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DARBOUX-TYPE TRANSFORMATIONS AND HYPERELLIPTIC CURVES FRITZ GESZTESY AND HELGE HOLDEN 9 9 Abstract. We systematically study Darboux-type transformations for the 9 KdV and AKNS hierarchies and provide a complete account of their effects 1 on hyperelliptic curves associated with algebro-geometric solutions of these hierarchies. n a J 3 2 1. Introduction 1 Since the early daysof completely integrable evolutionequations in the late six- v ties,Darboux-typetransformations(cf.,e.g.,[42]andthereferencestherein)played 8 animportantroleandturnedouttobeanintegralpartof(auto-)Bäcklundtransfor- 0 mationsbetweensolitonequations. ThecanonicalexampleinthiscontextisMiura’s 0 transformation [49] between the Korteweg-de Vries (KdV) hierarchy and modified 1 0 Korteweg-deVries(mKdV)hierarchy. Asdescribedin(2.35)–(2.42),Miura’strans- 9 formation, disregarding the time variable (i.e., considering the stationary case), is 9 effected by the classical Crum-Darboux transformation, which in turn is based on / t factorizing the second-order Lax (one-dimensional Schrödinger) operator n i d2 - L=− +V (1.1) v dx2 l o for the KdV hierarchy into a product of first-order differential expressions plus a s : shift, v i d d X L=AA++z , A= +φ, A+ =− +φ, V =φ2+φ +z . (1.2) 0 x 0 dx dx r a AssumingV tosatisfyoneofthestationaryKdVequationsandreversingtheorder of the two factors A and A+ produces a new Lax operator L, d2 L=A+A+z =− +V, V =φ2−φe+z , (1.3) 0 dx2 x 0 whose potential Ve is a new solution of oneeof thee equations in the stationary KdV hierarchy. In short, the transformation e V 7→V (1.4) e 1991 Mathematics Subject Classification. Primary 35Q53, 35Q55, 58F07; Secondary 35Q51, 35Q58. Keywordsandphrases. Darbouxtransformations,hyperellipticcurves,KdVhierarchy,AKNS hierarchy. Supported in part by the Research Council of Norway under grant 107510/410 and the Uni- versityofMissouriResearchBoardgrantRB-97-086. 1 2 GESZTESY AND HOLDEN represents a Darboux transformation, or equivalently, an auto-Bäcklund transformation of the stationary KdV hierarchy. Incidentally, ±φ in (1.2), (1.3) represent solutions of one of the stationary equations of the mKdV hierarchy and hence V 7→φ7→−φ7→V (1.5) represents a Bäcklund transformation from the (stationary) KdV to the mKdV e hierarchy(V 7→φ)aswellasauto-BäcklundtransformationsfortheKdV(V 7→V) and mKdV hierarchy (φ 7→ −φ). However, while φ and −φ satisfy the identical equation(s) within the stationary mKdV hierarchy, and hence e 0 A+ 0 −A D = 7→D = (1.6) A 0 −A+ 0 (cid:18) (cid:19) (cid:18) (cid:19) represents an isospectral deformation of De, V and V in general do not satisfy the same stationary equation of the KdV hierarchy, that is, e L7→L, (1.7) ingeneral,isnotanisospectraldeformationofL.Moreprecisely,eachsolutionV of e (one of) the nth stationary KdV equations is associated with a (possibly singular) hyperelliptic curve K of the type n 2n K :y2 = (z−E ), {E } ⊂C. (1.8) n m m m=0,...,2n m=0 Y Similarly, V corresponds to a curve K of the type n˜ 2n˜ e K :y2 = (z−eE ), {E } ⊂C (1.9) n˜ m m m=0,...,2n˜ m=0 Y e e e and hence V and V (resp. L and L) are isospectral if and only if K = K (i.e., n n˜ {E } ={E } , n=n˜). m m=0,...,2n m m=0,...,2n˜ Theprincipalaime ofthepaperisetore-examinetherelationbetweenK aendK , n n˜ depending on varioues choices of the function φ in (1.2), and to provide a complete, yet elementary solution of this problem. Historically, the first attempts to linkeK n and K were made by Drach [14], [15], [16]. The first solution of this problem was n˜ obtained by Ehlers and Kno¨rrer in 1982 on the basis of purely algebro-geometric techneiques. Anelementarybutquiteelaborateapproachwasrecentlydevelopedby Ohmiya [52] (based on two additional papers [51], [53]). Our proof of Theorem2.3 appears to be the only elementary and short one at this point. It seems worthwhile pointing out that the curves (1.8) and (1.9) may of course be singular, that is, some (or even all) of the E ’s may coincide. In fact, the class m of rational KdV solutions as discussed by Adler and Moser [2] (see also [50], [53]) arises in precisely this manner (with all E = 0). Similarly, the class of soliton m solutions and more generally, solitons relative to an algebro-geometricbackground potential, as described, for instance, in [11], [12], [17], [33], [35], [36], [43], [47], [48], App. A, [55] results in n pairs of coinciding E ’s, that is, {E } = m m m=0,...,2n {E ,E =E ,...,E =E } for an appropriate enumeration of the E ’s. 0 1 2 2n−1 2n m Finally, a quick description of the content of this paper. Section 2 is devoted to theKdVcaseandstartsbysummarizingthebasicformalismfortheKdVhierarchy and its algebro-geometric stationary solutions. Subsequently, we review Darboux transformations V 7→ V and finally derive the precise connection between K and n e DARBOUX-TYPE TRANSFORMATIONS 3 K in Theorem 2.3. Section 3 then presents the analogous results for the AKNS n˜ hierarchy. Besides of being an important hierarchy of soliton evolution equations, teheAKNShierarchyallowsustostudy hyperelliptic curveswithoutabranchpoint at infinity as opposed to the KdV case, which necessarily leads to hyperelliptic curves branched at infinity. The corresponding Theorem 3.3, detailing the connection between K and K in the AKNS case, to the best of our knowledge, appears n n˜ to be without precedent. e 2. The stationary KdV hierarchy In this section we study Bäcklund transformations of the KdV hierarchy. Our principalaimistoprovearesultbyEhlersandKno¨rrer[19]onhyperellipticcurves and Darboux transformations by entirely elementary means and at the same time offer a more detailed treatment (cf. Theorem 2.3). We start by introducing a polynomial recursion formalism following Al’ber [3], [4] (see also [13], Ch. 12, [26], [27]) presented in detail in [32], [36] (see also [29], [30], [37]). Suppose V : C 7→ C , with C = C∪{∞}, is meromorphic and consider the ∞ ∞ Schrödinger operator d2 L=− +V(x), x∈C. (2.1) dx2 Introducing {fj}j∈N0, with N0 =N∪{0}, recursively by 1 1 f =1, f =− f +Vf + V f , j ∈N, (2.2) 0 j,x j−1,xxx j−1,x x j−1 4 2 one finds explicitly, f =1, 0 f = 1V +c , 1 2 1 f =−1V + 3V2+c 1V +c , etc., (2.3) 2 8 xx 8 12 2 where {cℓ}ℓ∈N ⊂ C denote integration constants. The fj are well-known to be differentialpolynomialsinV (see,e.g.,[20]). GivenV andf onedefinesdifferential j expressions P of order 2n+1, 2n+1 n d 1 P = f (x) − f (x) Lj, n∈N , (2.4) 2n+1 n−j n−j,x 0 dx 2 j=0(cid:18) (cid:19) X and one verifies [P ,L]=2f , n∈N , (2.5) 2n+1 n+1,x 0 with[·, ·]thecommutator. ThestationaryKdVhierarchyisthendefinedinterms of the stationary Lax relations [P ,L]=2f =0, n∈N . (2.6) 2n+1 n+1,x 0 Explicitly, one finds n=0: V =0, x n=1: 1V − 3VV −c V =0, etc. (2.7) 4 xxx 2 x 1 x V(x) is called an algebro-geometric KdV potential if it satisfies one (and hence infinitely many) of the equations in the stationary KdV hierarchy (2.6). 4 GESZTESY AND HOLDEN Introducing the fundamental polynomial F (z,x) of degree n in z, n n F (z,x)= f (x)zj, (2.8) n n−j j=0 X equations (2.2) and (2.6), that is, f (x)=0, x∈C, imply n+1,x F −4(V −z)F −2V F =0. (2.9) n,xxx n,x x n Multiplying (2.9) by F and integrating yields n 1 1 F (z,x)F (z,x)− F (z,x)2−(V(x)−z)F (z,x)2 =R (z), (2.10) n,xx n n,x n 2n+1 2 4 wheretheintegrationconstantR (z)isamonicpolynomialinzofdegree2n+1, 2n+1 and hence of the form 2n R (z)= (z−E ), {E } ⊂C. (2.11) 2n+1 m m m=0,...,2n m=0 Y Introducing the algebraic eigenspace, ker(L−z)={ψ: C7→C meromorphic|(L−z)ψ =0}, z ∈C (2.12) ∞ one verifies d 1 P = F (z,x) − F (z,x) . (2.13) 2n+1 ker(L−z) n dx 2 n,x (cid:18) (cid:19)(cid:12)ker(L−z) (cid:12) (cid:12) Moreover,a celebrat(cid:12)ed resultby Burchnall–Chaundy[6], [7](cid:12), [8] (see also [25], [54], (cid:12) [55], [57], [59]) then yields P2 +R (L)=0. (2.14) 2n+1 2n+1 Equation (2.14) naturally leads to the hyperelliptic curve K˙ defined by n 2n K˙ : F (z,y)=y2−R (z)=0, R (z)= (z−E ). (2.15) n n 2n+1 2n+1 m m=0 Y The one-point compactification of K˙ by joining P , the point at infinity, is then n ∞ denoted by K . A general point P ∈ K \{P } will be denoted by P = (z,y), n n ∞ where F (z,y)=0. (2.16) n Moreover,we define the involution ∗ on K by n ∗: K 7→K , P =(z,y)7→P∗ =(z,−y), P∗ =P . (2.17) n n ∞ ∞ Introducing the polynomial H of degree n+1 in z, n+1 1 H (z,x)= F (z,x)+(z−V(x))F (z,x), (2.18) n+1 n,xx n 2 (2.10) implies 1 R (z)+ F (z,x)2 =F (z,x)H (z,x) (2.19) 2n+1 n,x n n+1 4 andwemaydefinethefollowingfundamentalmeromorphicfunctionφ(P,x) onK , n iy(P)+ 1F (z,x) φ(P,x)= 2 n,x (2.20a) F (z,x) n DARBOUX-TYPE TRANSFORMATIONS 5 −H (z,x) = n+1 , P =(z,y)∈K , x∈C, (2.20b) iy(P)− 1F (z,x) n 2 n,x where y(P) denotes the meromorphic function on K obtained upon solving y2 = n R (z) with P =(z,y). 2n+1 Introducing the stationaryBaker–Akhiezerfunctionψ(P,x,x )onK \{P } by 0 n ∞ x ψ(P,x,x )=exp dx′φ(P,x′) , P =(z,y)∈K \{P }, x,x ∈C, (2.21) 0 n ∞ 0 (cid:18)Zx0 (cid:19) choosing a smooth non-selfintersecting path from x to x which avoids singulari- 0 ties of φ(P,x), one easily verifies from (2.9), (2.10), (2.13), (2.20), and (2.21) the following result. Lemma 2.1. (see, e.g., [30], [32]) Suppose f (x) = 0 and let P = (z,y) ∈ n+1,x K \{P }, x,x ∈C. Then φ(P,x) satisfies the Riccati-type equation n ∞ 0 φ (P,x)+φ(P,x)2 =V(x)−z, (2.22) x and φ(P,x)φ(P∗,x)=H (z,x)/F (z,x), (2.23) n+1 n φ(P,x)+φ(P∗,x)=F (z,x)/F (z,x), (2.24) n,x n φ(P,x)−φ(P∗,x)=2iy(P)/F (z,x), (2.25) n φ(P,x)=iz−ny(P)+O(|z|−1/2) as P =(z,y)→P . (2.26) ∞ ψ(P,x,x ) satisfies 0 (L−z(P))ψ(P, ·,x )=0, (P −iy(P))ψ(P, ·,x )=0, (2.27) 0 n+1 0 and ψ(P,x,x )ψ(P∗,x,x )=F (z,x)/F (z,x ), (2.28) 0 0 n n 0 ψ (P,x,x )ψ (P∗,x,x )=H (z,x)/F (z,x ), (2.29) x 0 x 0 n+1 n 0 ψ(P,x,x )ψ (P∗,x,x )+ψ(P∗,x,x )ψ (P,x,x ) 0 x 0 0 x 0 =F (z,x)/F (z,x ), (2.30) n,x n 0 W(ψ(P, ·,x ),ψ(P∗, ·,x ))=−2iy(P)/F (z,x ), (2.31) 0 0 n 0 ψ(P,x,x )=exp iz−ny(P)(x−x ) (1+O(|z|−1/2)) as P =(z,y)→P . 0 0 ∞ (2.32) (cid:0) (cid:1) (Here W(f,g)(x)=f(x)g′(x)−f′(x)g(x) denotes the Wronskian of f and g.) In addition, we introduce the formal diagonal Green’s function G(P,x,x) associated with the differential expression L by ψ(P,x,x )ψ(P∗,x,x ) 0 0 G(P,x,x)= (2.33a) W(ψ(P, ·,x ),ψ(P∗, ·,x )) 0 0 iF (z,x) = n , P =(z,y)∈K \{P }, x∈C, (2.33b) n ∞ 2y(P) using φ = ψ /ψ, (2.25), and (2.28). Equations (2.10) and (2.33b) then yield the x universal equation −2G′′(P,x,x)G(P,x,x)+G′(P,x,x)2+4(V(x)−z)G(P,x,x)2 =1, P =(z,y)∈K \{P }. (2.34) n ∞ 6 GESZTESY AND HOLDEN TheseobservationscompleteourreviewofthestationaryKdVhierarchy. Thetime- dependent KdV hierarchy could readily be defined at this point but since it is not essential for our purpose we resist the temptation to do so (see, e.g., [30], [32] for the time-dependent formalism). NextweturntotheDarbouxtransformationsinconnectionwithL=−d2/dx2+ V. Define 1(1+σ)ψ(P,x,x )+ 1(1−σ)ψ(P∗,x,x ) for σ ∈C, ψ(P,x,x ,σ)= 2 0 2 0 0 (ψ(P,x,x0)−ψ(P∗,x,x0) for σ =∞, P ∈K \{P }, (2.35) n ∞ pick Q =(z ,y )∈K \{P }, and introduce the differential expressions 0 0 0 n ∞ d d A (Q )= +φ(Q ,x,σ), A+(Q )=− +φ(Q ,x,σ), σ ∈C , (2.36) σ 0 dx 0 σ 0 dx 0 ∞ where φ(P,x,σ)=ψ (P,x,x ,σ)/ψ(P,x,x ,σ), P ∈K \{P }, σ ∈C . (2.37) x 0 0 n ∞ ∞ One verifies (cf. (2.22)) d2 L=A (Q )A+(Q )+z =− +V, (2.38) σ 0 σ 0 0 dx2 with V(x)=φ(Q ,x,σ)2+φ (Q ,x,σ)+z , (2.39) 0 x 0 0 independent of the choice of σ ∈ C . Interchanging the order of the differential ∞ expressions A (Q ) and A+(Q ) in (2.38) then yields σ 0 σ 0 d2 L (Q )=A+(Q )A (Q )+z =− +V (x,Q ), (2.40) σ 0 σ 0 σ 0 0 dx2 σ 0 with e e V (x,Q )=φ(Q ,x,σ)2−φ (Q ,x,σ)+z σ 0 0 x 0 0 =V(x)−2(ln(ψ(Q ,x,x ,σ))) , σ ∈C . (2.41) 0 0 xx ∞ e The transformation V(x)7→V (x,Q ), Q ∈K \{P }, σ ∈C (2.42) σ 0 0 n ∞ ∞ is usually called the Darboux transformation (also Crum-Darboux transformation e or single commutation method) and goes back at least to Jacobi [40] and Dar- boux [10]. While we only aim at its properties from an algebraic point of view, its analytic properties in connection with spectral deformations (isospectral and non-isospectral ones) have received enormous attention in the context of spectral theory(especially,regardingtheinsertionofeigenvaluesintospectralgaps),inverse spectral theory, and Bäcklund tranformations for the (time-dependent) KdV hierarchy. A complete bibliography in this context being impossible, we just refer to [1], [9], [11], [12], [18], Ch. 4, [19], [21], [23], [24], [28], [33], [34], [35], [36], [44], [45], [46], [56], [58] and the extensive literature therein. From a historical point of view it is very interesting to note that Drach [14], [15], [16] in his 1919 studies of Darboux transformations (being a student of Darboux) not only introduced a set of nonlinear differential equations for V, whichtoday can be identified with the stationary KdV hierarchy, but also studied the effect of Darboux transformations ontheunderlyinghyperellipticcurve. Asaconsequence,heseemstohavebeenthe DARBOUX-TYPE TRANSFORMATIONS 7 first to explicitly establish the connection between integrable systems and spectral theory. For modern treatments of this connection see, for instance, [5], Ch. 3 and [30]. Before analyzing the Darboux tranformation (2.42) in some detail, we briefly return to the formal diagonal Green’s function G(P,x,x) in (2.33). In view of the possibility of linear combinations as displayedin the definition of ψ(P,x,x ,σ) 0 in (2.35), and our lack of natural boundary conditions (resp. L2-conditions) in connection with the (possibly singular) differential expression L, our definition of G(P,x,x) in (2.33) appears to be highly arbitrary. However, the following consid- erationswillshowthatourchoicein(2.33)is the unique onethatyields abounded Green’s function as P →P (for x∈C\{x } fixed). In fact, ∞ 0 izn G(P,x,x)= +O(|z|−1) as P =(z,y)→P . (2.43) ∞ 2y(P) Introducing ψ(P,x,x ,σ )ψ(P,x,x ,σ ) 0 1 0 2 H(P,x,x,x ,σ ,σ )= , 0 1 2 W(ψ(P, ·,x ,σ ),ψ(P, ·,x ,σ )) 0 1 0 2 σ ,σ ∈C , σ 6=σ , (2.44) 1 2 ∞ 1 2 one infers from Lemma 2.1, (2.32), and (2.35) H(P,x,x,x ,σ ,σ ) 0 1 2 i(1+σ )(1+σ )zn = 1 2 exp(2iz−ny(P)(x−x ))(1+O(|z|−1/2)) 0 4(σ −σ )y(P) 1 2 i(1−σ )(1−σ )zn + 1 2 exp(−2iz−ny(P)(x−x ))(1+O(|z|−1/2)) (2.45) 0 4(σ −σ )y(P) 1 2 i(1−σ σ )zn + 1 2 +O(|z|−1/2) as P =(z,y)→P ∞ 2(σ −σ )y(P) 1 2 for σ ,σ ∈ C, σ 6= σ , and similarly if σ = ∞, σ ∈ C, or σ ∈ C, σ = ∞. 1 2 1 2 1 2 1 2 Thus, H(P,x,x,x ,σ ,σ ) is uniformly bounded for P in a neighborhood of P 0 1 2 ∞ and x∈C\{x } fixed, if and only if 0 σ =−σ ∈{−1,1}. (2.46) 1 2 Since ψ(P,x,x ,−σ)=ψ(P∗,x,x ,σ), this shows that our choice for G(P,x,x) in 0 0 (2.32) yields the unique diagonal Green’s function of L which is bounded for P in a neighborhood of P for fixed x ∈ C\{x }. In addition, H(P,x,x,x ,σ ,σ ) is ∞ 0 0 1 2 independent of x if and only if (2.46) holds. 0 Next, assuming that ψ ∈ker(L−z), Lψ(z)=zψ(z), (2.47) one infers A+(Q )ψ(z)∈ker(L (Q )−z), σ 0 σ 0 L (Q )(A+(Q )ψ(z))=zA+(Q )ψ(z), (2.48) σ 0 eσ 0 σ 0 and e W(A+(Q )ψ (z),A+(Q )ψ (z))=(z−z )W(ψ (z),ψ (z)), σ 0 1 σ 0 2 0 1 2 ψ (z),ψ (z)∈ker(L−z). (2.49) 1 2 Since 8 GESZTESY AND HOLDEN (A+(Q )ψ(P, ·,x ))(x)= φ(Q ,x,σ)−φ(P,x) ψ(P,x,x ), σ 0 0 0 0 P ∈K \{Q ,P }, (2.50) (cid:0) (cid:1) n 0 ∞ we define ψ˜ (P,x,x ,Q )=(A+(Q )ψ(P, ·,x ))(x) σ 0 0 σ 0 0 = φ(Q ,x,σ)−φ(P,x) ψ(P,x,x ), (2.51) 0 0 P ∈K \{Q ,P }, σ ∈C . (cid:0) n 0 (cid:1)∞ ∞ Then (L (Q )ψ˜ (P, ·,x ,Q ))(x)=zψ˜ (P,x,x ,Q ), σ 0 σ 0 0 σ 0 0 P =(z,y)∈K \{Q ,P }, (2.52) n 0 ∞ e and we define in analogy to (2.33a) the diagonal Green’s function G (P,x,x,Q ) σ 0 of L (Q ) by σ 0 e ψ˜ (P,x,x ,Q )ψ˜ (P∗,x,x ,Q ) e σ 0 0 σ 0 0 G (P,x,x,Q )= , σ 0 W(ψ˜ (P, ·,x ,Q ),ψ˜ (P∗, ·,x ,Q )) σ 0 0 σ 0 0 e P =(z,y)∈Kn\{Q0,P∞}. (2.53) Lemma 2.2. Assume f (x) = 0 and let Q = (z ,y ) ∈ K \{P }, P = n+1,x 0 0 0 n ∞ (z,y)∈K \{Q ,P },σ ∈C . ThenthediagonalGreen’sfunctionG (P,x,x,Q ) n 0 ∞ ∞ σ 0 in (2.34) explicitly reads e H (z,x)+φ(Q ,x,σ)2F (z,x)−φ(Q ,x,σ)F (z,x) n+1 0 n 0 n,x G (P,x,x,Q )= σ 0 −2i(z−z )y(P) 0 (2.54a) e (φ(P,x)−φ(Q ,x,σ))(φ(P∗,x)−φ(Q ,x,σ))F (z,x) 0 0 n = (2.54b) −2i(z−z )y(P) 0 iF (z,x) σ,n˜ = , (2.54c) 2y˜(P) e ˙ where y˜(P) denotes the meromorphic solution on K (Q ) obtained upon solving σ,n˜ 0 y2 =R (z), P =(z,y) for some polynomial R (z) of degree 2n˜+1∈N σ,2n˜+1 σ,2n˜+1 0 (cf. (2.15) and (2.58)) and F (z,x) denotes a peolynomial with respect to z of σ,n˜ degreeen˜, with 0 ≤ n˜ ≤ n+1. In particular, theeDarboux transformation (2.42), V(x)7→V (x,Q ) maps the celass of algebro-geometric KdV potentials into itself. σ 0 Proof. (2.54a) and (2.54b) follow upon use of φ(P,x) = ψ (P,x,x )/ψ(P,x,x ), e x 0 0 (2.23), (2.24), (2.28)–(2.31), and (2.49). Since the numerator in (2.54a) is a polynomial in z, and izn G (P,x,x,Q )= +O(|z|−1) as P =(z,y)→P (2.55) σ 0 ∞ 2y(P) againby(2.54ea),oneconcludes(2.54c)and0≤n˜ ≤n+1. Byinspection,F (z,x) σ,n˜ satisfies equation (2.10) with V(x) replaced by V (x,Q ), n by n˜, and R (z) σ 0 2n+1 by R (z). As a consequence, V(x) being an algebro-geometric KdVepotential σ,2n˜+1 implies that V (x,Q ) is one as well. e σ 0 e e DARBOUX-TYPE TRANSFORMATIONS 9 Thefollowingtheorem,theprincipalresultofthissection,willclarifythedepen- dence of n˜ = n˜(n,Q ,σ) on its variables. This result was originally derived using 0 an entirely different algebro-geometricapproach by Ehlers and Kno¨rrer [19]. Theorem 2.3. Suppose f (x) =0 and let Q =(z ,y ) ∈K \{P }, n ∈ N , n+1,x 0 0 0 n ∞ 0 σ ∈C , and n˜ =n˜(n,Q ,σ) as in (2.54c). Then ∞ 0 n+1 for σ ∈C \{−1,1} and y 6=0, ∞ 0 n+1 for σ =∞ and y0 =0, n˜(n,Q0,σ)=nn ffoorr σσ ∈∈C{−,1y,01=} a0n,danyd0 6=R20n,+1,z(z0)6=0, (2.56) n−1 for σ ∈C, y =0, and R (z )=0,n∈N, and hence the hyperelliptic curve K˙ (0Q ) associate2dn+w1i,zth V0 (x,Q ) is of the type σ,n˜ 0 σ 0 ˙ K (Q ): F (z,y,Q )=y2−R (z,Q )=0, (2.57) σ,n˜ 0 σ,n˜ e 0 σ,2n˜+1 e0 with e e e R (z,Q ) σ,2n˜+1 0 (z−z )2R (z) for σ ∈C \{−1,1} and y 6=0, e 0 2n+1 ∞ 0 (z−z0)2R2n+1(z) for σ =∞ and y0 =0, =RR22nn++11((zz)) ffoorr σσ ∈∈C{−,1y,01=} a0n,danyd0 6=R20n,+1,z(z0)6=0, (2.58) (z−z )−2R (z) for σ ∈C, y =0, and R (z )=0,n∈N. Here 0 2n+1 0 2n+1,z 0 2n R (z)= (z−E ). (2.59) 2n+1 m m=0 Y Proof. Our starting point will be (2.54b) and a careful case distinction taking into account whether or not Q is a branch point, and distinguishing the cases σ ∈ 0 C\{−1,1},σ ∈{−1,1},and σ =∞. Case (i). σ ∈C \{−1,1} and y 6=0: One computes from (2.35) and (2.37), ∞ 0 (1+σ)ψx(Q0,x,x0)+(1−σ)ψx(Q∗0,x,x0) for σ ∈C\{−1,1}, φ(Q ,x,σ)= (1+σ)ψ(Q0,x,x0)+(1−σ)ψ(Q∗0,x,x0) (2.60) 0 ψx(Q0,x,x0)−ψx(Q∗0,x,x0) for σ =∞,  ψ(Q0,x,x0)−ψ(Q∗0,x,x0) and upon comparison with φ(Q ,x)6=φ(Q∗,x),  0 0 ψ (Q ,x,x ) ψ (Q∗,x,x ) φ(Q ,x)= x 0 0 , φ(Q∗,x)= x 0 0 , (2.61) 0 ψ(Q ,x,x ) 0 ψ(Q∗,x,x ) 0 0 0 0 oneconcludesthatnocancellationscanoccurin(2.54b),provingn˜(n,Q ,σ)=n+1 0 and the first statement in (2.58). Case (ii). σ =∞ and y =0: Combining (2.20a), (2.21), (2.25), and (2.35) one 0 computes φ(Q ,x,∞)= lim φ(P,x,∞) 0 P→Q0 φ(P,x)exp x dx′φ(P,x′) −φ(P∗,x)exp x dx′φ(P∗,x′) = lim x0 x0 P→Q0 exp(cid:0)Rxx0dx′φ(P,x′)(cid:1) −exp xx0dx′(cid:0)φR(P∗,x′) (cid:1)! (cid:0)R (cid:1) (cid:0)R (cid:1) 10 GESZTESY AND HOLDEN =φ(Q ,x) 0 φ(P,x)−φ(P∗,x) + lim × P→Q0 exp xx0dx′φ(P,x′) −exp xx0dx′φ(P∗,x′) (cid:0)R (cid:1) (cid:0)R x (cid:1) ×exp dx′φ(P∗,x′) (cid:18)Zx0 (cid:19)! x =φ(Q ,x)+exp dx′φ(Q ,x′) × 0 0 (cid:18)Zx0 (cid:19) 2iy(P)/F (z,x) n × lim × P→Q0 exp iy(P) xx0 Fnd(zx,′x′) −exp −iy(P) xx0 Fnd(zx,′x′) (cid:0) R (cid:1) (cid:0) 1 x RF (z,x′(cid:1)) ×exp − dx′ n,x 2 F (z,x′) (cid:18) Zx0 n (cid:19)! =φ(Q ,x) 0 1 2iy(P) +ψ(Q ,x,x ) lim 0 0 Fn(z0,x)ψ(Q0,x,x0)P→Q0(cid:18)2iy(P) xx0 Fnd(zx,′x′) +O(y(P)2)(cid:19) x dx′ −1 R =φ(Q ,x)+ F (z ,x) , x∈C\{x }, (2.62) 0 n 0 F (z,x′) 0 (cid:18) Zx0 n (cid:19) using lim y(P)=y(Q )=y =0. From P→Q0 0 0 1F (z ,x) n,x 0 φ(Q ,x)= (2.63) 0 2 F (z ,x) n 0 oneconcludesagainthatnocancellationscanoccurin(2.54b). Thusn˜(n,Q ,∞)= 0 n+1 and the second statement in (2.58) holds. The remainderofthe proofrequiresa morerefinedargument,the basisofwhich will be derived next. First, replacing V(x) by V(x)−z , we may assume without 0 loss of generality that z = 0 in the following. Moreover, from this point on we 0 exclude the trivial case V(x) = 0, x ∈ C (the case V(x) = E will be studied in 0 Example 2.7 below). Writing y(z)2 =R (z) = y2+y˜ z+y˜ z2+O(z3), (2.64) 2n+1 0 1 2 z→0 a comparison of the powers z0 and z1 in (2.10) yields 2f f =f2 +4Vf2+4y2 (2.65) n,xx n n,x n 0 and f f +f f −f f −4Vf f +2Vf2−2y˜ =0. (2.66) n−1,xx n n,xx n−1 n,x n−1,x n n−1 n 1 Inserting (2.65) into (2.66), a little algebra proves the basic identity f2(f /f ) +f f (f /f ) +2Vf2+4y2(f /f )−2y˜ =0. (2.67) n n−1 n xx n,x n n−1 n x n 0 n−1 n 1 Case (iii). σ ∈{−1,1} and y 6=0: Then (2.35) yields 0 φ(Q ,x,1)=φ(Q ,x), φ(Q ,x,−1)=φ(Q∗,x), (2.68) 0 0 0 0 withφ(Q ,x)6=φ(Q∗,x)sincey 6=0. Inthiscasethereisacancellationin(2.54b). 0 0 0 For instance, choosing σ =1 one computes from (2.8) and (2.20a), φ(P,x)−φ(Q ,x,1)=φ(P,x)−φ(Q ,x) 0 0