THE KDV HIERARCHY AND ASSOCIATED TRACE FORMULAS

THE KDV HIERARCHY AND ASSOCIATED TRACE FORMULAS F. GESZTESY, R. RATNASEELAN, AND G. TESCHL Abstract. A natural algebraic approach to the KdV hierarchy and its algebro-geometric

nite-gap solutions is developed. In addition, a new derivation of associated higher-order trace formulas in connection with one-dimensional Schrodinger operators is presented.

1. Introduction The purpose of this paper is to advocate a most natural algebraic approach to hierarchies of completely integrable evolution equations such as the AKNS and Toda hierarchies and a systematic treatment of associated trace formulas. Speci cally, we shall treat in great detail the simplest example of these completely integrable systems, the Korteweg-de Vries (KdV) hierarchy, and derive the corresponding higher-order trace formulas for onedimensional Schrodinger operators. Even though the main ingredients of our approach to the KdV hierarchy (to be outlined below) appear to be well-known, it seems to us that no systematic attempt to combine them all into a complete description of the KdV hierarchy and its algebro-geometric solutions has been undertaken in the literature thus far. The principal aim of this paper is to ll this gap and at the same time provide the intimate connection with general higher-order trace formulas for the associated Lax operator. The key ingredients just mentioned are a recursive approach to Lax pairs following Al'ber [1], [2] (see also [9], Ch. 12, [15]), naturally leading to the celebrated Burchnall-Chaundy polynomial [6], [7] and hence to hyperelliptic curves Kg of genus g 2 N 0 (= N [ f0g) and a classic representation of positive divisors of Kg of degree g due to Jacobi [27] and rst applied to the KdV case by Mumford [36], Section III a).1, with subsequent extensions due to McKean [33]. Finally, following a recent series of papers on trace formulas for Schrodinger operators [16]{ [19], [22]{[24] we present a new algorithm for deriving higher-order trace formulas associated with the KdV hierarchy.

F. GESZTESY, R. RATNASEELAN, AND G. TESCHL

In Section 2 we brie y review Al'ber's recursive approach to the KdV hierarchy. In particular, we illustrate the role of commuting dierential expressions of order 2g + 1, g 2 N 0 and 2, respectively, in connection with the Burchnall-Chaundy polynomial, hyperelliptic curves Kg of genus g branched at in nity, and the equations of the stationary (i.e., timeindependent) KdV hierarchy. Section 3 combines Al'ber's recursion formalism with Jacobi's representation of positive divisors of degree g of Kg as applied to the KdV case by Mumford and McKean and provides a detailed construction of the stationary KdV hierarchy and its algebro-geometric solutions. The principal new result of Section 3, summarized in (3.50){ (3.64), concern divisors of degree g +1 of Kg associated with Schrodinger-type operators with general boundary conditions of the type de ned in (3.45). In Section 4 we present a systematic extension of this body of ideas to the time-dependent KdV hierarchy going beyond the standard treatment in the literature. Especially, our t-dependent discussion in connection with divisors of degree g + 1 of Kg associated with the general eigenvalue problem (3.45) as presented in (4.36){(4.50) is without precedent. Moreover, our proof of the theta function representation (4.51) of the Baker-Akhiezer function (P; x; x0; t; t0) in Theorem 4.6, based on the fundamental meromorphic function (P; x; t) de ned in (4.15), is new. In Section 5 we turn to (higher-order) trace formulas for Schrodinger operators associated with general boundary conditions (cf. (5.3)), a key ingredient in the solution of inverse spectral problems. Unlike Sections 3 and 4, the approach in Section 5 applies to general (not necessarily algebrogeometric nite-gap) solutions of the KdV hierarchy. The principal new results of Section 5 are the (universally valid) nonlinear dierential equation (5.18) for ? (z; x), 2 R (de ned in (5.4)), the resulting recursion relation (5.21), and, in particular, our method of proof of Theorem 5.3 (i). In Appendix A we provide a brief summary on hyperelliptic curves of the KdV-type and their theta functions and establish our basic notation used in Sections 3 and 4. Finally, Appendix B provides an explicit illustration of the Riemann-Roch theorem in connection with hyperelliptic curves branched at in nity which appears to be of independent interest. We emphasize that the methods of this paper are widely applicable to 1 + 1-dimensional completely integrable systems. The corresponding account for the Toda and Kac-van Moerbeke hierarchy can be found in [5].

THE KDV HIERARCHY AND ASSOCIATED TRACE FORMULAS

2. The KdV Hierarchy, Recursion Relations, and Hyperelliptic Curves In this section we brie y review the construction of the KdV hierarchy using a recursive approach advocated by Al'ber [1], [2] (see also [9], Ch. 12, [15], [20]) and outline its connection with the Burchnall-Chaundy polynomial [6], [7] and associated hyperelliptic curves branched at in nity. Suppose

V (:; t) 2 C 1(R ); t 2 R ; V (x; :) 2 C 1(R ); x 2 R and consider the dierential expressions (Lax pair) d2 + V (x; t); L(t) = ? dx 2 g h X d ? 1 f (x; t)iL(t)g?j ; g 2 N ; (x; t) 2 R 2 ; P2g+1(t) = fj (x; t) dx 0 2 j;x j =0

where the ffj g0jg satisfy the recursion relation f0 = 1; fj;x = ? 41 fj?1;xxx + V fj?1;x + 21 Vxfj?1; 1  j  g: De ne in addition fg+1 by fg+1;x = ? 41 fg;xxx + V fg;x + 21 Vxfg : Then one computes [P2g+1; L] = 2fg+1;x; where [:; :] denotes the commutator. The Lax equation d L(t) ? [P (t); L(t)] = 0; t 2 R 2g+1 dt is then equivalent to KdVg (V ) = Vt ? 2fg+1;x; t 2 R :

(2.1)

(2.2) (2.3)

(2.4)

(2.5)

(2.6)

(2.7)

(2.8)

Varying g 2 N 0 yields the KdV hierarchy KdVg (V ) = 0; g 2 N 0 :

(2.9)

F. GESZTESY, R. RATNASEELAN, AND G. TESCHL

Explicitly, one obtains from (2.4),

f0 = 1 = f~0 ; f1 = 21 V + c1 = c1f~0 + f~1 ;

f2 = ? 81 Vxx + 38 V 2 + c1 21 V + c2 = c2 f~0 + c1 f~1 + f~2 ;

(2.10)

f3 = 321 Vxxxx ? 165 V Vxx ? 325 Vx2 + 165 V 3 + c2 21 V + c1[? 81 Vxx + 38 V 2 ] + c3 = c3f~0 + c2f~1 + c1f~2 + f~3; etc. Hence by (2.8), KdV0 (V ) = Vt ? Vx = 0; KdV1 (V ) = Vt + 41 Vxxx ? 23 V Vx ? c1 Vx;

(2.11)

KdV2 (V ) = Vt ? 161 Vxxxxx + 85 V Vxxx + 54 VxVxx ? 158 V 2 Vx ? c2Vx + c1 [ 41 Vxxx ? 23 V Vx]; etc. represent the rst few equations of the KdV hierarchy. Here c` denote integration constants which naturally arise when solving (2.4). Moreover, the corresponding homogeneous KdV equations, obtained by taking all integration constants equal to zero, c`  0, `  1 are then denoted by



^g (V ) := KdVg (V ) KdV c` 0; 1`g

(2.12)

and similarly we denote by P~2g+1 := P2g+1 (c`  0), f~j := fj (c`  0), etc. the corresponding homogeneous quantities. Before we turn to a discussion of the stationary KdV hierarchy we brie y sketch the main steps leading to (2.3){(2.8). Let Ker(L(t) ? z), z 2 C denote the two-dimensional nullspace of L(t) ? z (in the algebraic sense as opposed to the functional analytic one). We seek a representation of P2g+1 (t) on Ker(L(t) ? z) of the form

d + G (z; x; t); P2g+1 (t) Ker(L(t)?z) = Fg (z; x; t) dx g?1

(2.13)

THE KDV HIERARCHY AND ASSOCIATED TRACE FORMULAS

where Fg are polynomials in z of the type

Fg (z; x; t) = Gg?1(z; x; t) =

g X j =0 gX ?1 j =0

fg?j (x; t)zj ;

(2.14)

gg?j (x; t)zj :

(2.15)

The Lax equation (2.7) restricted to Ker(L(t) ? z) then yields = fL_ + (L ? z)P2g+1 g 0 = fL_ ? [P2g+1 ; L]g

Ker(L?z)

Ker(L?z)

n o = ? [Fg;xx + 2Gg?1;x] d + [Vt ? Fg Vx ? 2(V ? z)Fg;x ? Gg?1;xx] Ker(L?z) (2.16) dx

implying

Gg?1 = ?Fg;x=2

(2.17)

(neglecting a trivial integration constant) and (2.18) Vt = ? 21 Fg;xxx + 2(V ? z)Fg;x + VxFg : Insertion of (2.14) into (2.18) then yields (2.8). We omit further details and just record a few of the polynomials Fg , F0 = 1 = F~0 ;

F1 = c1 + 21 V + z = c1 F~0 + F~1;

F2 = c2 + c1 21 V ? 18 Vxx + 38 V 2 + (c1 + 21 V )z + z2 = c2 F~0 + c1 F~1 + F~2; etc. One veri es

P2g+1 =

g X m=0

cg?m P~2m+1 ; c0 = 1:

(2.19)

(2.20)

Finally, we specialize to the stationary KdV hierarchy characterized by Vt = 0 in (2.9) (respectively (2.8)), or more precisely, by commuting dierential expressions [P2g+1; L] = 0

(2.21)

of order 2g + 1 and 2, respectively. Eq. (2.18) then becomes

Fg;xxx ? 4(V ? z)Fg;x ? 2VxFg = 0

(2.22)

F. GESZTESY, R. RATNASEELAN, AND G. TESCHL

and upon multiplying by Fg and integrating one infers 1 F F ? 1 F 2 ? (V ? z)F 2 = R (z); 2g+1 g 2 g;xx g 4 g;x where R2g+1 (z) is of the form

(2.23)

(z ? En ); fEng0n2g  C :

(2.24)

R2g+1 (z) =

2g Y

n=0

Because of (2.21) one computes i2 h h i 2 ? (V ? z )F 2 P2g+1 Ker(L?z) = ? 21 Fg;xxFg ? 14 Fg;x g Ker(L?z) = ?R2g+1 (z ):

(2.25)

Since z 2 C is arbitrary, one obtains the Burchnall-Chaundy polynomial [6], [7] relating P2g+1 and L,

?P22g+1 = R2g+1 (L) =

2g Y

n=0

(L ? En):

(2.26)

The resulting hyperelliptic curve Kg of (arithmetic) genus g, obtained upon one-point compacti cation of the curve

y2 = R2g+1 (z) =

2g Y

n=0

(z ? En)

(2.27)

(cf. Appendix A), will be the basic ingredient in our algebro-geometric treatment of the KdV hierarchy in Sections 3 and 4. The spectral theoretic content of the polynomials Fg , Gg?1 is clearly displayed in (3.35), (3.37), (3.40){(3.44). 3. The Stationary Formalism Combining the recursion formalism of Section 2 with a polynomial approach to represent positive divisors of degree g of a hyperelliptic curve Kg of genus g originally developed by Jacobi [27] and applied to the KdV case by Mumford [36], Section III a).1 and McKean [33], we provide a detailed construction of the stationary KdV hierarchy and its algebrogeometric solutions. Our considerations (3.50){(3.64) in connection with the general boundary conditions for Schrodinger-type operators in (3.45) are new. As indicated at the end of Section 2, the stationary KdV hierarchy is intimately connected with pairs of commuting dierential expressions P2g+1 and L of orders 2g + 1 and

THE KDV HIERARCHY AND ASSOCIATED TRACE FORMULAS

2, respectively and hyperelliptic curves Kg obtained upon one-point compacti cation of the curve

y2 = R2g+1 (z) =

2g Y

n=0

(z ? En)

(3.1)

described in detail in Appendix A (whose results and notations we shall freely use in the remainder of this paper). Since we are interested in real-valued KdV solutions we now make the additional assumption

fEng0n2g  R; E0 < E1 <


< E2g ; g 2 N 0 : Writing

Fg (z; x) =

g X j =0

and combining (2.23) and (3.3) yields

0 (x)2 = ?4R j

2g+1 (j (x))

[z ? j (x)]

(3.3)

[j (x) ? k (x)]?2; 1  j  g; x 2 R :

(3.4)

fg?j (x)zj =

Yg k=1 k6=j

Yg

(3.2)

j =1

Integrating the nonlinear rst-order system (3.4) as a vector eld on the (complex) manifold Kg 


 Kg = Kgg , its solution is well-de ned as long as the 's do not collide. Since we focus on real-valued solutions V of the KdV hierarchy, we may restrict the vector eld to the g submanifold j=1~ ?1([E2j?1 ; E2j ]) which is isomorphic to the torus S 1 


 S 1 = T g . Thus

0 (x) = ?2iR1=2

Yg

[j (x) ? k (x)]?1 ; 1  j  g; x 2 R ;

(3.5)

f^j (x0 )g1jg  Kg ; ~ (^j (x0 )) = j (x0) 2 [E2j?1; E2j ]; 1  j  g

(3.6)

j

j (x)) 2g+1 (^

k=1 k6=j

with the initial conditions for some xed x0 2 R , has the unique solution f^j (x)g1jg  Kg satisfying

^j (:) 2 C 1(R ; Kg ); ~(^j (x)) 2 [E2j?1; E2j ]; 1  j  g; x 2 R :

(3.7)

These facts are veri ed using the charts (A.7), (A.8) which also shows that the solution ^j (x) changes sheets whenever it hits E2j?1 or E2j and its projection j (x) = ~(^j (x)) remains trapped in [E2j?1; E2j ] for all x 2 R .

F. GESZTESY, R. RATNASEELAN, AND G. TESCHL

Given (3.3), (3.5), and (2.17) one obtains g Yg X 1 1 0 Gg?1(z; x) = ? 2 Fg;x(z; x) = 2 j (x) [z ? k (x)] j =1 k=1 k6=j

Yg  z ? k (x)  = ?i R21g=2+1 (^j (x)) j =1 k=1 j (x) ? k (x) k6=j g X

and hence

R21g=2+1 (^j (x)) = j (x)R2g+1 (j (x))1=2 = iGg?1 (j (x); x); ^j (x) = (j (x); iGg?1(j (x); x)); 1  j  g:

Moreover, since



(3.8)

(3.9)

[R2g+1 (z) + Gg?1 (z; x)2 ] z=j (x) = 0; 1  j  g;

(3.10)

R2g+1 (z) + Gg?1(z; x)2 = Fg (z; x)Hg+1 (z; x)

(3.11)

one infers for some polynomial Hg+1 in z of degree g + 1,

Hg+1(z; x) =

Yg

[z ? `(x)]:

`=0

(3.12)

Eqs. (3.9), (3.11), and (3.12) suggest de ning f^`(x)g0`g  Kg by

R21g=2+1 (^`(x)) = ?iGg?1 (`(x); x); ^`(x) = (`(x); ?iGg?1 (`(x); x)); 0  `  g:

(3.13)

One veri es

0(x)  E0 ; ` (x) 2 [E2`?1; E2` ]; 1  `  g; x 2 R :

(3.14)

Next, we de ne the fundamental meromorphic function (P; x) on Kg ,

iR21g=2+1 (P ) ? Gg?1 (~(P ); x) iR21g=2+1 (P ) + 12 Fg;x(~(P ); x) = Fg (~(P ); x) Fg (~(P ); x) = 1=2 ?Hg+1 (~(P ); x) ; P = (~(P ); R21g=2+1(P )); x 2 R ; iR2g+1 (P ) + Gg?1(~(P ); x) with divisor ((:; x)) given by (P; x) =

((:; x)) = D^0(x)^ (x) ? DP1^(x) :

(3.15) (3.16)

THE KDV HIERARCHY AND ASSOCIATED TRACE FORMULAS

Here we abbreviated

^(x) = (^1(x); : : : ; ^g (x)); ^ (x) = (^1(x); : : : ; ^g (x)):

(3.17)

Given (P; x) we de ne the stationary Baker-Akhiezer (BA) function (P; x; x0), meromorphic on Kg nfP1g, by (P; x; x0 ) = exp

hZ x x0

i dy(P; y) ; (x; x0 ) 2 R 2 :

(3.18)

Properties of V (x), (P; x), and (P; x; x0) are summarized in the following

Lemma 3.1. Let P = (z; R2g+1 (z)1=2 ) = (~(P ); R21g=2+1(P )) 2 Kg nfP1g, (z; x; x0 ) 2 C  R2 .

Then

g X

[E2j?1 + E2j ? 2j (x)]:

(i).

V (x) = E0 +

(ii).

(P; x)satis es the Riccati-type equation

j =1

(3.19)

x(P; x) + (P; x)2 = V (x) ? z: (iii).

(3.20)

(P; x; x0) satis es the Schrodinger equation

?

xx (P; x; x0 ) + [V (x) ? z ]

(P; x; x0 ) = 0:

(3.21)

(iv).

(P; x)(P ; x) = Hg+1(z; x)=Fg (z; x):

(3.22)

(v).

(P; x) + (P ; x) = ?2Gg?1(z; x)=Fg (z; x) = Fg;x(z; x)=Fg (z; x):

(3.23)

(vi).

(P; x) ? (P ; x) = 2iR21g=2+1 (P )=Fg (z; x):

(3.24)

(vii).

(P; x; x0) (P ; x; x0 ) = Fg (z; x)=Fg (z; x0 ):

(viii).

x (P; x; x0 ) x (P

(ix).

(P; x; x0) = [Fg (z; x)=Fg (z; x0 )]1=2 exp iR21g=2+1 (P )

 ; x; x

0 ) = Hg+1 (z; x)=Fg (z; x0 ):

h

(3.25)

Zx x0

i dyFg (z; y)?1 :

(3.26) (3.27)

Proof. (i). Insert (3.3) into (2.23) and compare the coecient of z 2g . (ii). Combine (2.17), (2.23), and (3.15). (iii). Follows from xx= = x + 2 = V ? z. (iv). Multiply the rst and third expression in (3.15) replacing P by P  in one of the two factors. (v), (vi) are clear from (3.15). (vii). Combine (3.18) and (3.23). (viii). Use (3.22), (3.25), and x =  . (ix). Invoke (2.17), (3.15), and (3.18).

F. GESZTESY, R. RATNASEELAN, AND G. TESCHL

Eq. (3.19) represents a trace formula for the nite-gap potential V (x). The method of proof of Lemma 3.1 (i) indicates that higher-order trace formulas associated with the KdV hierarchy can be obtained from (3.3) and (2.23) comparing powers of z. Since we shall derive trace formulas for general potentials in Section 5, we postpone the special case of nite-gap potentials at this point and refer to Example 5.5. We also record

Lemma 3.2. Let (z; x) 2 C  R . Then (i). (ii).

Hg+1(z; x) = 21 Fg;xx(z; x) ? [V (x) ? z]Fg (z; x):

Hg+1;x(z; x) = ?2[V (x) ? z]Gg?1 (z; x):

(3.28) (3.29)

Proof. (i). By (2.17), (2.23), and (3.11),

? 21 Fg;xx = ? 14 Fg?2Fg;x ? (V ? z)Fg ? Fg?1R2g+1 = (V ? z)Fg ? Fg?1(R2g+1 + G2g?1) = ?(V ? z)Fg ? Hg+1: (ii). By (2.17), (2.22), and (3.28),

Hg+1;x = ?Gg?1;xx ? (V ? z)Fg;x ? VxFg = 21 Fg;xxx ? (V ? z)Fg;x ? VxFg = (V ? z)Fg;x = ?2(V ? z)Gg?1 : Explicitly, one computes from (2.4), (2.14), and (3.28), H1 = H~ 1 = ?V + z;

  H2 = ?c1 V + 14 Vxx ? 12 V 2 + c1 ? 12 V z + z2 = c1 H~ 1 + H~ 2;  H3 = ?c2 V + c1( 41 Vxx ? 12 V 2 ? 161 Vxxxx + 38 Vx2 + 21 V Vxx ? 38 V 3 i + [c2 ? c1 21 V + 18 Vxx ? 81 V 2 ]z + [c1 ? 12 V z2 + z3 = c2H~ 1 + c1 H~ 2 + H~ 3;

(3.30)

etc. We also mention the following well-known result connecting Dirichlet and Neumann eigenvalues.

Lemma 3.3. [33] Suppose j (x0) 2 fE2j?1; E2j g, 1  j  g. Then 0 (x0 ) = E0, j (x0 ) 2 fE2j?1; E2j gnfj (x0 )g, 1  j  g. Conversely, suppose j (x0 ) 2 fE2j?1; E2j g, 1  j  g. Then 0 (x0 ) = E0, j (x0 ) 2 fE2j?1 ; E2j gnfj (x0 )g, 1  j  g.

THE KDV HIERARCHY AND ASSOCIATED TRACE FORMULAS

Proof. If j (x0 ) 2 fE2j?1 ; E2j g, 1  j  g then Gg?1 (z; x0 ) = 0 in (3.11) yields R2g+1 (z) = Fg (z; x0 )Hg+1(z; x0 ) and hence proves the rst claim. Conversely, assuming j (x0 ) 2 fE2j?1; E2j g, 1  j  g one infers from (3.13) that Gg?1(j (x0 ); x0) = iR21g=2+1 (^j (x0 )) = 0, 1  j  g, i.e., again Gg?1(z; x0 ) = 0. Hence R2g+1(z) = Fg (z; x0 )Hg+1(z; x0 ) also proves the second claim.

Given the bounded potential V (x) in (3.19), consider the dierential expression  = ? dxd22 + V (x) and de ne the corresponding self-adjoint Schrodinger operator H in L2 (R) by d2 + V (x); x 2 R ; f 2 D(H ) = H 2;2(R); (3.31) Hf = f;  = ? dx 2 with H m;n(:) the usual Sobolev spaces. The resolvent of H reads ((H ? z)?1 f )(x) =

Z

R

dx0 G(z; x; x0 )f (x0); z 2 C n(H ); f 2 L2 (R );

(3.32)

where the Green's function G(z; x; x0 ) is explicitly given by

8 > < +(z; x; x0 ) ?(z; x0 ; x0 ); x  x0 0 ? 1 G(z; x; x ) = W ( +(z; :; x0 ); ?(z; :; x0 )) > ; : +(z; x0 ; x0 ) ?(z; x; x0 ); x  x0 (3.33) with W (f; g) = fg0 ?f 0 g the Wronskian of f and g and  (z; x; x0 ) the branches of (P; x; x0) in the charts (; ~ ). One computes W ( +(z; :; x0 ); ? (z; :; x0 )) = (2=i)R2g+1(z)1=2 Fg (z; x0 )?1

(3.34)

and

i Qgj=1[z ? j (x)] G(z; x; x) = 2R (z)1=2 = 2RiFg (z;(zx))1=2 ; (3.35) 2g+1 2g+1 taking into account our convention (A.3) for R2g+1 (z)1=2 . In particular, the spectrum (H ) of H is given by g[ ?1 (3.36) (H ) = [E2j ; E2j+1] [ [E2g ; 1): j =0

Eq. (3.35) illustrates the spectral theoretic content of the polynomial Fg (z; x). Moreover, the Weyl m-functions m (z; x0 ), associated with the restriction of  to (x0 ; 1) with a Dirichlet boundary condition at x0 , read

m (z; x0 ) = (z; x0 ) = [iR2g+1 (z)1=2 ? Gg?1(z; x0 )]Fg (z; x0 )?1;

(3.37)

F. GESZTESY, R. RATNASEELAN, AND G. TESCHL

where (z; x) denote the branches of (P; x) in the charts (; ~ ). As a consequence, the Weyl m-matrix M (z; x0 ) associated with H is given by (see, e.g., [32], Ch. 8)

1 0 m ( z;x ) m ( z;x ) [ m ( z;x )+ m ( z;x )] = 2 ? 0 + 0 C M (z; x0 ) =[m?(z;x0)?m+ (z;x0)]?1B A @ ? 0 + 0 [m? (z;x0 )+m+ (z;x0 )]=2 1 0 1 1 @ @ G(z; x0 ; x0 ) 2 (@1 + @2 )G(z; x0 ; x0 )C =B @1 1 2 A ( @ + @ ) G ( z; x ; x ) G ( z; x ; x ) 1 2 0 0 0 0 2 1 0 i B@ Hg+1(z; x0 ) ?Gg?1 (z; x0 )CA ; = 1 = 2 2R2g+1(z) ?Gg?1 (z; x0 ) Fg (z; x0 )

where

@1 G(z; x0 ; x0 ) = @xG(z; x; x0 ) x=x0 ; @2 G(z; x; x0 ) = @x0 G(z; x; x0 ) x0=x0 ; @1 @2 G(z; x0 ; x0 ) = @x@x0 G(z; x; x0 ) x=x0=x0 ; etc.

(3.38)

(3.39)

The corresponding self-adjoint spectral matrix (; x0 ), de ned by

Z Mp;q (z; x0 ) = (z ? )?1dp;q (; x0); R Z ?1 + d Im[M ( + i; x )]; p;q (; x0 ) ? p;q (; x0) = lim lim  p;q 0 #0 #0 +

;  2 R ; 1  p; q  2;

(3.40) (3.41)

explicitly reads (cf., e.g., [32], Ch. 8)

8 > Hg+1(; x0 ) ;  2 (H )o < 2R ()1=2 d1;1(; x0) = > 2g+1 ; > d > :0;  2 R n(H ) 8 > ?Gg?1 (; x0) ;  2 (H )o < 2R ()1=2 d1;2(; x0) = d2;1(; x0) = > 2g+1 ; > d d > :0;  2 R n(H ) 8 > Fg (; x0 ) ;  2 (H )o > < d2;2(; x0) = 2R2g+1 ()1=2 : > d > :0;  2 R n(H )

(Here Ao denotes the interior of A  R .)

(3.42) (3.43) (3.44)

THE KDV HIERARCHY AND ASSOCIATED TRACE FORMULAS

Closely associated with H is Hx 0 in L2 (R ) de ned by

Hx 0 f = f; 2 R [ f1g; x0 2 R ; f 2 D(Hx 0 ) = fg 2 L2 (R )jg; g0 2 AC ([x0 ; R]) for all R > 0;

(3.45)

lim [g0(x0  ) + g(x0  )] = 0; g 2 L2 (R )g; #0 with AC(loc)(I ) the set of (locally) absolutely continuous functions on I . Here, in obvious notation, = 1 denotes the Dirichlet Schrodinger operator HxD0 = Hx10 and = 0 the corresponding Neumann Schrodinger operator HxN0 = Hx00 . Moreover, Hx 0 decomposes into the direct sum of half-line operators

Hx 0 = H? ;x0  H+ ;x0 ; L2 (R ) = L2 ((?1; x0])  L2 ([x0 ; 1)):

(3.46)

The resolvent of Hx 0 reads ((Hx 0 ? z)?1 f )(x) =

Z R

dx0 G x0 (z; x; x0 )f (x0 ); z 2 C n(Hx 0 ); f 2 L2 (R );

G x0 (z; x; x0 ) = G(z; x; x0 ) ?

( + @2 )G(z; x; x0 )( + @1 )G(z; x0 ; x0 ) ; ( + @1 )( + @2 )G(z; x0 ; x0) 2 R ; z 2 C nf(Hx 0 ) [ (H )g;

0 0 0 ?1 G1 x0 (z; x; x ) = G(z; x; x ) ? G(z; x; x0 )G(z; x0 ; x )G(z; x0 ; x0 ) ;

z 2 C nf(Hx10 ) [ (H )g:

(3.47) (3.48) (3.49)

Next we de ne the polynomial Kg +1 (z; x), 2 R of degree g + 1 in z,

Kg +1(z; x) = Hg+1(z; x) ? 2 Gg?1(z; x) + 2Fg (z; x) =

Yg `=0

[z ?  ` (x)]; 2 R :

(3.50)

In particular,

Hg+1(z; x) = Kg0+1(z; x); `(x) = 0` (x); 0  `  g:

(3.51)

F. GESZTESY, R. RATNASEELAN, AND G. TESCHL

Explicitly, one computes

K1 = 2 ? V + z = K~ 1 ; K2 = c1 ( 2 ? V ) + 41 Vxx ? 12 V 2 + 21 Vx + 12 2V   + c1 + 2 ? 12 V z + z2 = c1 K~ 1 + K~ 2 ;   K3 = c2 ( 2 ? V ) + c1 21 2V + 12 Vx + 14 Vxx ? 12 V 2 ? 18 Vxxx + 43 V Vx ? 81 2Vxx + 38 2V 2 ? 161 Vxxxx + 83 Vx2 + 21 V Vxx ? 38 V 3

(3.52) + [ 21 Vx + 2c1 + 12 2V ? 21 c1V + c2 + 18 Vxx ? 18 V 2]z + [c1 + 2 ? 21 V ]z2 + z3 = c2 K~ 1 + c1 K~ 2 + K~ 3 ; etc. Then (3.35) and

iK (z; x) ( + @1 )( + @2 )G(z; x; x) = 2R g+1 (z)1=2 ; 2 R 2g+1

(3.53)

together with (3.47) and (3.48) yield

(Hx 0 ) = (H ) [ f ` (x0 )g0`g ; 2 R ;

(3.54)

(Hx10 ) = (H ) [ fj (x0 )g1jg ; j (x0 ) = 1 j (x0 ); 1  j  g;

(3.55)

with

 0 (x0 )  E0; 2 R ;  ` (x0) 2 [E2`?1 ; E2`]; 1  `  g; 2 R [ f1g:

(3.56)

Next, one veri es 2 iR21g=+1 (P ) ? Gg?1(~(P ); x) + Fg (~(P ); x) (P; x) + = Fg (~(P ); x) ? Kg +1 (~(P ); x) = 1=2 ; iR2g+1 (P ) + Gg?1 (~(P ); x) ? Fg (~(P ); x)

(3.57)

R2g+1 (z) + [Gg?1 (z; x) ? Fg (z; x)]2 = Fg (z; x)Kg +1 (z; x);

(3.58)

[(P; x) + ][(P ; x) + ] = Kg +1(z; x)=Fg (z; x);

(3.59)

[ x(P; x; x0) + (P; x; x0)][ x (P ; x; x0) + (P ; x; x0)] = Kg +1(z; x)=Fg (z; x0 ): (3.60)

THE KDV HIERARCHY AND ASSOCIATED TRACE FORMULAS

The divisor of (:; x) + , 2 R is given by ((:; x) + ) = D^ 0 (x)^ (x) ? DP1^(x) ;

(3.61)

R21g=2+1 (^ ` (x)) = ?iGg?1 ( ` (x); x) + i Fg ( ` (x); x); ^ ` (x) = ( ` (x); ?iGg?1 ( ` (x); x) + i Fg ( ` (x); x)); 0  `  g; 2 R :

(3.62)

with

The rst-order system of dierential equations for  ` (x), 2 R , i.e., the analog of (3.5) in the case = 1, will be derived in the next section (see (4.45) for r = 0). Here we only record the nal result for completeness,

 ` 0(x) = ?2i[ 2 ? V (x) +  ` (x)]R21g=2+1 (^ ` (x))

Yg m=0 m6=`

[ ` (x) ?  m (x)]?1;

~ (^ 0 (x)) =  0 (x)  E0 ; ~ (^ ` (x)) =  ` (x) 2 [E2`?1 ; E2`]; 1  `  g; ( ; x) 2 R 2 :(3.63) In particular, taking = 0 in (3.63) then yields the rst-order system of dierential equations for `(x), 0  `  g. (We remark that V (x) in (3.63) has been used for reasons of brevity only. In order to obtain a system of dierential equations for  ` (x) one needs to replace V (x) by the corresponding trace formula (see, e.g., (5.23), (5.30), and (5.34)).) We emphasize that due to our convention (A.3) for R21g=2+1 (P ), the dierential equations (3.5) and (3.63) exhibit the well-known monotonicity properties of j (x) and  j (x), 2 R , j  1 with respect to x 2 R . For instance, Dirichlet eigenvalues corresponding to the right (left) half axis (x; 1) ((?1; x)) associated with the decomposition (3.46) are always increasing (decreasing) with respect to x 2 R , etc. We conclude with the -function representation for (P; x), (P; x; x0 ), V (x) to be derived in Section 4 (cf. Theorem 4.6) in the general t-dependent case.

(P0 ? AP0 (P1) + P0 (^(x)))  ^ ( P ) +  ( x ))) (P0 ? AP0 1 P0 ( i ^ (x))) h Z P (3) P0 ? AP0 (P ) + P0 ( ; ! exp ?  ( ^ (P0 ? AP0 (P ) + P0 (^(x))) P0 P1;0 (x)

(P; x) = ? +

(3.64)

F. GESZTESY, R. RATNASEELAN, AND G. TESCHL

( ? A (P ) +  (^(x))) ( ? A (P ) +  (^(x ))) (P; x; x0) = ( P0? A P0(P ) + P0 (^(x))) (P0 ? AP0 (P1) +  P0(^(x 0)))  P0

h

P0

1

P0

 exp ? i(x ? x0 )

ZP P0

P0

P0

P0

i !P(2)1;0 ; P0 = (E0 ; 0); = R ;

0

(3.65)

with the linearizing property of the Abel map, (2)

P0 (^(x)) = P0 (^(x0 )) + U20 (x ? x0 ); (x; x0 ) 2 R 2 ; Z (2) (2) (2) (2) = ( U ; : : : ; U ) ; U = !P1;0; 1  j  g: U (2) 0 0;1 0;g 0;j bj

(3.66) (3.67)

The Its-Matveev formula [26], [4], Ch. 3, [38], Ch. II for V (x) then reads

V (x) = E0 + = E0 +

g X j =1

g X

j =1

(E2j?1 + E2j ? 2j ) ? 2@x2 ln[(P0 ? AP0 (P1) + P0 (^(x)))]

(E2j?1 + E2j ? 2j ) ? 2@x2 ln[(P0 + AP0 (^ 0 (x)) + P0 (^ (x)))]; (3.68)

where j 2 [E2j?1; E2j ], 1  j  g are determined from

!P(2)1;0 = ?[2R21g=2+1 (:)]?1

Yg

(~ ? j )d~ = [ ?2 + 0(1)] d near P1 !0

j =1

(3.69)

and the second equality in (3.68) is a consequence of the equivalence DP1^(x)  D^0 (x)^ (x) , i.e.,

AP0 (P1) + P0 (^(x)) = AP0 (^ 0 (x)) + P0 (^ (x)); x 2 R :

(3.70)

4. The Time-Dependent Formalism In this section we construct algebro-geometric solutions of the KdV hierarchy corresponding to g-gap initial values on the basis of a suitable time-dependent generalization of the polynomial approach developed in Chapters 2 and 3. Even though the nal results (4.51){ (4.55) are well-known, in fact, classical by now, the approach presented in this section, based on the fundamental meromorphic function (P; x; t) in (4.15), merits attention since it easily extends to general 1 + 1-dimensional completely integrable systems such as the AKNS and Toda hierarchies. (The corresponding approach to the Toda and Kac-van Moerbeke lattices is presented in detail in [5].) The results (4.36){(4.50) in connection with the general boundary condition in (3.45) and our strategy of proof of the theta function representation of the BA function (P; x; x0; t; t0) in (4.51), based on (4.15) and (4.16), are new.

THE KDV HIERARCHY AND ASSOCIATED TRACE FORMULAS

Our starting point will be a g-gap solution V (0) of the stationary KdVg equation,

V (0) (x) = E0 +

g X

(0) [E2j?1 + E2j ? 2(0) j (x)]; j (x) 2 [E2j ?1 ; E2j ]; 1  j  g

j =1

(4.1)

satisfying g X `=0

cg?`f`+1;x = 0; c0 = 1;

(4.2)

where f`+1 are given by (2.4) with V = V (0) . Our principal aim then is to construct the KdV ow KdVr (V ) = 0; V (x; t0) = V (0) (x); x 2 R for some r 2 N 0 . In terms of Lax operators this amounts to solving d L(t) ? [P (t); L(t)] = 0; t 2 R ; [P (t ); L(t )] = 0: 2r+1 2g+1 0 0 dt As a consequence one then obtains that [P2g+1(t); L(t)] = 0; t 2 R ;

?P2g+1 (t)2 = R2g+1 (L(t)) =

2g Y

(L(t) ? En ); t 2 R

n=0

(4.3)

(4.4)

(4.5) (4.6)

since the KdVr ows are isospectral deformations of L(t0 ). In this paper we shall base the explicit solution of (4.3) not directly on (4.4) and (4.5) but instead take the following equations as our point of departure, Vt = ? 21 F^r;xxx + 2(V ? z)F^r;x + VxF^r ; (x; t) 2 R 2 ; (4.7) 2 ? 2(V ? z )F 2 = 2R 2 (4.8) Fg;xxFg ? 21 Fg;x 2g+1 (z ); (x; t) 2 R ; g where

Fg (z; x; t) =

Yg

j =1

[z ? j (x; t)]

(4.9)

(cf. (2.14), (2.18), and (2.23)). In order to stress the fact that the integration constants c` used in Fg and Fr (cf. (2.10), (2.14)) in general can dier from each other, we explicitly

F. GESZTESY, R. RATNASEELAN, AND G. TESCHL

employ the notation Fg , Gg?1, Hg+1, Kg +1 , etc. and F^r , G^ r?1, H^ r+1, K^ r +1, etc. throughout this section. Similarly to (3.5){(3.8), (3.11), and (3.12) we have

Yg 1=2 j;x(x; t) = ?2iR2g+1 (^j (x; t)) [j (x; t) ? k (x; t)]?1; k=1 k6=j

1  j  g; (x; t) 2 R 2 ;

f^j (x0 ; t)g1jg  Kg ; ~ (^j (x0 ; t)) = j (x0 ; t) 2 [E2j?1 ; E2j ]; 1  j  g; t 2 R ;

(4.10)

(4.11)

!

g Yg X 1 z ? k (x; t) 1=2 Gg?1(z; x; t) = ? 2 Fg;x(z; x; t) = ?i R2g+1 (^j (x; t)) ; j =1 k=1 j (x; t) ? k (x; t) k6=j

R2g+1 (z) + Gg?1 (z; x; t)2 = Fg (z; x; t)Hg+1 (z; x; t); Yg Hg+1(z; x; t) = [z ? `(x; t)]:

(4.12)

(4.13) (4.14)

`=0

In analogy to (3.15) and (3.18) one then considers the meromorphic function (P; x; t) on Kg , 2 iR21g=+1 (P ) ? Gg?1(~(P ); x; t) ?Hg+1 ; (x; t) 2 R 2 (P; x; t) = = 1=2 Fg (~(P ); x; t) iR2g+1 (P ) + Gg?1(~(P ); x; t) (4.15)

and the t-dependent BA function (P; x; x0; t; t0), meromorphic on Kg nfP1g, (P; x; x0; t; t0 ) = exp

nZ t t0

ds[F^r (z; x0 ; s)(P; x0; s) + G^ r?1(z; x0 ; s)] +

Zx x0

o dy(P; y; t) ;

(x; x0 ; t; t0) 2 R 4 : (4.16)

THE KDV HIERARCHY AND ASSOCIATED TRACE FORMULAS

Lemma 4.1. Let P = (z; R2g+1 (z)1=2 ) = (~(P ); R21g=2+1(P )) 2 Kg nfP1g, (z; x; x0 ; t; t0) 2 C  R 4 , r 2 N 0 . Then (i). V (x; t) = E0 +

g X

[E2j?1 + E2j ? 2j (x; t)]:

(4.17)

j =1

(ii). (P; x; t) satis es

x(P; x; t) + (P; x; t)2 = V (x; t) ? z; t(P; x; t) = @x[F^r (z; x; t)(P; x; t) + G^ r?1(z; x; t)]:

(4.18) (4.19)

(iii). (P; x; x0 ; t; t0 ) satis es

?

xx (P; x; x0 ; t; t0 ) + [V (x; t) ? z ]

(P; x; x0 ; t; t0) = 0; (4.20) ^ r?1(z; x; t) (P; x; x0 ; t; t0 ): (4.21) t (P; x; x0 ; t; t0 ) = F^r (z; x; t) x (P; x; x0 ; t; t0 ) + G

(iv). (P; x; t)(P ; x; t) = Hg+1 (z; x; t)=Fg (z; x; t):

(4.22)

(v). (P; x; t) + (P  ; x; t) = ?2Gg?1 (z; x; t)=Fg (z; x; t) = Fg;x (z; x; t)=Fg (z; x; t): (4.23) (vi). (P; x; t) ? (P ; x; t) = 2iR21g=2+1 (P )=Fg (z; x; t):

(4.24)

Proof. The proof of (i), (4.18), (4.20), and (iv){(vi) is analogous to that in Lemma 3.1. In order to prove (4.19) one can argue as follows. By (4.7) and (4.18),

@t (x + 2) = tx + 2t = Vt = (F^r  + G^ r?1)xx + 2(F^r  + G^ r?1)x; which implies (@x + 2)[t ? (F^r  + G^ r?1)x] = 0 and hence t = (F^r  + G^ r?1 )x + Ce2

R x dy

;

where C is independent of x (but may depend on P and t). The high-energy behavior of (P; x; t) derived from (4.15) yields (P; x; t) z!1 = i(z)1=2 + 0(1), P 2  uniformly in (x; t) 2 R 2 and hence C = 0 proving (4.19). (4.21) then immediately follows from (4.16) and (4.19). In analogy to (3.28) we now introduce

H^ r+1(z; x; t) = 12 F^r;xx(z; x; t) ? [V (x; t) ? z]F^r (z; x; t):

(4.25)

F. GESZTESY, R. RATNASEELAN, AND G. TESCHL

>From (4.7) and (4.8) one then computes H^ r+1;x = 21 F^r;xxx ? (V ? z)F^r;x ? VxF^r = ?Vt ? 2(V ? z)G^ r?1 : The t-dependence of Fg , Gg?1, and Hg+1 is governed by

(4.26)

Lemma 4.2. Let (z; x; t) 2 C  R 2 , r 2 N 0 . Then (4.27)

(ii).

Fg;t(z; x; t) = 2[Fg (z; x; t)G^ r?1 (z; x; t) ? F^r (z; x; t)Gg?1 (z; x; t)]: Gg?1;t(z; x; t) = Fg (z; x; t)H^ r+1 (z; x; t) ? F^r (z; x; t)Hg+1 (z; x; t):

(iii).

Hg+1;t(z; x; t) = 2[H^ r+1(z; x; t)Gg?1 (z; x; t) ? Hg+1(z; x; t)G^ r?1 (z; x; t)]:

(4.29)

(i).

(4.28)

Proof. By (2.17), (4.19), and (4.24),

t (P ) ? t(P ) = ?2iR21g=2+1 (P )Fg?2Fg;t = @x [F^r ((P ) ? (P ))] = 2iR21g=2+1 (P )(Fg F^r;x ? Fg;xF^r )Fg?2; implying (4.27). Similarly, by (2.17) and (4.25),

Gg?1;t = ? 21 Fg;tx = F^r Gg?1;x ? Fg G^ r?1;x = Fg H^ r+1 ? F^r Hg+1: Finally, (2.17), (4.25){(4.28) yield

Hg+1;t = ?Gg?1;tx ? (V ? z)Fg;t ? VtFg = ?Fg;xH^ r+1 ? Fg H^ r+1;x + F^r;xHg+1 + F^r Hg+1;x ? 2(V ? z)(Fg G^ r?1 ? F^r Gg?1 ) ? Vt Fg = 2(Gg?1H^ r+1 ? 2G^ r?1Hg+1): As a consequence of (4.27) one obtains the following time-dependence of j (x; t).

Corollary 4.3. Let (x; t) 2 R 2 , r 2 N 0 . Then

Y j;t(x; t) = ?2iF^r (j (x; t); x)R21g=2+1 (^j (x; t)) [j (x; t) ? k (x; t)]?1; ^j (x; t0) = ^(0) j (x); 1  j  g;

g

k=1 k6=j

~ (^j (x; t)) = j (x; t) 2 [E2j?1; E2j ]; 1  j  g:

(4.30) (4.31)

Proof. Take z = j (x; t) in (4.27) and observe (4.1), (4.3), (4.9), and 2 R21g=+1 (^j (x; t)) = iGg?1(j (x; t); x; t); ^j (x; t) = (j (x; t); iGg?1(j (x; t); x; t));

(4.32)

THE KDV HIERARCHY AND ASSOCIATED TRACE FORMULAS

the latter fact following from (4.12) (as in (3.9)). One observes that the special case r = 0 (i.e., F^0 = 1) in (4.30) is equivalent to (4.10), (4.11). Next we record the remaining t-dependent analogs of Lemma 3.1 (vii){(ix).

Lemma 4.4. Let P = (z; R2g+1 (z)1=2 ) = (~(P ); R21g=2+1(P )) 2 Kg nfP1g, (z; x; x0 ; t; t0) 2 C  R 4 , r 2 N 0 . Then (i).

(P; x; x0 ; t; t0) (P ; x; x0; t; t0 ) = Fg (z; x; t)=Fg (z; x0 ; t0):

(ii).

x (P; x; x0 ; t; t0 ) x (P

(iii).

(P; x; x0 ; t; t0) = [Fg (z; x; t)=Fg (z; x0 ; t0 )]1=2 

 exp

hZ t 1=2 iR2g+1 (P ) t0

n

 ; x; x

(4.33)

0 ; t; t0 ) = Hg+1 (z; x; t)=Fg (z; x0 ; t0 ):

dsF^r (z; x0 ; s)Fg (z; x0 ; s)?1 +

Zx x0

(4.34)

io dyFg (z; y; t)?1 :

(4.35)

Proof. (i). Combining (4.16), (4.23), and (4.27) yields

(P; x; x0; t; t0 ) (P ; x; x0 ; t; t0) = exp

hZ t t0

dsFg;s(z; x0 ; s)Fg (z; x0 ; s)?1 +

Zx x0

dyFg;y (z; y; t)Fg (z; y; t)?1

i

= [Fg (z; x0 ; t)=Fg (z; x0 ; t0)][Fg (z; x; t)=Fg (z; x0 ; t)] = Fg (z; x; t)=Fg (z; x0 ; t0): (ii). (4.22), (4.33) and

x

=  imply

x (P; x; x0 ; t; t0 ) x (P

 ; x; x

0 ; t; t0 )

= [Hg+1(z; x; t)=Fg (z; x; t)][Fg (z; x; t)=Fg (z; x0 ; t0)] = Hg+1(z; x; t)=Fg (z; x0 ; t0 ): (iii). Follows from (4.15), (4.16), and (4.27). Remark 4.5. We emphasize that instead of taking (4.7) and (4.8) as our starting point for solving (4.3), and subsequently deriving the rst-order dierential system (4.10), (4.30), one could have started directly with the system (4.10), (4.30) and derived (4.7), (4.8) and the remaining facts of this section (cf. [5]).

Next, we turn to the t-dependent analog of (3.50){(3.63) and start by introducing

Kg +1(z; x; t) = Hg+1(z; x; t) ? 2 Gg?1(z; x; t) + 2Fg (z; x; t) =

Yg

[z ?  ` (x; t)]; 2 R ; `=0 (4.36)

F. GESZTESY, R. RATNASEELAN, AND G. TESCHL

with

Hg+1(z; x; t) = Kg0+1 (z; x; t); `(x; t) = 0` (x; t); 0  `  g:

(4.37)

One then veri es in analogy to (3.57){(3.62) that 2 iR21g=+1 (P ) ? Gg?1(~(P ); x; t) + Fg (~(P ); x; t) (P; x; t) + = Fg (~(P ); x; t) ?Kg +1 (~(P ); x; t) = 1=2 ; iR2g+1 (P ) + Gg?1(~(P ); x; t) ? Fg (~(P ); x; t)

(4.38)

R2g+1 (z) + [Gg?1 (z; x; t) ? Fg (z; x; t)]2 = Fg (z; x; t)Kg +1 (z; x; t);

(4.39)

[(P; x; t) + ][(P ; x; t) + ] = Kg +1(z; x; t)=Fg (z; x; t);

(4.40)

[ x(P; x; x0; t; t0) + (P; x; x0; t; t0 )][ x(P ; x; x0 ; t; t0) + (P ; x; x0 ; t; t0)]

with

= Kg +1(z; x; t)=Fg (z; x0 ; t0);

(4.41)

((:; x; t) + ) = D^ 0 (x;t)^ (x;t) ? DP1^(x;t) ;

(4.42)

R21g=2+1 (^ ` (x; t)) = ?iGg?1 ( ` (x; t); x; t) + i Fg ( ` (x; t); x; t); ^ ` (x; t) = ( ` (x; t); ?iGg?1 ( ` (x; t); x; t) + i Fg ( ` (x; t); x; t)); 0  `  g: (4.43)

Eq. (4.36) and Lemma 4.2 then yield Kg +1;t(z; x; t) = 2fK^ r +1(z; x; t)[Gg?1 (z; x; t) ? Fg (z; x; t)]

? Kg +1 (z; x; t)[G^ r?1(z; x; t) ? F^r (z; x; t)]g and in analogy to Corollary 4.3 one obtains from (4.44),

 `;t(x; t) = ?2iK^ r +1 ( ` (x; t); x; t)R21g=2+1 (^ ` (x; t)) ^ ` (x; t0 ) = ^ ;` (0) (x); 0  `  g; (x; t) 2 R 2 ;

Yg m=0 m6=`

(4.44)

[ ` (x; t) ?  m (x; t)]?1 ;

~ (^ 0 (x; t)) =  0 (x; t)  E0 ; ~ (^ ` (x; t)) =  ` (x; t) 2 [E2`?1; E2` ]; (x; t) 2 R 2 ;

(4.45)

(4.46)

THE KDV HIERARCHY AND ASSOCIATED TRACE FORMULAS

where f ;` (0)(y)g0`g are the corresponding eigenvalues of Hy ;(0) (cf. (3.45), (3.54), and (3.56)) associated with the initial value V (0) (x) in (4.1). In an analogous fashion one can analyze the behavior of  ` (x; t) as a function of 2 R . In fact, (4.36) yields

@ K (z; x; t) = ?2[G (z; x; t) ? F (z; x; t)] g?1 g @ g+1

(4.47)

and hence

h @ i Yg @ K (z; x; t) (x; t)]  ( x; t ) [  ( x; t ) ?  g +1 (x;t) = ? m ` ` z =  @ @ ` m=0 m6=`

= ?2[Gg?1 ( ` (x; t); x; t) ? Fg ( ` (x; t); x; t)] = ?2iR21g=2+1 (^ ` (x; t))

(4.48)

by (4.43). This implies for ( ; x; t) 2 R 3 , g @  (x; t) = 2iR1=2 (^ (x; t)) Y [ ` (x; t) ?  m (x; t)]?1; 0  `  g: 2g+1 ` @ ` m=0 m6=`

(4.49)

As in Section 3 we conclude with the -function representation of (P; x; t), (P; x; x0 ; t; t0), and V (x; t).

Theorem 4.6. Let P = (z; R2g+1 (z)1=2 ) 2 Kg nfP1g, (z; x; x0 ; t; t0) 2 (E0 ; 0). Then

(P; x; t) = ? +

C

 R 4 , P0 =

(P0 ? AP0 (P1) + P0 (^(x; t)))  (P0 ? AP0 (P1) + P0 (^ (x; t))) i ^ (x; t))) h Z P (3)  ( (  ? A ( P ) +  P P P 0 0 0  ( ? A (P ) +  (^(x; t))) exp ? P !P1;^ 0 (x;t) (4.50) 0 P0 P0 P0

and

( ? A (P ) +  (^(x; t))) (P; x; x0; t; t0 ) = ( P0? A P0(P ) + P0 (^(x; t)))  P0

P0

1

P0

Z P (2) Z P (2) i (P0 ? AP0 (P1) + P0 (^(x0; t0 ))) h  ( ? A (P ) +  (^(x ; t ))) exp ? i(x ? x0 ) P !P1;0 ? i(t ? t0 ) P P1;2r ; 0 0 0 0 P0 P0 P0 (4.51)

F. GESZTESY, R. RATNASEELAN, AND G. TESCHL

where (cf. (A.26))

(2) P1 ;2r =

r X s=0

cr?s(2s + 1)!P(2)1;2s;

(4.52)

(2) (2) P0 (^(x; t)) = P0 (^(x0 ; t0)) + U20 (x ? x0) + U22r (t ? t0 ); Z (2) (2) (2) (2)

P1;2r ; 1  j  g: = ( U ; : : : ; U ) ; U = U (2) 2r 2r;1 2r;g 2r;j

bj

(4.53) (4.54)

The Its-Matveev formula ([26], [4], Ch. 3, [38], Ch. II) for V (x; t) reads (cf. (3.68))

V (x; t) = E0 +

g X

(E2j?1 + E2j ? 2j ) ? 2@x2 ln[(P0 ? AP0 (P1) + P0 (^(x; t)))]: j =1 (4.55)

Sketch of Proof. Since (4.50) follows directly from (4.42) and (A.29), and (4.55) can be inferred from (4.51) and (4.20) upon expanding all quantities in (4.20) near P1 in a wellknown manner, we rst concentrate on the proof of (4.51). Let (P; x; x0; t; t0) be de ned as in (4.16) and denote the right-hand-side of (4.51) by (P; x; x0; t; t0 ). In order to prove that = , one rst observes from (4.10) and (4.30) that

F^r (~(P ); x0; s)(P; x0; s) = @s@ ln[j (x0 ; s) ? ~ (P )] + 0(1) for P near ^j (x0; s)

(4.56)

and

(P; y; t) = @y@ ln[j (y; t) ? ~(P )] + 0(1) for P near ^j (y; t):

(4.57)

Hence exp

nZ t t

ds[ @s@ ln(j (x0 ; s) ? ~ (P )) + 0(1)]

o

8 0 > > [j (x0 ; t) ? ~ (P )]0(1) for P near ^j (x0 ; t) 6= ^j (x0 ; t0 ) > < = >0(1) for P near ^j (x0 ; t) = ^j (x0 ; t0 ) > > :[j (x0 ; t0) ? ~ (P )]?10(1) for P near ^j (x0 ; t0) 6= ^j (x0 ; t)

(4.58)

THE KDV HIERARCHY AND ASSOCIATED TRACE FORMULAS

and exp

nZ x x

dy[ @y@ ln(j (y; t) ? ~ (P )) + 0(1)]

o

8 0 > > [j (x; t) ? ~ (P )]0(1) for P near ^j (x; t) 6= ^j (x0 ; t) > < = >0(1) for P near ^j (x; t) = ^j (x0 ; t) ; > > :[j (x0 ; t) ? ~ (P )]?10(1) for P near ^j (x0 ; t) 6= ^j (x; t)

(4.59)

where 0(1) 6= 0 in (4.58) and (4.59). Consequently, all zeros and poles of and on Kg nfP1g are simple and coincide. By an application of the Riemann-Roch theorem it remains to identify the essential singularity of and at P1. For that purpose we rst recall the known fact that the diagonal Green's function G(z; x; x; t) of H (t) satis es

G(z; x; x; t) = (i=2) !0

1 X p f~j (x; t) 2j ;  = 1= z;

j =0

(4.60)

with f~j (x; t) the homogeneous coecients as introduced in the context of (2.12) satisfying the recursion (2.4) for all j 2 N . Combining g (z; x; t) G(z; x; x; t) = 2iF (4.61) R2g+1 (z)1=2 (cf. (3.35)), (4.15), (4.16), and (4.60) then yields Zx Z x iR21g=2+1 (P ) + 0( 2) = i(x ? x0 )[ ?1 + 0(1)]; dy(P; y; t) = dy !0 !0 x0 Fg (~ (P ); y; t) x0 (4.62) which coincides with the singularity at P1 of the x-dependent term in the exponent of (4.51) taking into account (3.69). Finally, in order to identify the t-dependent essential singularity of and , we may allude to (2.20) and, without loss of generality, consider the homogeneous case where c0 = 1, cq = 0, 1  q  r. Invoking (4.27) then yields from (4.15) and (4.61)

Zt

ds[F~r (z; x0 ; s)(P; x0; s) + G~ r?1(z; x0 ; s)] Zt n o = ds F~r (z; x0 ; s)iR21g=2+1 (P )Fg (z; x0 ; s)?1 + 12 @s@ ln[Fg (z; x0 ; s)] t0 1 R t dsF~ (z; x ; s)G(z; x ; x ; s)?1 + 0(1);  = 1=pz: = ? r 0 0 0 t0 2  !0 t0

(4.63)

Comparing (2.14) (in the homogeneous case) and (4.60) implies

? 12 F~r (z; x0 ; s)G(z; x0 ; x0 ; s)?1 =!0 i ?2r?1 + 0(1)

(4.64)

F. GESZTESY, R. RATNASEELAN, AND G. TESCHL

and hence

Zt t0

ds[F~r (z; x0 ; s)(P; x0; s) + G~ r?1(z; x0 ; s)] = i(t ? t0)[ ?2r?1 + 0(1)]; !0

(4.65)

completing the proof of (4.51). The linearity of the Abel map with respect to x and t in (4.53) then follows by a standard argument considering the dierential (x; x0 ; t; t0) = d ln (:; x; x0 ; t; t0). 5. General Trace Formulas Following a recent series of papers on new trace formulas for Schrodinger operators [16]{ [19], [22]{[24], [39], we rst discuss appropriate Krein spectral shift functions, the key tool for general higher-order trace formulas. Subsequently, we develop a new method for deriving small-time heat kernel (respectively high-energy resolvent) expansion coecients associated with the general -boundary conditions in (5.3). Interest in these types of trace formulas stems from their crucial role in the solution of inverse spectral problems. Unlike Sections 3 and 4, where we focused on the special case of stationary nite-gap solutions of the KdV hierarchy (the natural extension of solitons as re ectionless potentials), we now turn to the general situation and consider potentials of the type

V 2 C 1(R ); V (x)  c; x 2 R ; V real-valued.

(5.1)

As in Section 3 we introduce the dierential expression  = ? dxd22 + V (x), x 2 R and de ne the self-adjoint operators H and Hx 0 in L2(R ),

Hf = f; f 2 D(H ) = fg 2 L2(R )jg; g0 2 ACloc (R); g 2 L2 (R )g

(5.2)

and for 2 R [ f1g; x0 2 R ,

Hx 0 f = f; f 2 D(Hx 0 ) =ff 2 L2 (R )jg; g0 2 AC ([x0 ; R]) for all R > 0; lim [g0(x0  ) + g(x0  )] = 0; g 2 L2 (R )g; #0

(5.3)

with Hx10 = HxD0 (Hx00 = HxN0 ) the corresponding Dirichlet (Neumann) Schrodinger operator. If G(z; x; x0 ) denotes the Green's function of H (as in (3.32), (3.33)), formulas (3.47){(3.49)

THE KDV HIERARCHY AND ASSOCIATED TRACE FORMULAS

for the resolvent of Hx 0 apply without change in the present general situation. In particular, de ning

8 > :G(z; x; x); =1

(5.4)

(cf. the notation introduction in (3.39)) one computes for 2 R [ f1g,

d ln[? (z; x)]; z 2 C nf(H ) [ (H )g: Tr[(Hx ? z)?1 ? (H ? z)?1 ] = ? dz x

(5.5)

Given hypothesis (5.1), one can prove the existence of asymptotic expansions of the type Tr[(Hx ? z)?1 ? (H ? z)?1 ] z!=i1 uniformly with respect to x 2 expansion

R

1 X j =0

rj (x)z?j ; 2 R [ f1g

(5.6)

(cf. [24]). In particular, one can derive the heat kernel

Tr[e?Hx1 ? e?H ] #0

1 X j =0

j s1 j (x) ; x 2 R ;

(5.7)

where j +1 ?1 1 s1 j (x) = (?1) (j !) rj (x); j 2 N 0

(5.8)

1 and s1 j (rj ) are the well-known invariants of the KdV hierarchy. In the special case of nite-gap potentials the connection of ? (z; s) in (5.4) with our polynomial approach in Section 3 is clearly demonstrated by (3.35) for = 1 and (3.53) for = R . Before describing a new constructive (i.e., recursive) approach to the coecients rj (x), 2 R , we recall the de nition of Krein's spectral shift function [30] associated with the pair (Hx ; H ) (cf. [19], [23], [24]). The rank-one resolvent dierence of Hx and H (cf. (3.47), (3.48)) is intimately connected with the fact that for each x 2 R , 2 R [ f1g,

? (z; x) is Herglotz with respect to z

(5.9)

F. GESZTESY, R. RATNASEELAN, AND G. TESCHL

(i.e., a holomorphic map C + ! C + , where C + = fz 2 C j Im(z) > 0g). The exponential Herglotz representation for ? (z; x) (cf. [3]) then reads for each x 2 R , ? (z; x) = exp

n Z o c + [( ? z)?1 ? (1 + 2)?1][ (; x) +  ] d ; R

8 > ; (5.10) :0; = 1

where, by Fatou's lemma,

 (; x) = ?1 lim Imfln[ + @1 )( + @2 )G( + i; x; x)]g ?  ; 2 R [ f1g #0

(5.11)

for a.e.  2 R . Moreover,

? 1   (; x)  0;  (; x) = 0;  < inf (Hx ); 2 R ; 0   1(; x)  1;  1(; x) = 0;  < inf (H )

(5.12)

for a.e.  2 R . As a consequence, one obtains (cf. [39]) Tr[f (Hx ) ? f (H )] =

Z

R

df 0() (; x); 2 R [ f1g; x 2 R

(5.13)

for any f 2 C 2 (R ) with (1 + 2)f (j) 2 L2((0; 1)), j = 1; 2 and for f () = ( ? z)?1 , z 2 C n[inf (Hx ); 1). In particular, (5.13) holds for traces of heat kernel and resolvent dierences, i.e., for any 2 R [ f1g, x 2 R ,

? e?H ] = ?

Z1

de? (; x);  > 0; (5.14) Z1 ? 1 ? 1 Tr[(Hx ? z) ? (H ? z) ] = ? d( ? z)?2  (; x); z 2 C nf(Hx ) [ (H )g; ex;0 (5.15) where 8 > < inf (Hx ); 2 R ex;0 = > : (5.16) : inf (H ); = 1 Tr[e?Hx

e x;0

Returning to a recursive approach for the expansion coecients rj (x) in (5.6) we rst consider the expansion ? (z; x)

= (i=2)

z!i1

1 X

j =?

j (x)z?j?1=2 ; 2 R [ f1g:

(5.17)

THE KDV HIERARCHY AND ASSOCIATED TRACE FORMULAS

(A comparison of (5.17) and (4.60) reveals that j1(x) = f~j (x), j 2 N 0 in the case = 1.) In order to obtain a recursion relation for j (x) one can use the following result.

Lemma 5.1. Let z 2 C n(H ), x 2 R . (i). Assume 2 R . Then ? (z; x) = ( + @1 )( + @2 )G(z; x; x) satis es 2[V (x) ? 2 ? z]? xx (z; x)? (z; x) ? [V (x) ? 2 ? z]? x (z; x)2 ? 2Vx(x)? x (z; x)? (z; x)

? 4f[V (x) ? z][V (x) ? 2 ? z] ? Vx(x)g? (z; x)2 = ?[V (x) ? z ? 2]3:

(5.18)

(ii). Assume = 1. Then ?1(z; x) = G(z; x; x) satis es 1 1 ?1 xxx(z; x) ? 4[V (x) ? z ]?x (z; x) ? 2Vx (x)? (z; x) = 0

(5.19)

?2?1xx(z; x)?1 (z; x) + ?1x (z; x)2 + 4[V (x) ? z]?1(z; x)2 = 1:

(5.20)

and

While the results (5.19) and (5.20) in the Dirichlet case = 1 are well-known, see, e.g., [14], the result (5.18) (with the exception of the Neumann case = 0 which was rst presented in [21]) is new. Unfortunately, we have no reasonably short derivation of the dierential equation (5.18). It can be veri ed (not without tears) after quite tedious though straightforward calculations (we recommend additional help in the form of symbolic computations). Insertion of the expansion (5.17) into (5.18) and (5.20) in Lemma 5.1 yields

Lemma 5.2. The coecients j (x) in (5.17) satisfy the following recursion relation. (i). Assume 2 R . Then

? 1 = 1; 0 = 2 ? 21 V; 1 = 21 2V + 21 Vx ? 81 V 2 + 18 Vxx;

2 = ? 161 V 3 + 38 2V 2 + 163 Vx(4 V + Vx) + 18 Vxx(V ? 2) ? 81 Vxxx ? 641 Vxxxx;

= 1 Pj [2(V ? 2) ? (V ? 2) j +1

`=1 `?1 j ?`;xx `?1;x j ?`;x ? 4 ` j ?`+1 ? 4V (V ? 2) ` ?1 j ?` ? 2Vx ` ?1 j ?`;x + ` ?1 j ?`] 2 + 81 Pj`=0[ `;x j ?`;x ? 2 ` j ?`;xx ? 4( ? 2V ) ` j ?`]; j  2: 8

(5.21)

F. GESZTESY, R. RATNASEELAN, AND G. TESCHL

(ii). Assume = 1. Then

01 = 1; 11 = 21 V;

1 = ? 1 Pj 1 1 j +1

2

1 1 1 1 Pj 1 1 1 1 1 `=1 ` j +1?` + 2 `=0 [V ` j ?` + 4 `;x j ?`;x ? 2 `;xx j ?`];

j  1: (5.22)

The nal result for rj (x) then reads

Theorem 5.3. The coecients rj (x) in (5.6) satisfy the following recursion relations. (i). Assume 2 R . Then ?1 r0 (x) = ? 21 ; r1 (x) = 2 ? 12 V (x); rj (x) = j j ?1(x) ? P`j=1 j ?`?1(x)r` (x); j  2:

(5.23)

(ii). Assume = 1. Then

?1 1 (x)r1(x); j  2: r01 = 21 ; r11(x) = 12 V (x); rj1(x) = j j1(x) ? P`j=1 j ?` `

(5.24)

Proof. It suces to combine (5.5), (5.6), (5.17), and the following well-known fact on as?j ?j ymptotic expansions: F (z) jzj!1 = P1 = P1 j =1 cj z implies ln[1 + F (z )] jzj!1 j =1 dj z , where ?1(`=j )c d , j  2. d1 = c1 , dj = cj ? P`j=1 j ?` `

Theorem 5.3 (i) has rst been derived (by using a dierent strategy) in [24]. The current derivation, based on the universal dierential equation (5.18), is new. Combined with (5.21), Theorem 5.3 (i) yields the most ecient algorithm to date for computing rj (x), 2 R . The connection between rj (x) and  (; x) is illustrated in the following result.

Theorem 5.4. [24] Let e x;0 = inf (Hx ), 2 R , e10 = inf (H ). (i). Assume 2 R . Then R 1 dzj+1( ? z)?j?1j (?)j?1[ 1 +  (; x)]; j 2 N : rj (x) = ? 21 (e x;0)j ? zlim 2 !i1 ex;0 (ii). Assume = 1. Then j rj1(x) = 21 (e1 0 ) + zlim !i1

R 1 dzj+1( ? z)?j?1j (?)j?1[ 1 ?  1(; x)]; j 2 N : e1 2 0

(5.25)

(5.26)

We conclude with an example that yields the higher-order trace formulas for periodic potentials which also applies to the (quasi-periodic) nite-gap potentials of Section 3.

THE KDV HIERARCHY AND ASSOCIATED TRACE FORMULAS

Example 5.5. Assume V is periodic, i.e., for some > 0, V (x + ) = V (x) for all x 2 R in addition to (5.1). Then Floquet theory implies

(H ) = (i). Assume 2 R . Then

1 [

n=1

[E2(n?1) ; E2n?1]; E0 < E1  E2 < E3 




(Hx ) = f n(x)gn2N0 [ (H );  0 (x)  E0;  n(x) 2 [E2n?1; E2n ]; n 2 N ;

(5.27)

(5.28)

8 > 0;  <  0 (x); E2n?1 <  <  n(x); n 2 N > > <  (; x) = > ?1;  0 (x) <  < E0;  n(x) <  < E2n ; n 2 N : > > : ? 1 ; E2(n?1) <  < E2n?1 ; n 2 N 2

(5.29)

j j j rj (x) = 21 E0j ?  0 (x)j + 21 P1 n=1 [E2n?1 + E2n ? 2n (x) ]; j 2 N :

(5.30)

Inserting (5.29) into (5.25) then yields the higher-order periodic trace formulas (ii). Assume = 1. Then

(Hx1) = fn(x)gn2N [ (H ); n(x) 2 [E2n?1 ; E2n]; n 2 N ; 8 > 0;  < E0; n(x) <  < E2n; n 2 N > > <  1(; x) = > 1; E2n?1 <  < n(x); n 2 N : > 1 > : ; E2(n?1) <  < E2n?1 ; n 2 N : 2 Insertion of (5.32) into (5.26) then yields r1(x) = 1 E j + 1 P1 [E j + E j ? 2 (x)j ]; j 2 N : j

2 0

2

n=1 2n?1

2n

n

(5.31) (5.32)

(5.33)

The results (5.29) and (5.32) remain valid in the algebro-geometric nite-gap situation discussed in Section 3 where

E2n+1 =  n(x) = E2n+2; n  g + 1; 2 R [ f1g:

(5.34)

Hence (5.30) and (5.33) apply to the stationary KdV solutions of Section 3 (e.g., (5.33) for j = 1 coincides with (3.19)) which, in general, are quasi-periodic with respect to x. Moreover, (5.30) and (5.33) also extend to certain classes of almost-periodic V (x), see, e.g., [8], [28], [31], [32], Chs. 9, 11.

F. GESZTESY, R. RATNASEELAN, AND G. TESCHL

The periodic Dirichlet trace formula (5.33) for j = 1 has been noticed by Hochstadt [25] and later on by Dubrovin [10]. The general case j = N appeared in McKean and van Moerbeke [34] and Flaschka [13]. More recent accounts of (5.33) can be found in [8], [28], [31], [32], Chs. 9, 11, [40]. The Neumann case = 0 in (5.30) is due to McKean and Trubowitz [35]. The general case 2 R was rst studied in [24]. Additional references on the subject of trace formulas and their use in connection with the inverse spectral problem can be found in the papers cited in this paragraph and in the ones listed at the beginning of this section. Appendix A. Hyperelliptic Curves of the KdV-type and Theta Functions

We brie y summarize our basic notation for hyperelliptic KdV-type curves (i.e., those branched at in nity) and their theta functions as employed in Sections 3 and 4. For details on this standard material we refer, e.g., to [11], [12], [29], [37]. Consider the points fEng0ng  R , E0 < E1 <


< E2g , g 2 N 0 and de ne the cut ?1 [E ; E ] [ [E ; 1) with the holomorphic function plane  = C n S gj=0 2j 2j +1 2g

8 > < !C 1 = 2 R2g+1 (:) : > h i : z ! Q2ng=0(z ? En) 1=2

(A.1)

on it. R2g+1 (:)1=2 is extended to all of C by

R2g+1 ()1=2 = lim R ( + i)1=2 ;  2 C n; #0 2g+1

(A.2)

with the sign of the square root chosen according to

8 > (?1)g ijR2g+1 ()1=2 j; > > > < (?1)g+j ijR2g+1()1=2 j; 1 = 2 R2g+1 () = > (?1)g+j jR2g+1()1=2 j; > > > : jR2g+1 ()1=2j;

 2 (?1; E0)  2 (E2j?1; E2j ); 1  j  g :  2 (E2j ; E2j+1); 0  j  g ? 1 (A.3)  2 (E2g ; 1)

Next we de ne the set

M = f(z; R2g+1 (z)1=2 )jz 2 C ;  2 f?; +gg [ fP1 = (1; 1)g

(A.4)

B = f(En; 0)g0n2g [ fP1 = (1; 1)g;

(A.5)

and

THE KDV HIERARCHY AND ASSOCIATED TRACE FORMULAS

the set of branch points. M becomes a compact Riemann surface upon introducing the charts (UP0 ; P0 ) de ned as follows

P0 = (z0; 0 R2g+1 (z)1=2 ) or P0 = P1;

(A.6)

P = (z; R2g+1 (z)1=2 ) 2 UP0  M; VP0 =  (UP0 )  C :

2 M nB : UP0 = fP 2 M j jz ? z0 j < C; R2g+1 (z)1=2 the branch obtained by straight line analytic continuation starting from z0g, C = minn jz0 ? Enj, P0

8 8 > > < UP0 ! VP0 < V ! UP0 ? 1  P0 : > ; P0 : > P0 : : P ! (z ? z0 ) :  ! (z0 + ; R2g+1(z0 +  )1=2)

P0

(A.7)

= (En0 ; 0):

8 > < min jEn ? En0 j; g 2 N UP0 = fP 2 M j jz ? En0 j < Cn0 g; Cn0 = > n6=n0 ; : 1; g=0 8 > < U ! V P0 1 = 2 VP0 = f 2 C j j j < Cn0 g; P0 : > P0 ; : P ! (z ? En0 )1=2 8 > < [0; 2); n0 even ) 1 = 2 1 = 2 ( i= 2) arg( z ? E n 0 ; arg(z ? En0 ) 2 > (z ? En0 ) = j(z ? En0 ) je ; : (?; ]; n0 odd (A.8) 8 > < V ! U P0 ? 1 P0 : > P0 ; :  ! (En0 +  2;  [Qn6=n0 (En0 ? En +  2)]1=2 h Y hY i1=2 i1=2 h i X (En ? En +  2) = (?1)g i?n0 (En ? En) 1 + 2?1 2 (En ? En)?1 + 0( 4) : n6=n0

0

n6=n0

0

n6=n0

0

P0

F. GESZTESY, R. RATNASEELAN, AND G. TESCHL

= P1 :

?1=2 UP0 = fP 2 M j jzj > C1g; C1 = max n jEn j; VP0 = f 2 C j j j < C1 g; 8 > > < UP0 ! VP0 1=2 z1=2 = jz1=2 je(i=2) arg(z) ; P0 : > P ! (1=z ) ; 0  arg(z) < 2; > : P1 ! 0 8 >VP ! UP > > < 0  0 h i  ? 1 P0 : > !  ?2;  ?2g?1 Qn(1 ?  2En) 1=2 ; > > :0 ! P 1 hY i1=2 X (1 ?  2En) = 1 ? 2?1 2 En + 0( 4): n

(A.9)

n

Upper and lower sheets   M with associated charts  are de ned by

8 > : :P ! z

(A.10)

The compact Riemann surface (curve) resulting from (A.4){(A.9) is denoted by Kg . Topologically, Kg is a sphere with g handles and hence has genus g. Next, de ne the holomorphic sheet exchange map (involution)

8 > g :(z; R2g+1 (z)1=2 ) ! (z; R2g+1 (z)1=2 ) = (z; ?R2g+1 (z)1=2 )

(A.11)

and the two meromorphic projection maps

8 8 > > > > Kg ! C [ f1g Kg ! C [ f1g > > < < ~ : >(z; R2g+1 (z)1=2 ) ! z ; R21g=2+1 : >(z; R2g+1 (z)1=2 ) ! R2g+1 (z)1=2 : > > > > (A.12) :P1 ! 1 :P1 ! 1

~ has a pole of order 2 at P1 and two simple zeros at (0; R2g+1 (z)1=2 ) if R2g+1 (0) 6= 0 or a double zero at (0; 0) if R2g+1 (0) = 0 (i.e., if 0 2 fEng0n2g ) and R21g=2+1 as a pole of order 2g + 1 at P1 and 2g + 1 simple zeros at (En; 0), 0  n  2g. Moreover, 2 (P  ) = ?R1=2 (P ); P 2 K : ~ (P ) = ~ (P ); R21g=+1 g 2g+1

(A.13)

THE KDV HIERARCHY AND ASSOCIATED TRACE FORMULAS

Thus Kg is a two-sheeted rami ed covering of the Riemann sphere C 1 ( = C [ f1g), in particular, Kg is compact and hyperelliptic. Using our local charts one infers that for g 2 N , d~ =R21g=2+1 is a holomorphic dierential on Kg with a zero of order 2(g ? 1) at P1 and hence

j = ~ j?1d~ =R21g=2+1(:); 1  j  g

(A.14)

form a basis for the space of holomorphic dierentials on Kg . Next we introduce a canonical homology basis faj ; bj g1jg for Kg as follows. The cycle a` starts near E2`?1 on +, surrounds E2` counterclockwise thereby changing to ?, and returns to the starting point encircling E2`?1 , changing sheets again. The cycle b` surrounds E0 , E2`?1 counterclockwise (once) on +. The cycles are chosen such that their intersection matrix reads aj  bk = j;k , 1  j; k  g. Introducing the invertible matrix C in C g ,

c = (cj;k )1j;kg; Cj;k = c(k) = (c1(k); : : : ; cg (k));

Z

ak

Z E2k

j = 2

E2k?1 ?1 ; cj (k) = Cj;k

zj?1 dz=R2g+1 (z)1=2 2 iR ; (A.15)

the normalized dierentials !j , 1  j  g,

!j =

g X `=1

cj (`)`;

Z

ak

!j = j;k ; 1  j; k  g

(A.16)

form a canonical basis for the space of holomorphic dierentials on Kg . The matrix  in C g of b-periods,

Z

!j ; 1  j; k  g

(A.17)

j;k = k;j ; 1  j; k  g;  = iT; T > 0:

(A.18)

 = (j;k )1j;kg; j;k =

bk

then satis es In the chart (UP1 ; P1 =  ) induced by 1=~ 1=2 near P1 one infers

hY i1=2 o c(j ) 2(g?j)= (1 ?  2En) d n j =1 h i o n = ?2 c(g) + 21 c(g) P2ng=0 En + c(g ? 1)  2 + 0( 4) d:

! = ?2

g nX

(A.19)

Associated with the homology basis faj ; bj g1jg we also recall the canonical dissection of Kg along its cycles yielding the simply connected interior K^ g of the fundamental polygon @ K^ g

F. GESZTESY, R. RATNASEELAN, AND G. TESCHL

given by @ K^ g = a1 b1 a?1 1b?1 1 a2 b2a?2 1 b?2 1


a?g 1 b?g 1 . The Riemann theta function associated with Kg is de ned by

(z ) =

X

n2Zg

exp[2i(n; z) + i(n; n)]; z = (z1; : : : ; zg ) 2 C g ;

(A.20)

where (u; v) = Pgj=1 uj vj denotes the scalar product in C g . It has the fundamental properties

(z1 ; : : : ; zj?1; ?zj ; zj+1; : : : ; zg ) = (z );

(A.21)

(z + m + n) = (z ) exp[?2i(n; z) ? i(n; n)]; m; n 2 Zg:

A divisor D on Kg is a map D : Kg ! Z, where D(P ) 6= 0 for only nitely-many P 2 Kg . The set of all divisors will be denoted by Div(Kg ). With Lg we denote the period lattice

Lg = fz 2 C g jz = m + n; m; n 2 Zgg

(A.22)

and the Jacobi variety J (Kg ) is de ned by

J (Kg ) = C g =Lg :

(A.23)

The Abel maps AP0 (:), respectively P0 (:) are de ned by

8 > > RP > :P !AP0 (P )= ! P0

mod (Lg )

8 > > P > :D!P0 (D)=

P 2Kg

D(P )AP0 (P )

;

(A.24)

with P0 2 Kg a xed base point. (In the main text we agree to x P0 = (E0 ; 0) for convenience.) Next, let M(Kg ) and M1(Kg ) denote the set of meromorphic functions (0-forms) and meromorphic dierentials (1-forms) on Kg . The residue of a meromorphic dierential  2 M1(Kg ) at a point Q0 2 Kg is de ned by resQ0 ( ) = (2i)?1 R Q0  , where Q0 is a counterclockwise oriented smooth simple closed contour encircling Q0 but no other pole of  . Holomorphic dierentials are also called (Abelian) dierentials of the rst kind (dfk); (Abelian) dierentials of the second kind (dsk) !(2) 2 M1(Kg ) are characterized by the property that all their residues vanish. They are normalized, e.g., by demanding that all their a-periods vanish, i.e.,

Z

aj

!(2) = 0; 1  j  g:

(A.25)

THE KDV HIERARCHY AND ASSOCIATED TRACE FORMULAS

If !P(2)1;n is a dsk on Kg whose only pole is P1 2 K^ g with principal part  ?n?2 d , n 2 N 0 near m P1 and !j = (P1 m=0 dj;m(P1 ) ) d near P1 , then

Z

bj

!P(2)1;n = [2i=(n + 1)]dj;n(P1):

(A.26)

A basis for dsk's on Kg , holomorphic on Kg nfP1g, is provided by

!n(2) = ~ g+1+nd~ =R21g=2+1(:); n 2 N 0 :

(A.27)

Any meromorphic dierential !(3) on Kg not of the rst or second kind is de ned to be of the third kind (dtk). A dtk !(3) 2 M1(Kg ) is usually normalized by the vanishing of its a-periods, i.e.,

Z

aj

!(3) = 0; 1  j  g:

(A.28)

A normal dtk !P(3)1;P2 associated with two points P1; P2 2 K^ g , P1 6= P2 by de nition has simple poles at P1 and P2 with residues +1 at P1 and ?1 at P2 and vanishing a-periods. If (3) !P;Q is a norma l dtk associated with P , Q 2 K^ g , holomorphic on Kg nfP; Qg, then

Z

bj

(3) !P;Q = 2i

ZP Q

!j ; 1  j  g;

(A.29)

where the path from Q to P lies in K^ g (i.e., does not touch any of the cycles aj , bj ). We shall always assume (without loss of generality) that all poles of dsk's and dtk's on Kg lie on K^ g (i.e., not on @ K^ g ). For f 2 M(Kg )nf0g, ! 2 M1(Kg )nf0g the divisors of f and ! are denoted by (f ) and (!), respectively. Two divisors D, E 2 Div(Kg ) are called equivalent, denoted by D  E , if and only if D ? E = (f ) for some f 2 M(Kg )nf0g. The divisor class [D] of D is then given by [D] = fE 2 Div(Kg )jE  Dg. We recall that deg((f )) = 0; deg((!)) = 2(g ? 1); f 2 M(Kg )nf0g; ! 2 M1(Kg )nf0g;

(A.30) where the degree deg(D) of D is given by deg(D) = PP 2Kg D(P ). One calls (f ) (respectively (!)) a principal (respectively canonical) divisor. Introducing the complex linear spaces

L(D) = ff 2 M(Kg )jf = 0 or (f )  Dg; r(D) = dimC L(D); L1(D) = f! 2 M1(Kg )j! = 0 or (!)  Dg; i(D) = dimC L1(D)

(A.31) (A.32)

F. GESZTESY, R. RATNASEELAN, AND G. TESCHL

(i(D) the index of speciality of D), one infers that deg(D), r(D), and i(D) only depend on the divisor class [D] of D. Moreover, we recall the following fundamental facts.

Theorem A.1. Let D 2 Div(Kg ), ! 2 M1(Kg )nf0g. Then (i).

i(D) = r(D ? (!)); g 2 N 0 :

(A.33)

r(?D) = deg(D) + i(D) ? g + 1; g 2 N 0 :

(A.34)

(ii). (Riemann-Roch theorem).

(iii). (Abel's theorem). D 2 Div(Kg ), g 2 N is principal if and only if

deg(D) = 0 and P0 (D) = 0:

(A.35)

(iv). (Jacobi's inversion theorem). Assume g 2 N , then P0 : Div(Kg ) ! J (Kg ) is surjective.

For notational convenience we agree to abbreviate

8 8 > > > > Kg ! f0; 1; : : : ; gg Kg ! f0; 1g > > 8 8 < < > > DQ : > > P ! P ! > > > > (A.36) : : :0 if P 62 fQ1 ; : : : ; Qg g :0; P 6= Q for Q = (Q1 ; : : : ; Qg ) 2 g Kg (nKg then n-th symmetric power of Kg ). Moreover, nKg can be identi ed with the set of positive divisors 0 < D 2 Div(Kg ) of degree n.

Lemma A.2. Let DQ 2 g Kg , Q = (Q1; : : : ; Qg ). Then 1  i(DQ) = s( g=2) if and only if there are s pairs of the type (P; P ) 2 fQ1 ; : : : ; Qg g (this includes, of course, branch points for which P = P  ).

We emphasize that most results in this appendix immediately extend to the case where fEng0n2g  C . (In this case  is no longer purely imaginary as stated in (A.18) but has a positive de nite imaginary part.)

THE KDV HIERARCHY AND ASSOCIATED TRACE FORMULAS

Appendix B. An Explicit Illustration of the Riemann-Roch Theorem

Finally, we give a brief illustration of the Riemann-Roch theorem in connection with KdV-type hyperelliptic curves, i.e., hyperelliptic curves branched at in nity, and explicitly determine a basis for the vector space L(?nDP1 ? D^(x0) ), n 2 N 0 . We freely use the notation introduced in Appendix A and refer, in particular, to the de nition (A.31) of L(D) and the Riemann-Roch theorem stated in Theorem A.1 (ii). In addition, we use the short-hand notation

nDP1 + D^(x0) =

n X

m=1

DP1 +

g X

j =1

D^j (x0 ) ; n 2 N 0 ; ^(x0 ) = (^1(x0 ); : : : ; ^g (x0 ))

(B.1)

and recall that

L(?nDP1 ? D^(x0 ) ) = ff 2 M(Kg )jf = 0 or (f ) + nDP1 + D^(x0 )  0g; n 2 N 0 :

(B.2)

With (P; x), (P; x; x0) de ned as in (3.15), (3.18) we obtain the following

Theorem B.1. Assume D^(x0) to be nonspecial (i.e., i(D^(x0) ) = 0) and of degree g 2 N . For n 2 N 0 , a basis for the vector space L(?nDP1 ? D^(x0 ) ) is given by

or equivalently,

8 > > f1g; n=0 > < f~ j g0j(n?1)=2 [ f~ j (:; x0)g0j(n?1)=2 ; n odd ; > > > :f~ j g0jn=2 [ f~ j (:; x0)g0j(n?2)=2 ; n even

(B.3)

o n @j L(?nDP1 ? D^(x0 )) = span @xj (:; x; x0 ) x=x0 0jn:

(B.4)

@ 2m+2 (P; x; x ) = (?~ )m+1 + R (P; x); 0 2m+1 @x2m+2 @ 2m+1 (P; x; x ) = (?~ )m @ (P; x; x ) + R (P; x); 0 0 2m @x2m+1 @x

(B.5)

Proof. The elements in (B.3) easily seen to be linearly independent and belonging to L(?nDP1 ? D^(x0 ) ). It remains to be shown that they are maximal. >From 0 = i(D^(x0 )) = i(DnP1 + D^(x0 ) ) and the Riemann-Roch theorem (A.34) one obtains r(?nDP1 ? D^(x0 )) = n + 1 proving (B.3). In order to prove (B.4), one repeatedly uses the Schrodinger equation (3.21) to prove inductively that

F. GESZTESY, R. RATNASEELAN, AND G. TESCHL

where Rn(:; x0 ) 2 L(?nDP1 ? D^(x0) ). Acknowledgments . F. G. would like to thank the organizers for their kind invitation to a most stimulating conference. References [1] S. I. Al'ber, Investigation of equations of Korteweg-de Vries type by the method of recurrence relations , J. London Math. Soc. (2) 19, 467{480 (1979) (Russian). [2] S. I. Al'ber, On stationary problems for equations of Korteweg-de Vries type , Commun. Pure Appl. Math. 34, 259{272 (1981). [3] N. Aronszajn and W. F. Donoghue, On exponential representations of analytic functions in the upper half-plane with positive imaginary part , J. Anal. Math. 5, 321{388 (1956{57). [4] E. D. Belokolos, A. I. Bobenko, V. Z. Enol'skii, A. R. Its, and V. B. Matveev, \Algebro-Geometric Approach to Nonlinear Integrable Equations ", Springer, Berlin, 1994. [5] W. Bulla, F. Gesztesy, H. Holden, and G. Teschl, Algebro-geometric quasi-periodic nite-gap solutions of the Toda and Kac-van Moerbeke hierarchy , preprint, 1995. [6] J. L. Burchnall and T. W. Chaundy, Commutative ordinary dierential operators , Proc. London Math. Soc. (2), 21, 420{440 (1923). [7] J. L. Burchnall and T. W. Chaundy, Commutative ordinary dierential operators , Proc. Roy. Soc. London A118, 557{583 (1928). [8] W. Craig, The trace formula for Schrodinger operators on the line , Commun. Math. Phys. 126, 379{407 (1989). [9] L. A. Dickey, \Soliton Equations and Hamiltonian Systems ", World Scienti c, Singapore, 1991. [10] B. A. Dubrovin, Periodic problems for the Korteweg-de Vries equation in the class of nite band potentials , Theoret. Math. Phys. 9, 215{223 (1975). [11] H. M. Farkas and I. Kra, \Riemann Surfaces ", 2nd ed., Springer, New York, 1992. [12] J. D. Fay, \Theta Functions on Riemann Surfaces ", Lecture Notes in Mathematics 352, Springer, Berlin, 1973. [13] H. Flaschka, On the inverse problem for Hill's operator , Arch. Rat. Math. Anal. 59, 293{309 (1975). [14] I. M. Gel'fand and L. A. Dikii, Asymptotic behaviour of the resolvent of Sturm-Liouville equations and the algebra of the Korteweg-de Vries equations , Russian Math. Surv. 30:5, 77{113 (1975). [15] I. M. Gel'fand and L. A. Dikii, Integrable nonlinear equations and the Liouville theorem , Funct. Anal. Appl. 13, 6{15 (1979). [16] F. Gesztesy, New trace formulas for Schrodinger operators , in \Evolution Equations ", G. Ferreyra, G. R. Goldstein, and F. Neubrander (eds.), Marcel Dekker, New York, 1975, pp. 201{221.

THE KDV HIERARCHY AND ASSOCIATED TRACE FORMULAS [17] F. Gesztesy and H. Holden, Trace formulas and conservation laws for nonlinear evolution equations , Rev. Math. Phys. 6, 51{95 (1994). [18] F. Gesztesy and H. Holden, On new trace formulae for Schrodinger operators , Acta Appl. Math. (to appear). [19] F. Gesztesy and B. Simon, The xi function, Acta Math. (to appear). [20] F. Gesztesy and R. Weikard, Spectral deformations and Soliton equations , in \Dierential Equations with Applications to Mathematical Physics ", W. F. Ames, E. M. Harrell II, and J. V. Herod (eds.), Academic Press, Boston, 1993, pp. 101{139. [21] F. Gesztesy and R. Weikard, Picard potentials and Hill's equation on a Torus , preprint, 1994. [22] F. Gesztesy, H. Holden, and B. Simon, Absolute summability of the trace relation for certain Schrodinger operators , Commun. Math. Phys. (to appear). [23] F. Gesztesy, H. Holden, B. Simon, and Z. Zhao, Trace formulae and inverse spectral theory for Schrodinger operators , Bull. Amer. Math. Soc. 29, 250{255 (1993). [24] F. Gesztesy, H. Holden, B. Simon, and Z. Zhao, Higher order trace relations for Schrodinger operators , Rev. Math. Phys. (to appear). [25] H. Hochstadt, On the determination of a Hill's equation from its spectrum , Arch. Rat. Mech. Anal. 19, 353{362 (1965). [26] A. R. Its and V. B. Matveev, Schrodinger operators with nite-gap spectrum and N -soliton solutions of the Korteweg-de Vries equation , Theoret. Math. Phys. 23, 343{355 (1975).  eine neue Methode zur Integration der hyperelliptischen Dierentialgleichungen [27] C. G. T. Jacobi, Uber und uber die rationale From ihrer vollstandigen algebraischen Integralgleichungen , J. Reine Angew. Math. 32, 220{226 (1846). [28] S. Kotani and M. Krishna, Almost periodicity of some random potentials , J. Funct. Anal. 78, 390{405 (1988). [29] A. Krazer, \Lehrbuch der Thetafunktionen ", Chelsea, New York, 1970. [30] M. G. Krein, Perturbation determinants and a formula for the traces of unitary and self-adjoint operators , Sov. Math. Dokl. 3, 707{710 (1962). [31] B. M. Levitan, On the closure of the set of nite-zone potentials , Math. USSR Sbornik 51, 67{89 (1985). [32] B. M. Levitan, \Inverse Sturm-Liouville Problems ", VNU Science Press, Utrecht, 1987. [33] H. P. McKean, Variation on a theme of Jacobi , Commun. Pure Appl. Math. 38, 669{678 (1985). [34] H. P. McKean and P. van Moerbeke, The spectrum of Hill's equation , Invent. Math. 30, 217{274 (1975). [35] H. P. McKean and E. Trubowitz, Hill's operator and hyperelliptic function theory in the presence of in nitely many branch points , Commun. Pure Appl. Math. 29, 143{226 (1976). [36] D. Mumford, \Tata Lectures on Theta II ", Birkhauser, Boston, 1984.

F. GESZTESY, R. RATNASEELAN, AND G. TESCHL [37] R. Narasimhan, \Compact Riemann Surfaces ", Birkhauser, Basel, 1992. [38] S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, \Theory of Solitons ", Consultants Bureau, New York, 1984. [39] B. Simon, Spectral analysis of rank one perturbations and applications, proceedings, \Mathematical Quantum Theory II: Schrodinger Operators ", J. Feldman, R. Froese, and L. M. Rosen (eds.), Amer. Math. Soc., Providence, RI, to appear. [40] E. Trubowitz, The inverse problem for periodic potentials , Commun. Pure Appl. Math. 30, 321{337 (1977). Department of Mathematics, University of Missouri, Columbia, MO 65211

E-mail address :

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

E-mail address :

[email protected]

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

E-mail address :

[email protected]