j = 3. Applying Theorem 5.2 then proves that both Q1 and Q3 are algebro-geometric. This implies that Q itself is algebro-geometric. The following corollary slightly extends the class of AKNS potentials Q(x) considered thus far in order to include some cases which are not elliptic but very closely related to elliptic Q(x). Such cases have recently been considered by Smirnov 89]. Corollary 6.5. Suppose ?2(ax+b) (6.11) Q(x) = ip(x)e02(ax+b) ?iq(x)e 0

35

0 ~ Q = ip ?0iq : (6.13) ~ Moreover, L = E is equivalent to L(T ) = (E ? ia)(T ). Hence the equation ~ L = (E ? ia) has a meromorphic fundamental system of solutions for all E . ~ Consequently, Theorem 6.4 applies and yields that Q is an algebro-geometric AKNS ~ potential. Thus, for some n there exists a di erential expression P of order n +1 with ~ ~ ~ leading coe cient ?J n such that P; L] = 0. De ne P = T ?1 P T . The expression P is a di erential expression of order n + 1 with leading coe cient ?J n and satis es ~ ~ P; L] = T ?1 P; L + iaI ]T = 0, that is, Q is an algebro-geometric AKNS potential. The converse follows by reversing the above proof. We add a series of remarks further illustrating the signi cance of Theorem 6.4. Remark 6.6. While an explicit proof of the algebro-geometric property of (p; q) is in general highly nontrivial (see, e.g., the references cited in connection with special cases such as the Lame-Ince and Treibich-Verdier potentials in the introduction), the fact of whether or not J 0 (x) + Q(x) (x) = E (x) has a fundamental system of solutions meromorphic in x for all but nitely many values of the spectral parameter E 2 C can be decided by means of an elementary Frobenius-type analysis (see, e.g., 39] and 40]). To date, Theorem 6.4 appears to be the only e ective tool to identify general elliptic algebro-geometric solutions of the AKNS hierarchy. Remark 6.7. Theorem 6.4 complements Picard's Theorem 6.1 in the special case where A(x) = ?J (Q(x) ? E ) in the sense that it determines the elliptic matrix functions Q which satisfy the hypothesis of the theorem precisely as (elliptic) algebrogeometric solutions of the stationary AKNS hierarchy. Remark 6.8. Theorem 6.4 is also relevant in the context of the Weierstrass theory of reduction of Abelian to elliptic integrals, a subject that attracted considerable interest, see, for instance, 7], 8], 9], Ch. 7, 10], 11], 21], 29]{ 31], 53], 56], 58],

36

where a; b 2 C and p; q are elliptic functions with a common period lattice. Then Q is an algebro-geometric AKNS potential if and only if J 0 + Q = E has a meromorphic fundamental system of solutions (with respect to the independent variable) for all values of the spectral parameter E 2 C . Proof. Suppose that for all values of E the equation L = J 0 + Q = E has a meromorphic fundamental system of solutions. Let ax+b 0 (6.12) T = e 0 e?ax+b : ~ ~ Then T LT ?1 = L + iaI = Jd=dx + QiaI , where

63], 83], 84], 90]. In particular, the theta functions corresponding to the hyperelliptic curves derived from the Burchnall-Chaundy polynomials (2.25), associated with Picard potentials, reduce to one-dimensional theta functions. 7. Examples With the exception of the studies by Christiansen, Eilbeck, Enol'skii, and Kostov in 21] and Smirnov in 89], not too many examples of elliptic solutions (p; q) of the AKNS hierarchy associated with higher (arithmetic) genus curves of the type (2.25) have been worked out in detail. The genus n = 1 case is considered, for example, in 52], 75]. Moreover, examples for low genus n for special cases such as the nonlinear Schrodinger and mKdV equation (see (2.46) and (2.49)) are considered, for instance, in 5], 8], 61], 64], 74], 88]. Subsequently we will illustrate how the Frobenius method, whose essence is captured by Proposition 3.2, can be used to establish existence of meromorphic solutions and hence, by Theorem 6.4, proves their algebro-geometric property. The notation established in the beginning of Section 3 will be used freely in the following. Example 7.1. Let p(x) = q(x) = n( (x) ? (x ? !2) ? 2 ); (7.1) where n 2 N . The potential (p; q) has two poles in the fundamental period parallelogram. Consider rst the pole x = 0. In this case we have 0 R= n n ; (7.2) 0 whose eigenvalues are n, that is, = n. Moreover, since p = q is odd, we have p2j?1 = q2j?1 = 0. One proves by induction that b(2) is a multiple of (1; 1)t and that 2j (2) t . Hence b(2) is a multiple of (1; ?1)t , that is, it is in the b2j?1 is a multiple of (1; ?1) 2n?1 range of R ? n. Hence every solution of L = E is meromorphic at zero regardless of E . Next consider the pole x = !2 and shift coordinates by introducing = x ? !2. Then we have p(x) = q(x) = n( ( + !2) ? ( ) ? 2 ) = ?p( ) and hence 0 R = ?n ?n : (7.3) 0 One can use again a proof by induction to show that b(2)?1 is in the range of R ? n, 2n which is spanned by (1; 1)t. Hence we have shown that the matrix ( Q(x) = in( (x) ? (0 ? ! ) ? ) ?in( (x) ? 0x ? !2 ) ? 2 ) (7.4) x is a Picard-AKNS and therefore an algebro-geometric AKNS potential.

37

2

2

Example 7.2. Here we let p = 1 and q = n(n + 1)}(x), where n 2 N . Then we have

just one pole in the fundamental period parallelogram. In this case we obtain R = 1 n(n0+ 1) (7.5) 1 and = n + 1. Since q is even we infer that q2j?1 = 0. A proof by induction then shows that b(2) is a multiple of (n ? 2j; 1)t and that b(2)?1 is a multiple of (1; 0)t. In 2j 2j (2) t , which spans the range of R ? . This shows particular, b2n is a multiple of (?n; 1) that Q(x) = 0 ?in(n + 1)}(x) (7.6) i 0 is a Picard-AKNS and hence an algebro-geometric AKNS potential. Incidentally, if p = 1, then J 0 + Q = E is equivalent to the scalar equation 00 2 t 0 2 ? q 2 = ?E 2 where = ( 1 ; 2 ) and 1 = 2 ? iE 2 . Therefore, if ?q is an elliptic algebro-geometric potential of the KdV hierarchy then by Theorem 5.7 of 44] is meromorphic for all values of E 2 is meromorphic for all values of E . Hence and therefore Q is a Picard-AKNS and hence an algebro-geometric AKNS potential. Conversely if Q is an algebro-geometric AKNS potential with p = 1 then ?q is an algebro-geometric potential of the KdV hierarchy (cf. (2.48)). In particular, q(x) = n(n + 1)}(x) is the celebrated class of Lame potentials associated with the KdV hierarchy (cf., e.g., 39] and the references therein). Example 7.3. Suppose e2 = 0 and hence g2 = 4e2 and g3 = 0. Let u(x) = 1 ?}0 (x)=(2e1). Then, near x = 0, u(x) = e 1 3 ? e51 x + O(x3); (7.7) 1x and near x = !2 , e3 u(x) = e1(x !2 ) ? 351 (x !2)5 + O((x !2)7): (7.8) Now let p(x) = 3u(x) and q(x) = u(x ? !2): Then p has a third-order pole at 0 and a simple zero at !2 while q has a simple zero at zero and a third-order pole at !2. Let us rst consider the point x = 0. We have R = 3?2 e01 ; (7.9) =e1 and hence = 1. Moreover, p2 = q2 = 0, p4 = ?3e1 =5, and q4 = ?3e3=5. Since = 1 1 we have to show that b(2) is a multiple of (q0; ?1)t . We get, using p2 = q2 = 0, 3 b(2) = (q0 E 4=6 + q4 ; ?E 4=6 ? q0p4 )t; (7.10) 3

38

2 which is a multiple of (q0 ; ?1)t if and only if q4 = p4q0 , a relationship which is indeed satis ed in our example. Next consider the point x = !2. Changing variables to = x ? !2 and using the periodic properties of u we nd that p(x) = 3q( ) and q(x) = p( )=3. Thus q has a pole at = 0 and one obtains m = 2, p0 = 3e1, q0 = 1=e1, p2 = q2 = 0, p4 = ?9e3 =5, and q4 = ?e1 =5. Since = 3, we have to compute again b(2) and nd, 3 1 using p2 = q2 = 0,

b(2) = (?q0 E 4 =6 + 3q4 ; ?E 4=2 ? q0p4 )t; 3

(7.11)

2 which is a multiple of (q0; ?3)t if and only if 9q4 = p4q0 , precisely what we need. 0 (x)=(2e1 ), then Hence, if e2 = 0 and u(x) = ?}

Q(x) = 3iu0(x) ?iu(x0? !2) (7.12) is a Picard-AKNS and therefore an algebro-geometric AKNS potential. 2 Example 7.4. Again let e2 = 0. De ne p(x) = 3 (}00(x) ? e2 ) and q(x) = ?}(x ? 1 2 . First consider x = 0. We have m = ?3, p = 4, q = 1, p = q = 0, !2)=e1 0 0 2 2 p4 = ?2e2 =5, and q4 = ?e2 =5. This yields = 1 and we need to show that b(2) is a 1 1 4 multiple of (1; ?1)t. We nd, using p2 = q2 = 0 and q0 = = 1, b(2) = i(?E 5 =24 ? p4=4 ? q4 ; E 5=24 + 5p4=4 ? q4 )t : 4

(7.13) This is a multiple of (1; ?1)t if 2q4 = p4 , which is indeed satis ed. Next consider x = !2. Now q has a second-order pole, that is, we have m = 1. Moreover, 2 (7.14) q(x) = ?21 ( (x ?1! )2 + e51 (x ? !2)2 + O((x ? !2)2 ) e1 2 and

p(x) = ?2e2 + 96e4(x ? !2)4 + O((x ? !2)6): 1 1

(7.15)

We now need b(2) to be a multiple of (q0; ?2)t , which is satis ed for q2 = p2 = 0. 2 Hence, if e2 = 0, then

2 0 Q(x) = 2i(}00 (x) ? e2)=3 i}(x ?0!2 )=e1 1 is a Picard-AKNS and thus an algebro-geometric AKNS potential.

39

(7.16)

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1

E-mail address :

2

Department of Mathematics, University of Missouri, Columbia, MO 65211, USA.

fritz@@math.missouri.edu

Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294{1170, USA.

E-mail address :

rudi@@math.uab.edu

44

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