The periodic defocusing Ablowitz-Ladik equation and the geometry of Floquet CMV matrices

THE PERIODIC DEFOCUSING AB LO WITZ-LADIK EQUA TION AND THE GEOMETR Y OF FLOQUET CMV MA TRICES Luen-Chau Li an d Irina Nenciu Abstract. In this w ork, we show that the p erio dic defo cusing Ablowitz -Ladik equa- tion can be expressed as an isospectral deformation of Floq uet CMV matrices. W e then introduce a Poisson Lie group whose underlying group is a lo op group and show that the set of Floq uet CMV matrices is a Coxeter dressing orbit o f this Pois- son Lie group. By u sing the group-theoretic fra mework, we establish the Liouville in tegrabilit y of the equation by constructing action-angle v ariables, we also solve the Hamiltonian equations generated by the commuting ﬂ ows via R iemann-Hilbert factorization problems. 1. In tro duction. The defo cusin g Ablo witz-Ladik (AL) equ ation (a.k.a. defo cusing discrete n on- linear Sc h r¨ odinger equation) is the system of diﬀeren tial-diﬀerence equations giv en b y − i ˙ α n = α n +1 − 2 α n + α n − 1 − | α n | 2 ( α n − 1 + α n +1 ) (1 . 1) where α n is a sequence of n um b ers inside the unit disk D . It w as int ro duced by Ablo witz and Ladik in [AL] as a sp atial discretization of the d efo cusing nonlinear Sc h r¨ odinger equation and since then h as b een the sub ject of n umerous s tudies. In particular, muc h atten tion has b een fo cused on the inv erse scattering metho d for solving the equation in the t w o-sided case where the index n r anges ov er the set of integ ers. (See [APT] and the references therein.) By con trast, the literature on the p erio d ic prob lem is r elativ ely spars e. (See, for example, [K], [MEKL], [N], [GHMT].) In rece n t years, one of the in teresting dev elopments in the arena of th e defo cusin g AL equatio n has b een the connection with the th eory of orthogonal p olynomials on the circle (OPUC) and the so-calle d C MV m atrices [S2], [N], and our w ork here is a con tinuatio n of this d ev elopment. Since we are dealing with the p erio dic defo cusing AL equation here, let us b egin w ith a set of V erblunsk y co eﬃcien ts { α j } ∞ j =0 satisfying the p erio d icit y condition α j + p = α j , j = 0 , 1 , 2 , · · · . Without loss of generalit y , w e m a y assum e p is ev en. In [S 2], Simon in tro duced the discriminant ∆( z ) asso ciated with { α j } ∞ j =0 and together with th e second author, they obtained the f ollo wing in v olution theorem [S2], [N] { ∆( z ) , ∆( w ) } AL = 0 , { P , ∆( z ) } AL = 0 (1 . 2) Typeset by A M S -T E X 1 2 L.-C. LI, I. N E NCIU b y calculating with W all p olynomials. Here {· , ·} AL is the Ablo witz-Ladik Poisson brac ket [KM], [S2] and P = Q p − 1 j =0 ρ j , w here ρ j = p 1 − | α j | 2 . This p rompted the searc h f or an in tegrable system whic h is related to OPUC in the same w ay the T o da lattice is related to orthogonal p olynomials on the line. As it tur ned out, the sough t-after in tegrable sys tem is th e p erio dic defo cusing AL equation [N]. In [N], the Lax equations for the comm uting ﬂows were expressed in terms of the extended CMV matrix E with p erio dic V erblunsky co eﬃcien ts. Ho we v er, as ∆( z ) is related to th e Flo qu et CMV matrix E ( h ) (wh ic h is a unitary lo op with sp ectral p arameter h ) through the c h aracteristic p olynomial det( z I − E ( h )) , it is natural to ask if the same equations can b e rewritten as isosp ectral deformations of E ( h ) . As the reader will see in Sectio n 2, this is indeed the case and the r esult is the point of departure in this wo rk. More precisely , the resu lt not only suggests that th e set of p × p Flo quet CMV matrices should hav e some P oisson geometric meaning, bu t also p oin ts to the linearization of suc h ﬂo ws on geometric ob jects r elated to the Jacobi v arieties of the un derlying sp ectral curves. Th us our goal in th is w ork is t w o-fold. First of all, w e will link th e Flo qu et CMV matrices to Poisson Lie groups, analogous to what we did in our earlier work on ﬁnite CMV matrices. (See [L1] and [KN].) Secondly , by using the group-theoretic fr amework, w e will study the defo cusin g AL equation with regard to action-a ngle v ariables. W e will also solv e the comm uting Hamiltonian ﬂ o ws via Riemann-Hilb ert fact orization problems. At this ju ncture, let us men tion some earlier w orks r elated to the in tegratio n of th e p erio dic d efo cusing AL equation whic h is part of our second goal here. T o start with , it has b een kno wn for quite some time that the defo cusing AL equation (1.1) can b e represented as a Lax sys tem on a lattice (or discr ete zero curv ature represen tation), wh ere the Lax op erator L j ( z ) asso ciated to site j of the lattice is given by (see, for example, [AL1], [AL2] and [FT]) L j ( z ) =  z ¯ α j α j z − 1  . (1 . 3) Therefore, in the p erio dic case with p erio d p, the mono dromy matrix M ( z ) = L p − 1 ( z ) · · · L 0 ( z ) undergo es an isosp ectral deformation which means that an equa- tion in Lax pair form (and d iﬀeren t fr om the one we are using here) is kn o w n for the p erio d ic defo cusing AL equation. In [MEKL], a trans f ormation of a natural gener- alizati on of (1.3 ) was disco v ered and the result was applied in the construction of ﬁnite gen u s solutions of a more general version of the AL equation. In particular, the authors in [MEKL] were able to write do wn the solution of the initial v alue problem for the p erio dic d efo cusing AL equation itself. On the other hand, from a diﬀerent d irection, the authors in [GHMT] considered a more general v ersion of the AL hierarc hy , and discussed the problem of solving the r -th AL ﬂo w when the initial data is the stationary solution of the p -th equ ation of the hierarc hy . As the reader will see in Section 6 b elo w, our approac h in solving the comm uting Hamil- tonian ﬂ o ws asso ciated with the p erio dic defo cusing AL equation is quite diﬀerent from those in these earlier works. The pap er is organized as follo ws. In Section 2, w e b egin by recalling the n otion of CMV m atrices, extended CMV matrices and Floqu et CMV matrices. Then we THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 3 sho w ho w to rewrite the Lax equations for E of the comm u ting ﬂ o ws asso ciated with the p erio dic defo cusing AL equ ation as isosp ectral deformations of E ( h ) . T o prepare for what w e need in subsequent sections, w e also discuss the structure of the p ow ers of E ( h ) . In Section 3, we ha v e t wo main goals. Th e ﬁr st goal is to sh o w that the set of p × p Floqu et CMV matrices is a sym p lectic leaf of a P oisson Lie group whose underlyin g group is a loop group e G R w . Indeed, as one w ould exp ect from results in [L1] and [KN] concerning the ﬁ nite dimensional case, the P oisson structure here is also a Skly anin structure. In fact, it is the Sklya nin structure asso ciated with the Iwasa w a decomposition of th e lo op group e G R w : e G R w = e K w e B w . (The decomp osition w as established in [GW].) Ho wev er, in order to wr ite do w n this Skly anin structure {· , ·} J ♯ , w e ﬁ nd it necessary to restrict ourselv es to a sub class of functions F ( e G R w ) of C ∞ ( e G R w ) . F ortunately , F ( e G R w ) forms an algebra of functions under ordinary multiplicatio n and is closed under {· , ·} J ♯ . Hence {· , ·} J ♯ d eﬁnes a P oisson brac k et on F ( e G R w ) . No w note that although w e are dealing w ith a restricted class of fu nctions here, it can b e c heck ed that the notion of P oisson Lie groups can b e extended to this in ﬁnite dimensional cont ext in a rigorous w a y . Moreo v er, we can c heck b y hand that the symplectic lea v es of ( e G R w , {· , ·} J ♯ ) are still g iv en by th e orbits of the dressing act ion. With this prep aration, th e tec hn iqu e in [L1] can b e naturally extended to sh o w that the set of p × p Floq u et CMV matrices is a dressin g orb it through a Co xeter elemen t x f of the aﬃne W eyl group W af f . In deed, the indu ced P oisson stru cture on e K w is a lo op group analog of th e Bruhat P oisson structur e in [L W] and [Soi] and we ca n sho w th at the set of p × p Flo quet CMV matrices is a pro duct of tw o dimensional orbits. In the rest of the section, our goal is to clarify the r elation b et wee n the AL brac k et and the Skly anin brac ket {· , ·} J ♯ , and to describ e the Hamilto nian equations generated b y the central functions on e G w , th u s connecting the group -theoretic f ramew ork with the equations in Section 2. In Section 4, we study the analytical p rop erties of the Blo c h solution of E u = z u, whic h pla y an imp ortan t role in subsequent sections. Since E deﬁnes a (p en tadiag- onal) p erio dic diﬀerence op erator, the semin al work of v an Mo erb ek e and Mumford [MM] comes to mind. How ev er, w e note that neither E nor its factors L and M in the th eta-facto rization of E satisfy the genericit y assumption in [MM]. So the analysis in this section is more delicate th an the standard case [MM], [AM]. In Section 5, we start with a simp le pro of of th e in volution theorem in (1.2), w hic h is p ossible b ecause of the group-theoretic s etup in Section 3. Then we pro ceed to con- struct the angle v ariables. T o compute the P oisson b rac kets b et ween the conserv ed quan tities and the v arious quan tities related to the pu tativ e angles, we make use of a device in tro duced in [DL T]. W e would like to p oin t out that in general, su c h computations could b e diﬃcult b ecause th ey ma y require detailed in f ormation on the asymptotics of the norm alized eig en vect ors in neigh b orho o ds of the p oin ts at inﬁnity of the Riemann surface. In our case, asymp totics b ey ond the leading ord er are diﬃcult to get b ecause w e are in a non-generic situation, bu t fortun ately w e are sa ve d by some sp ecial structure. Finally , in Section 6 w e solv e th e comm ut- ing Hamiltonian ﬂows via Riemann -Hilb ert factorizatio n p roblems, whic h again are suggested by the group -theoretic framewo rk. W e remark that it is in this v ery last 4 L.-C. LI, I. N E NCIU section that we ﬁnd it adv an tageous to think of ou r ﬂo ws on E ( h ) as ﬂo ws on the factors g e and g o ( h ) in the theta-factoriza tion of E ( h ) . This is p r ecisely the reason wh y we in tro duce Lax systems on a p erio d 2 lattice in Section 3. 2. Preliminaries. In this sect ion, for the con v enience of the reader, w e b egin with some backg round material on CMV matrices and the in v olution theorem of Nenciu-Simon. (Go o d references are [S2] and [S3].) Then we will sho w h o w to rewrite th e Lax equ ation in [N] for the p erio dic defo cusing AL equation (in terms of th e extended C MV matrix E ) as an isosp ectral deform ation of th e Floqu et CMV m atrix E ( h ). W e will also present a result on the structure of the p ow ers of E ( h ) which w e will use in Sections 5 and 6. The CMV matrices are the unitary analogs of Jac obi matrices [S3] and made their debut in th e numerica l linear a lgebra literat ure. (See [B-GE] and in particular [W].) Subsequ ently , they were redisco v ered b y Can tero, Moral and V al´ azquez [CMV] in the con text of the theory of orthogo nal p olynomials on the circle (OPUC). T o intro- duce th ese ob jects, let D = { z ∈ C | | z | < 1 } , and let dµ b e a nontrivial probabilit y measure on ∂ D , then one can pr o duce an orthonormal basis of L 2 ( ∂ D , dµ ) b y ap- plying the Gram-Schmidt pro cess to 1 , z , z − 1 , z 2 , z − 2 , · · · . As it turns out [C MV], the matrix repr esen tatio n of the op erator f ( z ) 7→ z f ( z ) in L 2 ( ∂ D , dµ ) with resp ect to this orthonormal b asis is the inﬁn ite CMV matrix C =      ¯ α 0 ρ 0 ¯ α 1 ρ 0 ρ 1 0 0 · · · ρ 0 − α 0 ¯ α 1 − α 0 ρ 1 0 0 · · · 0 ρ 1 ¯ α 2 − α 1 ¯ α 2 ρ 2 ¯ α 3 ρ 2 ρ 3 · · · 0 ρ 1 ρ 2 − α 1 ρ 2 − α 2 ¯ α 3 − α 2 ρ 3 · · · · · · · · · · · · · · · · · · · · ·      = e L f M (2 . 1) where α j ∈ D a re the so-calle d V erblu nsky co eﬃcien ts, ρ j = p 1 − | α j | 2 , j = 0 , 1 , · · · , and where e L = diag( θ 0 , θ 2 , · · · ) , f M = d iag(1 , θ 1 , θ 3 , · · · ) , with θ j =  ¯ α j ρ j ρ j − α j  , j = 0 , 1 , · · · . (2 . 2) The factoriz ation on the r ight h and s id e of (2.1) is calle d the theta -facto rization and lends itself to generalization. Indeed, if we no w ha v e a t wo-sided sequence { α j } ∞ j = − ∞ with α j ∈ D for all j, then we can deﬁne the extended (t wo-sided) CMV m atrix E by extending e L and f M to doub ly-inﬁnite m atrices in the obvious w a y . (Of course, the 1 × 1 blo c k will not app ear in this extension.) In this work, w e are mainly inte rested in the case where the sequence { α j } ∞ j =0 of V erb lunsky co eﬃcien ts is p erio dic of p erio d p and in th is con text, it is con v enien t to extend the one-sided sequence to a tw o-sided seqence { α j } ∞ j = − ∞ satisfying the p erio dicit y condition α j + p = α j for all j ∈ Z . T h us corresp ondingly , w e h a ve an extended CMV matrix with p erio dic V erblun ksky coeﬃcients and such matrices ha v e b een THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 5 used to formulate the Lax equation for the p erio dic d efo cusing AL equation in [N]. No w supp ose E is an extend ed CMV matrix with p erio dic V erblun sky co eﬃcien ts with p erio d p. Without loss of generalit y , we will assume from no w on wards that p is ev en (otherwise, we just r eplace p b y 2 p ). Note that if T is th e op erator on l ∞ ( Z ) deﬁned by ( T u ) j = u j + p , then E T = T E . Therefore, if for h ∈ ∂ D , w e deﬁn e X ( h ) = { u ∈ l ∞ ( Z ) | T u = h − 1 u } , (2 . 3) then the ﬁ n ite dimensional space X ( h ) is inv arian t un der E . A basis of X ( h ) is giv en b y the v ectors δ k = ∞ X j = − ∞ h − j e j p + k , k = 0 , · · · , p − 1 (2 . 4) where e j is the v ector in l ∞ ( Z ) with j -th comp onen t equal to 1 and zeros elsewhere. By deﬁnition, the Flo quet CMV matrix E ( h ) is the matrix of E | X ( h ) with resp ect to the ordered basis ( δ 0 , · · · , δ p − 1 ) , i,e. E δ k = p − 1 X j =0 ( E ( h )) j k δ j , k = 0 , · · · , p − 1 . (2 . 5) F om this, it is clear that the matrix of E n | X ( h ) with resp ect to the same ordered basis is E ( h ) n . T h us the en tries of E ( h ) n are r elated to those of E n b y the form u la ( E ( h ) n ) j k = X l ∈ Z h − l ( E n ) j,k + lp (2 . 6) for 0 ≤ j, k ≤ p − 1 . Finally , the Flo quet CMV matrix E ( h ) also has a theta- factorizat ion E ( h ) = g e g o ( h ), w here g e = diag( θ 0 , θ 2 , · · · , θ p − 2 ) , (2 . 7) and g o ( h ) =       − α p − 1 0 · · · 0 ρ p − 1 h 0 θ 1 0 . . . . . . . . . 0 θ p − 3 0 ρ p − 1 h − 1 0 · · · 0 ¯ α p − 1       . (2 . 8) This is of course a consequence of the factorizatio n for the corresp onding extended CMV matrix E . In case we wan t to emphasize th e d ep end en ce of g e and g o on α = ( α o , · · · , α p − 1 ) ∈ D p , we also w rite g e = g e ( α ) and g o = g o ( α ) . Another v ery imp ortan t n otion asso ciated with p erio dic V erblu n sky co eﬃcients is that of the discriminan t introd u ced in [S2]: ∆( z ) = z − p/ 2 tr ( T p ( z )) , z ∈ C \ { 0 } (2 . 9) 6 L.-C. LI, I. N E NCIU where T p ( z ) = 1 Q p − 1 j =0 ρ j  z − ¯ α p − 1 − α p − 1 z 1  · · ·  z − ¯ α 0 − α 0 z 1  (2 . 10) is the trans fer matrix. In [S2], b y seeking a Poisson br ac ket on D p so that the mo dulu s P = Q p − 1 j =0 ρ j generates th e Aleksand r o v ﬂow, th e author arriv es at the Ablo witz-Ladik brack et (see [KM] for more general ve rsions of this structure) { f 1 , f 2 } AL = 2 i p − 1 X j =0 ρ 2 j  ∂ f 1 ∂ α j ∂ f 2 ∂ ¯ α j − ∂ f 1 ∂ ¯ α j ∂ f 2 ∂ α j  . (2 . 11) W e recall the in v olution theorem of Nenciu-Simon whic h was obtained b y calculat- ing w ith W all p olynomials. Theorem 2.1 [N] , [S2] . F or al l z , w ∈ C \ { 0 } , { ∆( z ) , ∆( w ) } AL = 0 , { P , ∆( z ) } AL = 0 . (2 . 12) Henc e if P · ∆( z ) = P p/ 2 j = − p/ 2 I j z j , the f u nctions P , I 0 , R e I j , Im I j , j = 1 , · · · , p/ 2 − 1 Poisson c ommute with e ach other. This result, when com b in ed with the pro of that the functions P , I 0 , Re I j , Im I j , j = 1 , · · · , p/ 2 − 1 are f unctionally indep endent on an op en dense subset of D p [S2], sho w s that an y of the f unctions in the ab o v e list generates a completely in tegrable Hamiltonian sys tem. T o write d own the Lax pairs, the author in [N] actually considered a diﬀerent , but equiv alent set of comm uting Hamiltonians, whic h are constructed from the r eal and imaginary parts of K n = 1 n p − 1 X k =0 ( E n ) k k , 1 ≤ n ≤ p/ 2 − 1 , together w ith K p/ 2 and P . Indeed, an easy computation shows that { Re( K 1 ) , α j } AL = iρ 2 j ( α j − 1 + α j +1 ) , { log( P ) , α j } AL = iα j (2 . 13) for all 0 ≤ j ≤ p − 1 . Hence the p erio dic defo cusing AL equation is generated b y the Hamiltonian Re( K 1 ) − 2 log ( P ) . T o relat e the K n ’s to the Flo quet CMV matrix, and to the coeﬃcien ts of P · ∆( z ) , ﬁrst r ecall that [S2] det( z I − E ( h )) =   p − 1 Y j =0 ρ j   z p 2 [∆( z ) − ( h + h − 1 )] . (2 . 14) In view of (2.13), it is clear w e m ust consider the stru cture of the p ow ers of E . W e will skip th e pro of of the follo wing r esult whic h can b e established by in duction on n. THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 7 Lemma 2.2. F or n ≥ 1 , ( E n ) j,k is identic al ly zer o if one of the fol lowing holds: (a) | j − k | ≥ 2 n + 1 , or (b) j − k = 2 n, wher e j and k ar e b oth even, or (c) j − k = − 2 n wher e j and k ar e b oth o dd. As a consequence of this result, n ote that for n ≤ p 2 − 1 and for 0 ≤ k ≤ p − 1 , w e hav e ( E ( h ) n ) k k = ( E n δ k ) k = X q ∈ Z h − q ( E n ) k ,q p + k = ( E n ) k k (2 . 15) since | k − ( pq + k ) | ≥ p ≥ 2 n + 2 f or all q 6 = 0 . Hence it follo ws that (cf. (5.4)) K n = 1 n I | h | =1 tr( E ( h ) n ) dh 2 π ih , n = 1 , · · · , p/ 2 − 1 . (2 . 16) F r om the relation in (2.14), we see that the in tegral in (2.16) for n = p/ 2 are also relev an t, as this is related to I 0 and P . But b y an indu ction argument sim ilar to the p ro of of Lemma 2.2 ab o v e, we can sho w that ( E m ) k ,k +2 m = k +2 m − 1 Y j = k ρ j if k is ev en (2 . 17) and ( E m ) k ,k − 2 m = k − 1 Y j = k − 2 m ρ j if k is o dd . (2 . 18) Since P = Q k + p − 1 j = k ρ j for any k ∈ Z , it follo ws th at ( E ( h ) p/ 2 ) k k =  ( E p/ 2 ) k k + h − 1 · P if k is ev en ( E p/ 2 ) k k + h · P if k is o dd (2 . 19) and therefore tr( E ( h ) p/ 2 ) = tr( E p/ 2 ) + p 2 ( h + h − 1 ) P . (2 . 20) Th us it follo ws from (2.20 ) and (2.15) that K p/ 2 = 2 p I | h | =1 tr( E ( h ) p/ 2 ) dh 2 π ih (2 . 21) and so the set of Hamiltonians in Theorem 2.1 is equiv alen t to P , Re K j , Im K j , j = 1 , · · · , p/ 2 − 1 , K p/ 2 . The next result giv es the Lax equations of the Hamiltonian systems generated b y the ab o ve set of fu nctions an d is the cent ral result of [N]. W e will use the follo win g notation: for an inﬁn ite tw o-sided m atrix A , Π u ( A ) = 1 2 A 0 + A + , where A + is the upp er triangular part of A and A 0 is th e diagonal part. 8 L.-C. LI, I. N E NCIU Theorem 2.3. F or 1 ≤ n ≤ p / 2 , (a) the Hamiltonian e quation gener ate d by R e K n c an b e expr esse d as ˙ E = [ E , i Π u ( E n ) + i ((Π u ( E n )) ∗ ] , (2 . 22) (b) the H amiltonian e quation gener ate d by Im K n c an b e expr esse d as ˙ E = [ E , Π u ( E n ) − ((Π u ( E n )) ∗ ] . (2 . 23) Our next goal is to compu te the evo lution of E ( h ) un der (2.22) an d (2.23). In order to do this, w e hav e to establish the follo wing r esult. Prop osition 2.4. L et n ≤ p / 2 − 1 , then the structur e of E ( h ) n as a L aur ent p olynom ial in h is given by E ( h ) n = A 0 ( n ) + h A 1 ( n ) + h − 1 A − 1 ( n ) , (2 . 24) wher e A 1 ( n ) is strictly upp e r triangular and A − 1 ( n ) is strictly lower triangular. Pr o of. In order to pro ve (2.24), let 0 ≤ j, k ≤ p − 1 b e tw o indices. Then as in (2.15) , we ha ve  E ( h ) n  j k = X q ∈ Z h − q  E n  j,k + pq . (2 . 25) No w, note th at for | q | ≥ 2 and any j, k , we hav e the inequalit y   j − ( k + pq )   ≥ 2 p − | j − k | ≥ 2 p − ( p − 1) = p + 1 ≥ 2 n + 1 and therefore  E n  j,k + pq ≡ 0. Hence form ula (2.24) h olds for some matrices A ± 1 ( n ). T o ﬁnd the structur e of the matrices A ± 1 ( n ), note that for j ≥ k , w e ha ve   j − ( k − p )   = p + ( j − k ) ≥ p ≥ 2 n + 1 and so it follo ws from Lemma 2.2 that  E n  j,k − p ≡ 0. Consequent ly ,  E ( h ) n  j k =  E n  j k + h − 1  E n  j,k + p for j ≥ k . (2 . 26) A similar argumen t shows that  E ( h ) n  j k =  E n  j k + h  E n  j,k − p for j ≤ k . (2 . 27) These last t wo equations then establish our claim ab out the triangularit y of A ± 1 ( n ). In fact, w e ﬁnd that  A 0 ( n )  j k =  E n  j k for all 0 ≤ j, k ≤ p − 1 , ( 2 . 28)  A 1 ( n )  j k =   E n  j,k − p if j < k ; 0 if j ≥ k , (2 . 29) and  A − 1 ( n )  j k =   E n  j,k + p if j > k ; 0 if j ≤ k . (2 . 30) This completes the p ro of.  W e are now ready to giv e the r esu lt alluded to ab o ve whic h is th e p oint of departure in this w ork. W e will make use of the pro j ections Π k and Π e k w in tro duced in Section 3 b elo w. (See the second paragraph of Section 3 and (3.12).) THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 9 Prop osition 2.5. (a) If E e v olves ac c or ding to (2.22), then ˙ E ( h ) = [ E ( h ) , Π e k w ( i E ( h ) n ) ] . (2 . 31) (b) If E evolves ac c or ding to (2.23), then ˙ E ( h ) = [ E ( h ) , Π e k w ( E ( h ) n ) ] . (2 . 32) Pr o of. Let Q n ( h ) b e the matrix of Π u ( E n ) | X ( h ) with resp ect to the ord er ed basis ( δ 0 , · · · , δ p − 1 ) . T o establish (2.31 ) and (2.32), it su ﬃces to pro ve that Q n ( h ) = 1 2 ( A 0 ( n )) 0 + ( A 0 ( n )) + + h − 1 A − 1 ( n ) , (2 . 33) where A 0 ( n ) and A − 1 ( n ) are the matrices in (2.24), and where ( A 0 ( n )) + and ( A 0 ( n )) 0 are resp ectiv ely the u pp er triangular part and diagonal part of A 0 ( n ) . But by a direct calculatio n and making use of Lemma 2.2, we ﬁnd ( E n ) + δ k = h − 1 p − 1 X j =0 ( E n ) j − p,k δ j + k − 1 X j =0 ( E n ) j k δ j . (2 . 34) Hence from (2.28) and (2.30), we conclude that the matrix of ( E n ) + | X ( h ) with resp ect to the ordered basis ( δ 0 , · · · , δ p − 1 ) is give n by h − 1 A − 1 ( n ) + A 0 ( n ) + . On the other hand, it is easy to see th at th e matrix of ( E n ) 0 | X ( h ) with r esp ect to the same b asis is ( A 0 ( n )) 0 . Hence (2.3 3) follo ws.  W e close this section with an imp ortan t remark ab out the p erio dic defo cusing AL equation itself. Namely , if ˙ α j = iρ 2 j ( α j − 1 + α j +1 ) , or equ iv alen tly , ˙ E ( h ) = [ E ( h ) , Π e k w ( i E ( h )) ] , then β j ( t ) = e − 2 it α j ( t ) satisﬁes the p erio dic d efo cusing AL equation and vice versa. Th u s in ord er to solv e the p erio dic defo cu sing AL equa- tion, it suﬃces to solv e ˙ E ( h ) = [ E ( h ) , Π e k w ( i E ( h )) ] , whic h is generated b y the Hamil - tonian Re K 1 = − Re I p/ 2 − 1 ( E ( h )) . (Comp are (2.16) and (5.4). ) W e w ill solv e th e Hamiltonian equations generated by the conserv ed quan tities in T heorem 2.1 in Section 6 b elo w. 3. Flo quet CMV matrices, dressing orbits and Hamilt onian ﬂo w s. The ﬁrst goal of this section is to sho w th at the set of Floqu et CMV matrices is a symplectic leaf of a Po isson Lie group whose un d erlying group is a lo op group. Indeed, by follo win g the metho d of inv estigation in [L1], we will show that there exist symplectic lea ve s of a more elemen tary nature in terms of whic h w e can describ e the collectio n of Flo quet CMV matrices. As explained in Section 2 ab o v e, we can assume that p is ev en. Le t G R b e GL ( p, C ) considered as a real Lie grou p , and let K and B b e resp ective ly the unitary 10 L.-C. LI, I. N E NCIU group U ( p ) and the lo wer triangular group of p × p matrices with p ositiv e d iagonal en tr ies. It is w ell-kno wn that G R admits the Iwa sa wa decomp osition G R = K B . On the Lie algebra lev el, th is corresp onds to g R = k ⊕ b (with asso ciated pr o j ections Π k , Π b ), where g R , k and b are, resp ectiv ely , the Lie algebras of G R , K and B . Later on in the sectio n, w e will also need the maximal torus T of K, consisting of unitary d iagonal matrices. Let ˜ G R = C ∞ ( S 1 , G R ) b e the smo oth lo op group w ith the C ∞ top ology . ˜ G R is a F r¨ ec h et Lie group with the Lie algebra ˜ g R = C ∞ ( S 1 , g R ) . W e w ill use the follo wing nondegenerate ad-inv ariant pairin g on ˜ g R : ( X, Y ) = I m I | h | =1 tr( X ( h ) Y ( h )) dh 2 π ih . (3 . 1) As the reader will see, this c hoice is critical for what we ha v e in m in d. F ollo wing [GW], choose a symmetric w eight function w : Z − → R + , wh ic h is rapidly in creasing in the sense that lim n →∞ w ( n ) n − s = ∞ , ∀ s > 0 . ( 3 . 2) Also, assume that w is of non-analytic type: lim n →∞ w ( n ) 1 /n = 1 . (3 . 3) F or X ∈ ˜ g R giv en by X ( h ) = P ∞ j = − ∞ X j h j , we deﬁn e k X k w = ∞ X j = − ∞ k X j k w ( j ) , (3 . 4) where k · k is a norm on g R . Also, set ( P + X )( h ) = X j > 0 X j h j , ( P − X )( h ) = X j < 0 X j h j , ( P 0 X )( h ) = X 0 . (3 . 5 ) Consider the Banac h Lie group ˜ G R w = { g ∈ ˜ G R | k g k w < ∞ } (3 . 6) with Lie algebra ˜ g R w = { X ∈ ˜ g R | k X k w < ∞ } . (3 . 7) F r om [GW], we ha ve the Iw asa wa decomp osition for the lo op group ˜ G R w and its Lie algebra ˜ G R w = ˜ K w · ˜ B w , ˜ g R w = ˜ k w ⊕ ˜ b w (3 . 8) where ˜ K w = { g ∈ ˜ G R w | g ∗ g = I } , ˜ B w = { g ∈ ˜ G R w | P − g = 0 , P 0 g ∈ B } (3 . 9) THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 11 are Banac h Lie subgroups of ˜ G R w and ˜ k w , ˜ b w are their resp ectiv e Lie algebras. Denote b y k : ˜ G R w − → ˜ K w , b : ˜ G R w − → ˜ B w the analytic m ap s deﬁned by the factoriza tion g = k ( g ) b ( g ) − 1 , g ∈ ˜ G R w . Also, d en ote by Π ˜ k w and Π ˜ b w the pro jection maps relativ e to the sp litting ˜ g R w = ˜ k w ⊕ ˜ b w . Then from stand ard classical r-matrix theory [S TS1], [STS2], J ♯ = Π ˜ k w − Π ˜ b w (3 . 10) is a solution of the mo diﬁed Y ang-Baxter equ ation (mYBE). Hence w e can equip ˜ g R w with the J ♯ -brac ket [ X, Y ] J ♯ = 1 2 ([ J ♯ X, Y ] + [ X , J ♯ Y ]) . (3 . 11) W e will d enote the vec tor space ˜ g R w equipp ed with the Lie brack et [ · , · ] J ♯ b y ( ˜ g R w ) J ♯ . In fact, it is easy to chec k fr om (3.11) that ( ˜ g R w ) J ♯ = ˜ k w ⊖ ˜ b w (Lie algebra antidirect sum). Note that explicitly , th e pro jection maps Π ˜ k w , Π ˜ b w are giv en by the f orm ulas Π ˜ k w X = P − X + Π k X 0 − ( P − X ) ∗ Π ˜ b w X = P + X + Π b X 0 + ( P − X ) ∗ , (3 . 12) where P ± are d eﬁned in (3.5) . In order to introd uce the Poisson stru cture on ˜ G R w , it is necessary to restrict ourselv es to a sub class of fun ctions in C ∞ ( ˜ G R w ) . W e sa y that a function ϕ ∈ C ∞ ( ˜ G R w ) is smo oth at g ∈ ˜ G R w iﬀ there exists D ϕ ( g ) ∈ ˜ g R w (called the right gradient of ϕ at g ) suc h that d dt    t =0 ϕ ( e tX g ) = ( D ϕ ( g ) , X ) , X ∈ ˜ g R w (3 . 13) where ( · , · ) is the pairing in (3.1). If ϕ ∈ C ∞ ( ˜ G R w ) is smo oth at g for all g ∈ ˜ G R w , then we sa y it is sm o oth on ˜ G R w . Note that the nondegeneracy of ( · , · ) implies that the m ap i : ˜ g R w − → ( ˜ g R w ) ∗ , X 7→ ( X , · ) (3 . 14) is an isomorph ism on to a subs p ace of ( ˜ g R w ) ∗ whic h we will call the smo oth p art of ( ˜ g R w ) ∗ . Th us ϕ ∈ C ∞ ( ˜ G R w ) is smo oth at g iﬀ T ∗ e r g dϕ ( g ) is in the smo oth part of ( ˜ g R w ) ∗ and we can deﬁne the left gradient of su c h a f u nction at g by d dt    t =0 ϕ ( ge tX ) = ( D ′ ϕ ( g ) , X ) , X ∈ ˜ g R w . (3 . 15) F or eac h g ∈ ˜ G R w , we will denote the collection of all smo oth functions at g by F g ( ˜ G R w ) and we set F ( ˜ G R w ) = ∩ g ∈ ˜ G R w F g ( ˜ G R w ) . With the ab ov e considerations, it is easy to chec k th at F ( ˜ G R w ) is non-empty and forms an algebra u nder ordinary m u ltiplication of fun ctions. 12 L.-C. LI, I. N E NCIU Prop osition 3.1. (a) F or ϕ, ψ ∈ F ( ˜ G R w ) and g ∈ ˜ G R w , deﬁne { ϕ, ψ } J ♯ ( g ) = 1 2 ( J ♯ ( D ′ ϕ ( g )) , D ′ ψ ( g )) − 1 2 ( J ♯ ( D ϕ ( g )) , D ψ ( g )) . (3 . 16) Then { ϕ, ψ } J ♯ ∈ F ( ˜ G R w ) and henc e {· , ·} J ♯ deﬁnes a P oisson br acket on F ( ˜ G R w ) . (b) The H amiltonian e quation of motion gener ate d by ϕ ∈ F ( ˜ G R w ) is g iven by ˙ g = g (Π ˜ k w ( D ′ ϕ ( g ))) − (Π ˜ k w ( D ϕ ( g ))) g = (Π ˜ b w ( D ϕ ( g ))) g − g (Π ˜ b w ( D ′ ϕ ( g ))) . (3 . 17) Pr o of. (a) A straight forward calculation sh o ws that f or ϕ, ψ ∈ F ( ˜ G R w ) , D { ϕ, ψ } J ♯ ( g ) exists for eac h g ∈ ˜ G R w and is giv en by D { ϕ, ψ } J ♯ ( g ) = Ad g [ D ′ ϕ ( g ) , D ′ ψ ( g )] J ♯ + Ad g d ds | s =0 D ′ ϕ ( e sη ( g ) Dϕ ( g ) g ) − Ad g d ds | s =0 D ′ ψ ( e sη ( g ) Dϕ ( g ) g ) , (3 . 18) where η ( g ) = 1 2 Ad g ◦ J ♯ ◦ Ad g − 1 − 1 2 J ♯ . (3 . 19) This sho w s { ϕ, ψ } J ♯ ∈ F ( ˜ G R w ) . T o pr o ve the second half of (a), ﬁr st note that T r( X ( h ) Y ( h )) ∈ R for X , Y ∈ ˜ k w . F rom this, it follo ws that ( X, Y ) = 0 . Conse- quen tly , ˜ k w is an isotropic s u balgebra of ˜ g R w relativ e to the pairing ( · , · ) . O n the other h and, if X , Y ∈ ˜ b w , we ha ve I | h | =1 tr( X ( h ) Y ( h )) dh 2 π ih = tr( X 0 Y 0 ) ∈ R b ecause X 0 , Y 0 are lo w er triangular with real diagonal en tries. So ˜ b w is also an isotropic subalgebra of ˜ g R w . Com b ining these tw o facts, we can no w conclud e that J ♯ is skew-symmetric relativ e to ( · , · ) . Finally , the Jacobi iden tit y and the deriv a- tion prop erty no w follo w fr om standard calculatio ns in [STS2] wh ic h work w ithout c hange in our inﬁnite dimensional cont ext. (b) T his deriv ation of the Lax equation from the P oisson structure is standard .  Note that although w e are dealing with a restricted class of functions, the notion of Poisson s ubmanifolds can b e deﬁned analogously to the standard case. Now it is easy to c hec k that if ϕ ∈ F ( ˜ G R w ) , then so are ϕ ◦ r g and ϕ ◦ l g for all g ∈ ˜ G R w , where r g and l g denote right and left tr anslation by g , resp ectiv ely . Hence the notion of Poi sson Lie group can b e extended to this inﬁnite dimensional conte xt and ( ˜ G R w , {· , ·} J ♯ ) is a cob oundary P oisson Lie group . On the inﬁnitesimal lev el, w e will call ( ˜ g R w , ( ˜ g R w ) J ♯ ) the tangen t Lie bialgebra of ( ˜ G R w , {· , ·} J ♯ ) as the map ρ : ˜ g R w − → L ( ˜ g R w , ˜ g R w ) ( L ( ˜ g R w , ˜ g R w ) is th e space of linear maps on ˜ g R w ) giv en by ρ ( X ) = 1 2 ( ad X ◦ J ♯ − J ♯ ◦ ad X ) and satisfying the relation ([ Y , Z ] J ♯ , X ) = ( Z, ρ ( X )( Y )) is a 1-coboun dary (and h ence a 1-co cycle) with resp ect to the adj oint repr esen tation. Th us in sp eaking of a Lie bialgebra h er e, the u n derlying ve ctor spaces of the p air of Lie algebras inv olv ed are only required to b e in dualit y with resp ect to ( · , · ) and this is what we will con tinue to d o. THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 13 Corollary 3.2. (a) ˜ K w is a P oisson Lie sub gr oup of ( ˜ G R w , {· , ·} J ♯ ) in the sense that ˜ K w is a Lie sub gr oup of ˜ G R w which is also a Poisson submanifold of ( ˜ G R w , {· , ·} J ♯ ) . Mor e over, the tangent Li e bialgebr a ( ˜ k w , ( ˜ g R w ) J ♯ / ˜ k ⊥ w ) of ˜ K w (wher e ˜ k ⊥ w is deﬁne d r elative to ( · , · ) ) is isomorphic to ( ˜ k w , ˜ b − w ) wher e ˜ b − w is ˜ b w but e quipp e d with the − br acket. (b) The underlying gr oup of the Poisson gr oup ( ˜ G R w ) J ♯ dual to ( ˜ G R w , {· , ·} J ♯ ) c onsists of ˜ G R w e quipp e d with the multiplic ation g ∗ h ≡ k ( g ) h b ( g ) − 1 . (3 . 20) Pr o of. (a) T o sho w that ˜ K w is a Poisson Lie subgroup , it is enough to chec k that ˜ K w is a P oisson su b manifold of ( ˜ G R w , {· , ·} J ♯ ) and this can b e done b y using the expression for the Hamiltonian vect or ﬁeld in P r op osition 3.1(b). T o sh ow that the tangen t Lie b ialgebra ( ˜ k w , ( ˜ g R w ) J ♯ / ˜ k ⊥ w ) of ˜ K w is isomorphic to ( ˜ k w , ˜ b − w ) , ﬁrst n ote that ˜ k ⊥ w = ˜ k w . Hence we hav e X + ˜ k ⊥ w = Π ˜ b w X + ˜ k w for all X ∈ ( g R w ) J ♯ . Therefore, the in duced Lie brac ket on ( ˜ g R w ) J ♯ / ˜ k ⊥ w is given by [ X + ˜ k w , Y + ˜ k w ] = [Π ˜ b w X, Π ˜ b w Y ] J ♯ + ˜ k w = − [Π ˜ b w X, Π ˜ b w Y ] + ˜ k w . Conseqen tly , the m ap X + ˜ k w 7→ Π ˜ b w X is an isomorphism, when ˜ b w is equipp ed with the − b rac ket. (b) Th e formula is a consequence of the fact that ˜ g R w = ˜ k w ⊖ ˜ b w and can b e veriﬁed easily .  R emark 3.3 . In view of the second h alf of Corollary 3.2 (a), the induced stru cture on ˜ K w is the lo op group an alog of the Bruhat Po isson s tructure in [L W] and [Soi]. W e next turn to the description of the sy m plectic lea v es of the Skly anin structure in (3.16). Unfortunately , w e cannot assume the general resu lts in [STS2] and [L W] apply to our case without some veriﬁcati on, b ecause the analysis in th ese w orks is for ﬁn ite dimensional Po isson Lie groups. In this regard, let us also remark that as far as we know, the integ rabilit y of the c haracteristic distr ib ution of a Po isson structure on an inﬁnite d im en sional manifold is by no means automatic b ecause an analog of the Stefan-Sussm an n result [St, S u] is not a v ailable. In th e follo win g, w e will chec k things by hand. So let us d eﬁ ne S x = { X ϕ ( x ) | ϕ ∈ F x ( ˜ G R w ) } (3 . 21) for eac h x ∈ ˜ G R w . Then the c h aracteristic distribution of the P oisson brac k et {· , ·} J ♯ is give n by S = ∪ x ∈ ˜ G R w S x . On the other hand, if ( ˜ G R w ) o J ♯ denotes the identit y comp onent of ( ˜ G R w ) J ♯ , the r igh t dr essin g action [S T S2] of ( ˜ G R w ) o J ♯ on ˜ G R w is deﬁn ed b y the form u la Φ g ( x ) = k ( g ) − 1 x k ( x − 1 g x ) = b ( g ) − 1 x b ( x − 1 g x ) (3 . 22) 14 L.-C. LI, I. N E NCIU and the inﬁ nitesimal generator of this action corresp ondin g to ξ ∈ ˜ g R w is the ve ctor ﬁeld ξ ˜ G R w on ˜ G R w , w here ξ ˜ G R w ( x ) = 1 2 xJ ♯ ( x − 1 ξ x ) − 1 2 J ♯ ( ξ ) x. (3 . 23) F or eac h x ∈ ˜ G R w , let F x b e the subspace of T x ˜ G R w spanned b y the ve ctors ξ ˜ G R w ( x ) . Then F = ∪ x ∈ ˜ G R w F x is an in tegrable generalize d distribution w h ose lea v es are the orbits of the dr essin g action Φ . Prop osition 3.4. F or e ach x ∈ ˜ G R w , T x O = S x , wher e O is the orbit of the dr essing action c ontaining x. Henc e the char acteristic distribution S is inte gr able and the le aves of this distribution ar e gi v en by the orbits of the dr essing action Φ . Pr o of. F rom (3.7), X ϕ ( x ) = 1 2 xJ ♯ ( x − 1 D ϕ ( x ) x ) − 1 2 J ♯ ( D ϕ ( x )) x. (3 . 24) T o sho w that S x = F x , it suﬃ ces to sho w that for eac h ξ ∈ ˜ g R w , there exists ϕ ∈ F x ( ˜ g R w ) suc h that D ϕ ( x ) = ξ . T o d o so, w e use the fact that the e xp onentia l map exp : ˜ g R w − → ˜ G R w is a diﬀeomorphism of a n eigh b orho o d U of 0 ont o a neigh b orho o d V of the iden tity element of ˜ G R w [GW]. Clearly , V x is a n eigh b orho o d of x and in this neighborh o o d, deﬁne ¯ ϕ ξ ( g ) = ( ξ , log( g x − 1 )) and extend this to a function ϕ ξ ∈ C ∞ ( ˜ G R w ) . Then ϕ ξ ∈ F x ( ˜ G R w ) w ith D ϕ ξ ( x ) = ξ . This completes the pro of.  Let O b e a dressing orbit, as in the pr op osition ab o v e. W e next sh o w that O is a symplectic leaf of ( ˜ G R w , {· , ·} J ♯ ) , i.e., there exists a weak (resp. s trong) symplectic form (see, for example, [OR] for s u c h matters) ω O on O consistent with the P oisson b rac ket {· , ·} J ♯ in the case w hen O is inﬁn ite (resp. ﬁ nite) di- mensional. F or this pu rp ose, let ( ˜ g R w ) ∗ S denote the smo oth part of ( ˜ g R w ) ∗ and let ( T ∗ ˜ G R w ) S = ∪ g ∈ ˜ G R w T ∗ g r g − 1 ( ˜ g R w ) ∗ S . Also, denote b y π ♯ : ( T ∗ ˜ G R w ) S − → T ˜ G R w the bu n - dle map corresp onding to {· , ·} J ♯ and let j b e the left in v erse of i : ˜ g R w − → ( ˜ g R w ) ∗ . F or eac h x ∈ O , we deﬁne a skew-symmetric bilinear form ω x on S x b y the formula ω x ( π ♯ ( x )( α ) , π ♯ ( x )( β )) = ( T x r x − 1 π ♯ ( x )( α ) , j ( T ∗ e r x β )) = − ( j ( T ∗ e r x α ) , T x r x − 1 π ♯ ( x )( β )) . (3 . 25) Clearly , the v alue of the ab o ve exp r ession dep ends only on th e v alues of π ♯ ( x )( α ) and π ♯ ( x )( β ) . Thus ω x is a w ell-deﬁned skew-symmetric bilinear form on S x . No w supp ose ω x ( π ♯ ( x )( α ) , π ♯ ( x )( β )) = 0 for all π ♯ ( x )( β ) . Then from (3.25) and the non- degeneracy of ( · , · ), it foll o ws that j ( T ∗ e r x α ) = 0 wh ic h in term implies π ♯ ( x )( α ) = 0 . So this establishes the nondegeneracy of ω x . T h us there exists a 2-form ω O on O suc h th at ω O ( x ) = ω x for eac h x ∈ O . No w the argument that ω O is d iﬀeren tiable and closed follo ws as in the ﬁ nite dimensional case in [Ko]. Cons eqently , ω O deﬁnes a Poisson structure {· , ·} O on O and we ha ve { ϕ, ψ } J ♯ | O = { ϕ | O , ψ | O } O (3 . 26) for all ϕ, ψ ∈ F ( ˜ G R w ) . Hence w e ha v e established the follo wing result. THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 15 Prop osition 3.5. The symple ctic le aves of ( ˜ G R w , {· , ·} J ♯ ) ar e g iven by the orbits of the dr essing action Φ . In our n ext step, w e will make the connection b et ween certain s ymplectic lea v es of ( ˜ G R w , {· , ·} J ♯ ) (these are also those of the P oisson Lie subgroup ˜ K w ) and the Flo q u et C MV matrices. W e b egin with some notations. As in [L1], we w ill d enote b y g e an y p × p blo c k diagonal matrices with 2 × 2 diagonal b lo c ks of the form  ¯ α ρ ρ − α  , with α ∈ D and ρ = p 1 − | α | 2 . (3 . 27) W e will den ote th e collection of such matrices b y T e . O n the other hand, we w ill denote by g o ( h ) an y loops in ˜ K w of th e form       ¯ α 0 · · · 0 ρh 0 0 . . . ˜ b . . . 0 0 ρh − 1 0 · · · 0 − α       , (3 . 28) where α ∈ D , ρ = p 1 − | α | 2 and ˜ b is a ( p − 2) × ( p − 2) blo ck diagonal matrix with 2 × 2 b lo c ks of the same kind as in g e . W e will denote the collection of suc h unitary lo ops by T 0 . Clearly , for giv en g e ∈ T e and g o ∈ T o , the pro duct g e g o ( h ) is a Floqu et CMV matrix. Indeed, the map m | T e × T o : T e × T o − → { p × p Flo q u et C MV matrices } ( g e , g o ) 7→ g e g o (3 . 29) is a diﬀeomorph ism, wh ere m : ˜ K w × ˜ K w − → ˜ K w is the multiplicati on map of the Po isson Lie sub group ˜ K w . Finally , we w ill denote the d r essing orbit through x ∈ ˜ G R w b y O x . In analogy to formula (2.21) in [L1], w e in tro d uce the follo w ing sp ecial Flo q u et CMV matrix x f ( h ) = x e f x o f ( h ) (3 . 30) corresp ondin g to α = (0 , 0 , · · · , 0) . (3 . 31) In other w ord s, x e f = diag( w ∗ , w ∗ , · · · ) (3 . 32) and x o f ( h ) =       0 0 . . . 0 h 0 w ∗ . . . 0 0 . . . . . . . . . . . . . . . 0 0 . . . w ∗ 0 h − 1 0 . . . 0 0       (3 . 33) 16 L.-C. LI, I. N E NCIU where w ∗ =  0 1 1 0  . (3 . 34) These matrices are elemen ts of the aﬃne W eyl group W af f = W ⋉ ˇ T [PS ], w here W = N ( T ) /T is th e W eyl group of K, and ˇ T is the lattice of h omomorphisms S 1 − → T . Indeed, if h is the C artan sub algebra of g consisting of d iagonal matrices, λ i − λ j , i 6 = j are the r o ots corresp onding to the pair ( g , h ), then in terms of the simple ro ots α i = λ i − λ i +1 , i = 1 , · · · , p − 1 and th e highest ro ot θ = λ 1 − λ p , we ha ve x e f = w α 1 w α 3 · · · w α p − 1 , (3 . 35) and x o f ( h ) = w α 2 w α 4 · · · w α p − 2 exp( h ( E 00 − E p − 1 ,p − 1 )) w θ . (3 . 36) Here for eac h j = 1 , · · · , p − 1 , w α j = diag( I j − 1 , w ∗ , I p − j − 1 ) is the elemen t in W whic h corresp onds to the simple reﬂection s α j , while w θ =       0 0 · · · 0 1 0 0 . . . I p − 2 . . . 0 0 1 0 · · · 0 0       (3 . 37) is the elemen t in W wh ic h corresp onds to the reﬂection s θ . Finally , the elemen t exp( h ( E 00 − E p − 1 ,p − 1 )) is in ˇ T , w here E j j denote the diagonal matrix with a 1 in the ( j, j ) p osition an d zeros elsewhere. But no w recall th at there is an additonal elemen t α 0 = δ − θ in the simple system of aﬃne ro ots in addition to those give n b y the extensions of the α j ’s, j = 1 , · · · , p − 1 . (see, for example, [Mac] for details on aﬃne Lie algebras). With α 0 , we can in terpr et the pr o duct w α 0 = exp( h ( E 00 − E p − 1 ,p − 1 )) w θ as corresp onding to s α 0 . Th erefore, by (3.35) and (3.36), w e conclude that the CMV m atrix x f ( h ) in tr o duced ab o ve is a C oxete r elemen t of the aﬃne W eyl group. With this bac kground , w e are no w ready to accomplish our ﬁr st goal of th is section. Theorem 3.6. (a) O x e f = ˜ K w ∩ ˜ B w x e f ˜ B w = T e . (b) O x o f = ˜ K w ∩ ˜ B w x o f ˜ B w = T o . (c) E quip ˜ K w × ˜ K w with the pr o duct structur e, then O x e f × O x o f is a symple ctic le af of ˜ K w × ˜ K w . Mor e over, the c ol le ction of p × p Flo quet CMV matric es is the image of O x e f × O x o f under the P oisson autom orphism m | O x e f × O x o f : O x e f × O x o f − → { p × p Flo quet CMV matric es } , wher e m is the multiplic ation map of ˜ K w . Henc e { p × p Flo quet CMV matric es } = O x f = ˜ K w ∩ ˜ B w x f ˜ B w , a Coxeter dr essing orbit. Pr o of. (a) T ake an arbitrary elemen t a = k ( g ) − 1 x e f k (( x e f ) − 1 g x e f ) = b ( g ) − 1 x e f b (( x e f ) − 1 g x e f ) (3 . 38) THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 17 in the dressing orbit th rough x e f . F rom th e ﬁrst line of the ab ov e expr ession, it is clear that a is unitary . On the other hand , it f ollo ws from the second lin e of the ab o ve expression that P − a = 0 and that P 0 a is blo c k lo w er triangular with 2 × 2 blo c ks on the diagonal. Moreo ver, from the fact that the elemen ts in B ha ve p ositiv e d iagonal en tries, it foll o ws that eac h of the 2 × 2 b lo c ks on the main d iagonal of P 0 a has the follo wing prop erties: (i) the entry in the upp er righ t hand corner is p ositiv e, (ii) the determinan t is negativ e (since det ( w ∗ ) = − 1). But P − a = 0 implies P − a − 1 = 0 . As P − a − 1 = P − a ∗ = ( P + a ) ∗ , w e conclude that P + a = 0 and h ence a = P 0 a. But then ( a ∗ ) − 1 = (( P 0 a ) ∗ ) − 1 is up p er b lo c k triangular with diagonal blo cks ha ving the same prop er ties. Since a = ( a ∗ ) − 1 , it follo w s that a m u st b e blo ck diagonal, i.e., a = d iag( φ 0 , φ 2 , · · · , φ p − 2 ) (3 . 39) where for eac h j , φ 2 j is a unitary matrix with a p ositiv e en try in th e u pp er righ t hand corner and wh ose determinant is − 1 . Consequen tly , φ 2 j m u st b e of th e form φ 2 j =  ¯ α 2 j ρ 2 j ρ 2 j − α 2 j  (3 . 40) for some α 2 j ∈ D , where ρ 2 j = (1 − | α 2 j | 2 ) 1 2 . Hence we ha ve sho wn that O x e f ⊂ T e . The reverse inclusion T e ⊂ O x e f follo ws exactly as in the pro of of the corresp onding assertion in Theorem 2.4 (a) of [L1]. (b) T ak e an arbitrary elemen t b = k ( g ) − 1 x o f k (( x o f ) − 1 g x o f ) = b ( g ) − 1 x o f b (( x o f ) − 1 g x o f ) (3 . 41) in the dr essing orbit through x o f . F rom the ﬁr st line of the ab o ve expression, b is unitary . On the other hand, it follo w s from the second line of the s ame expression that b ( h ) = ∞ X j = − 1 b j h j . (3 . 42) No w, b − 1 = ( b (( x o f ) − 1 g x o p )) − 1 ( x o f ) ∗ b ( g ) (3 . 4 3) since x o f is u nitary . F r om this, it follo ws that ( P + ( h − 1 b )) ∗ = P − (( ¯ hb ) ∗ ) = P − ( hb − 1 ) = 0 . (3 . 44) Therefore, wh en we com b ine this with (3.42), we conclude that b ( h ) = b − 1 h − 1 + b 0 + b 1 h. (3 . 45) 18 L.-C. LI, I. N E NCIU Considering th e co eﬃcien t of h − 1 in the second line of (3.41 ), w e see that b − 1 =  b ( g ) − 1  0 ( x o f ) − 1  b (( x o f ) − 1 g x o p )  0 , as th e ﬁrst and last factors in the second line of (3. 39) only con tain nonn egativ e p ow ers of h . S in ce ( x o f ) − 1 has on ly one nonzero ent ry in its b ottom left corner, and it is multiplied by lo wer triangular m atrices with p ositiv e diagonal entries, we see that all the entries of b − 1 are zero, except ( b − 1 ) p − 1 , 0 = γ > 0. F u rther note that, if x ( h ) is un itary for all h ∈ S 1 , then an elemen t y ( h ) ∈ O x if and only if y ( h ) ∗ ∈ O x ∗ . In deed, y ( h ) = k ( g ) − 1 x k ( x − 1 g x ) ⇐ ⇒ y ( h ) ∗ = k ( g ′ ) − 1 x ∗ k ( xg ′ x ∗ ) , where g ′ = x − 1 g x . In our case, ( x o f ) ∗ = x o f , and hence b ( h ) ∗ = b ∗ 1 h − 1 + b ∗ 0 + b ∗ − 1 h ∈ O x o f . So we see, b y the ab o ve argumen t applied to b ∗ 1 , that b 1 m u st hav e only one n onzero en tr y , in its up p er right -hand corner: ( b 1 ) 0 ,p − 1 = δ > 0. It r emains to und erstand b 0 . Recall that b ( h ) is unitary and expand the r igh t- hand side of 1 = b ( h ) ∗ · b ( h ) = b ( h ) · b ( h ) ∗ in p o wers of h . Giv en the structure of the matrices b ± 1 , it is immediate that the coeﬃcients of h ± 2 are b oth zero. The co eﬃcien ts of h ± 1 m u st also b e zero. T his translates in to b ∗ 1 b 0 + b ∗ 0 b − 1 = 0 and b − 1 b ∗ 0 + b 0 b ∗ 1 = 0 . T aking in to accoun t the shap e of b ± 1 , and writing d o w n explicitly these relations leads to the fact that 0 − th ro w of b 0 = ( η , 0 . . . , 0) , (3 . 46) ( p − 1) − th row of b 0 = (0 , . . . , 0 , ν ) , (3 . 47) where δ η + γ ¯ ν = 0 . (3 . 48) Therefore, when we com b ine our analysis ab o ve on b ± 1 , together with (3.46) and (3.47) , our conclusion is that b ( h ) looks lik e b ( h ) =       η 0 · · · 0 δ h 0 0 . . . ˜ b . . . 0 0 γ h − 1 0 · · · 0 ν       , (3 . 49) THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 19 where ˜ b is a ( p − 2) × ( p − 2) matrix wh ic h is indep en den t of h . But a unitary matrix can only b e a dir ect su m of un itary matrices, and b ( h ) is unitary . Unitarit y of th e 2 × 2 piece at h = 1, together with the fact that γ , δ > 0, m eans that  η δ h γ h − 1 ν  =  − α p − 1 ρ p − 1 h ρ p − 1 h − 1 ¯ α p − 1  (3 . 50) for some complex num b er α p − 1 ∈ D and ρ p − 1 = p 1 − | α p − 1 | 2 . The last step requires that w e und er s tand the shap e of ˜ b . This is ac h iev ed by writing the co eﬃcient of h 0 in b ( h ) and the second line of (3.41). Again u sing the fact that the co eﬃcients of h 0 in the extern al factors on th e right-hand side of (3.41) are lo wer triangular with p ositiv e diagonal en tries, and k eeping trac k of all the (p oten tially) nonzero ent ries, w e obtain that ˜ b is a u n itary matrix whic h is blo c k-lo we r triangular, eac h 2 × 2 d iagonal blo ck having a p ositive en try in its upp er right-hand corner. Hence the same argumen t as in p art (a) of th is theorem sho w s that ˜ b = d iag( φ 1 , φ 3 , . . . ) , (3 . 51) where φ k =  ¯ α k ρ k ρ k − α k  . (3 . 52) Inserting this in to (3.49) and us in g (3.50) sho w s that b ∈ T o . The pro of is ﬁnished once we pr o ve that T o ⊂ O x o f . This is done as in the previous cases. More precisely , if b = θ 1 ⊕ θ 3 ⊕ · · · ⊕ θ p − 1 ( h ) is an elemen t of T o , c ho ose a matrix g = l 1 ⊕ · · · ⊕ l p − 3 ⊕ l p − 1 ( h ), w here, for 0 ≤ j ≤ ( p − 4) / 2, l 2 j +1 =  ρ 2 j +1 0 − α 2 j +1 1  (3 . 53) is a 2 × 2 blo ck situated b et ween the ro w s and columns 2 j + 1 and 2 j + 2, and l p − 1 ( h ) =  1 ¯ α p − 1 h 0 ρ p − 1  . (3 . 54) Note that l 2 j +1 are all indep enden t of h and lo w er triangular w ith p ositiv e diagonal en tr ies. F urther note that l p − 1 ( h ) =  1 0 0 ρ p − 1  + h  0 ¯ α p − 1 0 0  . (3 . 55) In p articular, this implies that g ∈ ˜ B w and h ence k ( g ) − 1 x o f k (( x o f ) − 1 g x o f ) = k ( g x o f ) . (3 . 56) F u rthermore, w e ha ve the factorizatio ns l 2 j +1 w ∗ = θ 2 j +1  ρ 2 j +1 0 − ¯ α 2 j +1 1  (3 . 57) 20 L.-C. LI, I. N E NCIU for 0 ≤ j ≤ ( p − 4) / 2, and l p − 1 ( h ) ·  0 h h − 1 0  = θ p − 1 ( h ) ·  1 α p − 1 h 0 ρ p − 1  . (3 . 58) In other w ords g · x o f = b · ˜ g , wh ere ˜ g is in ˜ B w for the same reason as g is. W e therefore conclud e that k ( g ) − 1 x o f k (( x o p ) − 1 g x o f ) = k ( g x o f ) = b (3 . 59) is indeed an element of O x o f . (c) It is easy to see that O x p = O x e f · O x o f . T he rest of the assertion is clear from what we ha v e already done.  R emark 3.7 . By essent ially follo wing the same argumen t, we can also establish the follo wing fact: { p × p Flo quet CMV m atrices } = O w α 1 · · · O w α p − 1 · O w α 2 · · · O w α p − 2 · O w α 0 , (3 . 60) where eac h orbit on the right hand sid e is t wo-dimensional. In the n ext resu lt, w e clarify the relation b et w een th e Ablowitz -Ladik brac ket in (2.11) and the S k lyanin brac k et {· , ·} J ♯ . Theorem 3.8. The map D p − → e G R w , α = ( α 0 , · · · , α p − 1 ) 7→ E ( h ) = g e ( α ) g o ( α )( h ) (3 . 61) is a Poisson emb e dding, when D p is e quipp e d with the Ablo witz-L adik br acket, and e G R w is e quipp e d with Sklyanin structur e {· , ·} J ♯ . Pr o of. As th e m ultiplication map of e G R w is a Po isson map , it is enough to sho w that the map α 7→ ( g e ( α ) , g o ( α )) is P oisson, when e G R w × e G R w is equipp ed with th e pro du ct structur e. F or this purp ose, denote b y E ab the p × p matrix whose ( a, b ) en tr y is equal to 1 and whose other en tries are zero. F or j eve n , j ∈ { 0 , · · · , p − 1 } and for l o dd , l ∈ { 0 , · · · , p − 1 } , intro d uce the follo wing fu nctions on e G R w × e G R w : F j ( A, B ) = Im I | h | =1 tr  A ( h ) E j j  dh 2 π ih , G j ( A, B ) = Re I | h | =1 tr  A ( h ) E j j  dh 2 π ih , F l ( A, B ) = Im I | h | =1 tr  B ( h ) E ll  dh 2 π ih , G l ( A, B ) = Re I | h | =1 tr  B ( h ) E j j  dh 2 π ih . (3 . 62) Then we ha ve F j ( g e , g o ) = − Im α j , G j ( g e , g o ) = Re α j F l ( g e , g o ) = − I m α l , G l ( g e , g o ) = Re α l . (3 . 63) THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 21 In view of this, it suﬃces to compute the P oisson brac k ets of these functions at ( g e , g o ) ∈ e G R w × e G R w . F or a function ϕ on e G R w × e G R w whic h is smo oth in b oth v ariables, w e denote by D i ϕ ( A, B ) (resp. D ′ i ϕ ( A, B )) its righ t gradien t (resp . left gradien t) with resp ect to the i -th v ariable, i = 1 , 2 . Then we ha v e D 1 F j ( g e , g o ) = g e E j j = ¯ α j E j j + ρ j E j +1 , j , D ′ 1 F j ( g e , g o ) = E j j g e = ¯ α j E j j + ρ j E j,j +1 , D 2 F j ( g e , g o ) = 0 , D ′ 2 F j ( g e , g o ) = 0 , (3 . 64) and similarly , D 2 F l ( g e , g o ) = g o ( h ) E ll = ¯ α l E ll + ρ l E l +1 ,l , D ′ 2 F l ( g e , g o ) = E ll g o ( h ) = ¯ α l E ll + ρ l E l,l +1 , l o d d , 1 ≤ l ≤ p − 3 , D 2 F p − 1 ( g e , g o ) = g o ( h ) E p − 1 ,p − 1 = ρ p − 1 hE 0 ,p − 1 + ¯ α p − 1 E p − 1 ,p − 1 , D ′ 2 F p − 1 ( g e , g o ) = E p − 1 ,p − 1 g o ( h ) = ρ p − 1 h − 1 E p − 1 , 0 + ¯ α p − 1 E p − 1 ,p − 1 , D 1 F l ( g e , g o ) = 0 , D ′ 1 F l ( g e , g o ) = 0 , l od d , 1 ≤ l ≤ p − 1 . (3 . 65) On th e other hand, it is clear that D 1 G j ( g e , g o ) = iD 1 F j ( g e , g o ) , D ′ 1 G j ( g e , g o ) = iD ′ 1 F j ( g e , g o ) , D 2 G j ( g e , g o ) = 0 , D ′ 2 G j ( g e , g o ) = 0 , (3 . 66) and similarly , D 2 G l ( g e , g o ) = iD 2 F l ( g e , g o ) , D ′ 2 G l ( g e , g o ) = iD ′ 2 F l ( g e , g o ) D 1 G l ( g e , g o ) = 0 , D ′ 1 G l ( g e , g o ) = 0 . (3 . 67) In what follo ws, the indices j and k are even, w hile the ind ices l and m are o dd, and all indices are from { 0 , · · · , p − 1 } . Let {· , ·} ∗ denote the pro duct structure on ˜ G R w × ˜ G R w , then it is immediate from the d eﬁnition of {· , ·} ∗ that { F j , F l } ∗ = 0 , { G j , G l } ∗ = 0 , { F j , G l } ∗ = 0 , { G j , F l } ∗ = 0 . (3 . 68 ) No w, from (3.64), D 1 F j ( g e , g o ) and D ′ 1 F j ( g e , g o ) are constan t lo ops, s o it follo ws from (3.12) and equation (2.6) of [L1] that J ♯ ( D 1 F j ( g e , g o )) = − α j E j j − ρ j E j +1 , j , J ♯ ( D ′ 1 F j ( g e , g o )) = ρ j E j,j +1 − 2 ρ j E j +1 , j − α j E j j . (3 . 69) Therefore, on using (3.69 ) and (3.64), w e ﬁnd for j 6 = k that tr ( J ♯ ( D ′ 1 F j ( g e , g o )) D ′ 1 F k ( g e , g o )) =( D ′ 1 F k ( g e , g o )) k k ( J ♯ ( D ′ 1 F j ( g e , g o )) k k + ( D ′ 1 F k ( g e , g o )) k ,k +1 ( J ♯ ( D ′ 1 F j ( g e , g o )) k +1 ,k = 0 , (3 . 70) 22 L.-C. LI, I. N E NCIU and similarly , tr ( J ♯ ( D 1 F j ( g e , g o )) D 1 F k ( g e , g o )) =( D 1 F k ( g e , g o )) k k ( J ♯ ( D 1 F j ( g e , g o )) k k + ( D 1 F k ( g e , g o )) k +1 ,k ( J ♯ ( D 1 F j ( g e , g o )) k ,k +1 = 0 . (3 . 71) Th us it follo ws from (3.70 ),(3.7 1) and (3.66) th at { F j , F k } ∗ ( g e , g o ) = 0 , { G j , G k } ∗ ( g e , g o ) = 0 , { F j , G k } ∗ ( g e , g o ) = 0 , for j 6 = k . (3 . 72) In a similar fashion, b y usin g (3.64), (3.66 ). and (3.69 ), w e ﬁn d { F j , G j } ∗ ( g e , g o ) = 1 2 tr ( i J ♯ ( D ′ 1 F j ( g e , g o )) D ′ 1 F j ( g e , g o )) − 1 2 tr ( i J ♯ ( D 1 F j ( g e , g o )) D 1 F j ( g e , g o )) = − ρ 2 j . (3 . 73) Analogously , for the o d d indices l, m , w ith l , m ∈ { 0 , · · · , p − 3 } , we hav e { F l , F m } ∗ ( g e , g o ) = 0 , { G l , G m } ∗ ( g e , g o ) = 0 , { F l , G m } ∗ ( g e , g o ) = 0 , for l 6 = m. (3 . 74) Also, { F l , G l } ∗ ( g e , g o ) = − ρ 2 l . (3 . 75) Next, we consider brac kets with the quantitie s F p − 1 and G p − 1 . T o this end, w e note the f orm ulas J ♯ ( D 2 F p − 1 ( g e , g o )) = − α p − 1 E p − 1 ,p − 1 − ρ p − 1 hE 0 ,p − 1 , J ♯ ( D ′ 2 F p − 1 ( g e , g o )) = ρ p − 1 h − 1 E p − 1 , 0 − α p − 1 E p − 1 ,p − 1 − 2 ρ p − 1 hE 0 ,p − 1 . (3 . 76) Therefore, if l is o dd , l ≤ p − 3 , a calculation similar to (3.70) and (3.71) ab o ve sho w s that { F p − 1 , F l } ∗ ( g e , g o ) = 0 , { G p − 1 , G l } ∗ ( g e , g o ) = 0 , { F p − 1 , G l } ∗ ( g e , g o ) = 0 . (3 . 7 7) No w, by (3.76), (3.67) and (3.65), w e hav e tr ( J ♯ ( D 2 F p − 1 ( g e , g o )) D 2 G p − 1 ( g e , g o )) = − i | α p − 1 | 2 , (3 . 78) while tr ( J ♯ ( D ′ 2 F p − 1 ( g e , g o )) D ′ 2 G p − 1 ( g e , g o )) = − 2 iρ 2 p − 1 − i | α p − 1 | 2 . (3 . 79) Consequent ly , { F p − 1 , G p − 1 } ∗ ( g e , g o ) = − ρ 2 p − 1 . (3 . 80) THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 23 Assem b ling the calculations, we conclude that for all a, b ∈ { 0 , · · · , p − 1 } , we hav e { F a , F b } ∗ ( g e , g o ) = 0 , { G a , G b } ∗ ( g e , g o ) = 0 , { F a , G b } ∗ ( g e , g o ) = 0 , if a 6 = b, { F a , G a } ∗ ( g e , g o ) = − ρ 2 a . (3 . 81) So ﬁn ally , w e obtain the follo w ing P oisson br ac ket relations { G a − iG a , G b − iF b } ∗ ( g e , g o ) = 0 , { G a + iF a , G b + iF b } ∗ ( g e , g o ) = 0 , if a 6 = b, { G a − iF a , G b + iF b } ∗ ( g e , g o ) = 2 iδ ab ρ 2 a , (3 . 82) as d esired.  Finally w e describ e the Hamiltonian equ ations generated by cen tral f unctions on ˜ G R w in the ab o ve f r amew ork . W e also introdu ce th e kind of equations wh ic h w e w ill need to use in Section 6 b elo w . Prop osition 3.9. (a) The Hamiltonian e quation of motion gener ate d by a c entr al function ϕ on ˜ G R w is given by the L ax e quation ˙ g = g (Π ˜ k w ( D ϕ ( g ))) − (Π ˜ k w ( D ϕ ( g ))) g = (Π ˜ b w ( D ϕ ( g ))) g − g (Π ˜ b w ( D ϕ ( g ))) (3 . 83) (b) Consider ( ˜ G R w , {· , ·} J ♯ ) and e qu ip the gr oup ˜ G R w × ˜ G R w with the pr o duct Poisson structur e. If ϕ is a c entr al function on ˜ G R w , then the L ax system ˙ g 1 = g 1  Π ˜ k w ( D ϕ ( g 2 g 1 ))  −  Π ˜ k w ( D ϕ ( g 1 g 2 ))  g 1 =  Π ˜ b w ( D ϕ ( g 1 g 2 ))  g 1 − g 1  Π ˜ b w ( D ϕ ( g 2 g 1 ))  , ˙ g 2 = g 2  Π ˜ k w ( D ϕ ( g 1 g 2 ))  −  Π ˜ k w ( D ϕ ( g 2 g 1 ))  g 2 =  Π ˜ b w ( D ϕ ( g 2 g 1 ))  g 2 − g 2  Π ˜ b w ( D ϕ ( g 1 g 2 ))  . (3 . 84) is the H amiltonian e quation on ˜ G R w × ˜ G R w gener ate d by H ϕ ( g 1 , g 2 ) = ϕ ( g 1 g 2 ) . Mor e- over, under the Hamiltonian ﬂow deﬁne d by (3.84), g = g 1 g 2 evolves ac c or ding to (3.83). (c) If k i ( t ) , b i ( t ) , i = 1 , 2 ar e the solutions of the factorizatio n pr oblems e tDϕ ( g 1 (0) g 2 (0)) = k 1 ( t ) b − 1 1 ( t ) , e tDϕ ( g 2 (0) g 1 (0)) = k 2 ( t ) b − 1 2 ( t ) , (3 . 85) wher e k i ( t ) ∈ ˜ K w , b i ( t ) ∈ ˜ B w , then the ﬂow deﬁne d by (3.84) i s given by g 1 ( t ) = k 1 ( t ) − 1 g 1 (0) k 2 ( t ) = b 1 ( t ) − 1 g 1 (0) b 2 ( t ) , g 2 ( t ) = k 2 ( t ) − 1 g 2 (0) k 1 ( t ) = b 2 ( t ) − 1 g 2 (0) b 1 ( t ) . (3 . 86) T o conclude th is section, w e r emark that equations of the t yp e in (3.84) are a sp ecial case of so-called Lax systems on a p erio dic lattice or diﬀerence Lax equations and w e refer the r eader to [ST S2] and [LP] for th e general theory . In Section 6 b elo w, w e will show ho w to solve the factorizatio n pr oblems f or the ﬂo ws generated by the comm utin g int egrals of th e p erio dic defo cusing Ablowitz -Ladik equation b y means of Riemann theta fu nctions asso ciated with a hyperelliptic curv e. 24 L.-C. LI, I. N E NCIU 4. Analytical prop erties of the Blo c h solution. F or any z ∈ C , consider the equation E u = z u, (4 . 1) where E is the extended CMV matrix with p erio dic V erb lu nsky co eﬃcients with p erio d p , as in Section 2. S ince E admits a θ -factorizatio n E = L M [S2], as in the one-sided case, it follo ws that (4.1) is equiv alent to M u = L ∗ u. (4 . 2) In terms of the comp onents of u and the entrie s of L an d M , (4.2) give s the three- term recurr ence r elations ρ 2 j − 1 u 2 j − 1 − α 2 j − 1 u 2 j = z ( α 2 j u 2 j + ρ 2 j u 2 j +1 ) ¯ α 2 j +1 u 2 j +1 + ρ 2 j +1 u 2 j +2 = z ( ρ 2 j u 2 j − ¯ α 2 j u 2 j +1 ) (4 . 3) for all j ∈ Z . Due to the equiv alen t form in (4.2), the space of s olutions of (4.1) is t wo dimensional. Indeed, it is clear from (4.3) that for giv en v alues of u − 1 and u 0 , w e can determine all other v alues of u n b y recursion. F or our analysis, we w ill ﬁx a b asis with the follo wing initial conditions: φ − 1 ( z ) = 1 , φ 0 ( z ) = 0 , ψ − 1 ( z ) = 0 , ψ 0 ( z ) = 1 . (4 . 4) By u sing the ﬁrst relation in (4.3) corresp ond ing to j = 0 , w e ha v e φ 1 ( z ) = ρ p − 1 z ρ 0 . (4 . 5) In general, an easy induction using (4.3) giv es the follo wing result. Prop osition 4.1. F or al l j ≥ 1 , φ 2 j ( z ) = − ¯ α 0 ρ p − 1 ρ 0 · · · ρ 2 j − 1 z j − 1 − · · · − ¯ α 2 j − 1 ρ p − 1 ρ 0 · · · ρ 2 j − 1 1 z j , φ 2 j +1 ( z ) = ¯ α 0 α 2 j ρ p − 1 ρ 0 · · · ρ 2 j z j − 1 + · · · + ρ p − 1 ρ 0 · · · ρ 2 j 1 z j +1 . (4 . 6) In p articular, φ p − 1 ( z ) = ¯ α 0 α p − 2 ρ p − 1 ρ 0 · · · ρ p − 2 z p/ 2 − 2 + · · · + ρ p − 1 ρ 0 · · · ρ p − 2 1 z p/ 2 . (4 . 7) Similarly , we hav e ψ 1 ( z ) = − α 0 ρ 0 − α p − 1 ρ 0 1 z , (4 . 8) and by induction, we obtain the follo wing an alog of Prop osition 4.1. THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 25 Prop osition 4.2. F or al l j ≥ 1 , ψ 2 j ( z ) = z j ρ 0 · · · ρ 2 j − 1 + · · · + ¯ α 2 j − 1 α p − 1 ρ 0 · · · ρ 2 j − 1 1 z j , ψ 2 j +1 ( z ) = − α 2 j ρ 0 · · · ρ 2 j z j − · · · − α p − 1 ρ 0 · · · ρ 2 j 1 z j +1 . (4 . 9) In p articular, ψ p − 1 ( z ) = − α p − 2 ρ 0 · · · ρ p − 2 z p/ 2 − 1 − · · · − α p − 1 ρ 0 · · · ρ p − 2 1 z p/ 2 , (4 . 10) and ψ p ( z ) = z p/ 2 ρ 0 · · · ρ p − 1 + · · · + | α p − 1 | 2 ρ 0 · · · ρ p − 1 1 z p/ 2 . (4 . 11) No w by th e p erio dicit y of the V erblunsky coeﬃcient s, we ha ve ( φ j + p ( z ) ψ j + p ( z ) ) = ( φ j ( z ) ψ j ( z ) ) M ( z ) (4 . 12) for all j, w here M ( z ) =  φ p − 1 ( z ) ψ p − 1 ( z ) φ p ( z ) ψ p ( z )  (4 . 13) is the mono dromy matrix. Prop osition 4.3. F or al l z , d et M ( z ) = 1 . Pr o of. Let W j ( z ) = ρ j ( φ j ( z ) ψ j +1 ( z ) − φ j +1 ( z ) ψ j ( z )) . Then from the ﬁr st relation in (4.3), w e ha v e W 2 j − 1 ( z ) = − z W 2 j ( z ) (4 . 14) for all j. S imilarly , f r om the second relation in (4.3), w e ﬁn d that W 2 j +1 ( z ) = − z W 2 j ( z ) (4 . 15) for all j. As the right hand sides of (4.14) and (4.15) are equ al, it follo ws that W 2 j − 1 ( z ) is in d ep end en t of j and consequen tly W p − 1 ( z ) = W − 1 ( z ) from wh ic h the assertion follo ws.  F r om Pr op osition 4 .3, the eigen v alues of the mono d rom y matrix (i .e., the Flo quet m u ltipliers) are the r o ots of the characte ristic p olynomial h 2 − tr M ( z ) h + 1 = 0 . (4 . 16) If T : l ∞ ( Z ) − → l ∞ ( Z ) denote the shif t op erator deﬁn ed by ( T u ) j = u j + p , th en the u nique solution of the problem E f = z f , T f = h − 1 f , f p − 1 = 1 (4 . 17) 26 L.-C. LI, I. N E NCIU is called the Blo ch solution and ﬁn ding th is solution is equiv alen t to considerin g the s p ectrum of the corresp onding Flo quet CMV matrix E ( h ), E ( h ) b v = z b v (4 . 18) where b v =     f 0 . . . f p − 2 1     . (4 . 19) Hence the ordered pair ( z , h ) in (4.17) must ob ey the equation det( z I − E ( h )) =   p − 1 Y j =0 ρ j   z p 2 [∆( z ) − ( h + h − 1 )] = 0 , (4 . 20) where the discriminan t ∆( z ) is related to the transfer matrix T p ( z ) = 1 Q p − 1 j =0 ρ j  z − ¯ α p − 1 − α p − 1 z 1  · · ·  z − ¯ α 0 − α 0 z 1  (4 . 21) b y the form u la [S2] ∆( z ) = z − p/ 2 tr T p ( z ) . (4 . 22) By comparin g (4.16) and (4.20), w e therefore conclud e th at tr M ( z ) = ∆ ( z ) and this relates th e m ultiplier curve and the sp ectral cu rv e. W e will make the genericit y assumption ( GA ) 1 the r o ots of P ( z ) =  Q p − 1 i =0 ρ i  2 (∆( z ) 2 z p − 4 z p ) = Q 2 p i =1 ( z − λ i ) are d istinct. Then it is straigh tforw ard to chec k that th e aﬃne curve as d eﬁ ned by th e equati on I ( h, z ) := h · det( z I − E ( h )) = 0 (4 . 23) is smo oth with branch p oints lo cated at λ 1 , · · · , λ 2 p . W e will denote by C the h yp erelliptic Riemann su rface of gen us g = p − 1 corr esp onding to this aﬃn e curve . In ord er to ﬁnd the divisor structure of f j , j = 0 , · · · , p − 2 on C, let ( z ) = Q + + Q − − P + − P − (4 . 24) where P + , Q + are on the + s h eet and P − , Q − are on the − sheet of C. (Th e ± - sheets corresp ond to the choice of sign in front of th e radical in the ﬁrst line of (4.25) .) S olving for h in terms of of z from (4.20 ), w e ﬁnd h ( z ) = z p/ 2 ∆( z ) ± p ∆( z ) 2 z p − 4 z p 2 z p/ 2 = z p/ 2 ∆( z ) ± z p/ 2 ∆( z ) h 1 − 2 z p z p ∆( z ) 2 + · · · i 2 z p/ 2 for z near ∞ , = z p/ 2 Q p − 1 j =0 ρ j + · · · for z near ∞ on the + sheet , = Q p − 1 j =0 ρ j z p/ 2 + · · · for z near ∞ on the − sheet . (4 . 25) THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 27 On th e other hand, it is clear from (5.1) that P z p/ 2 ∆( z ) ∼ 1 as z → 0 . Therefore, it follo ws from the ﬁrst line of (4.25 ) that h ( z ) ∼ 1 P z p/ 2 as z → 0 on the + sheet, while h ( z ) ∼ P z p/ 2 as z → 0 on the − sheet. Thus w e ha ve ( h ) = − p 2 P + + p 2 P − − p 2 Q + + p 2 Q − . (4 . 26) Prop osition 4.4. F or e ach 0 ≤ j ≤ p − 2 , f j ( P ) = h ( P ) φ j ( z ( P )) + 1 − h ( P ) φ p − 1 ( z ( P )) ψ p − 1 ( z ( P )) ψ j ( z ( P )) , P ∈ C. (4 . 27) Pr o of. Sin ce φ and ψ form a basis of solutions of the equation E u = z u, we must ha ve f j = c 1 φ j + c 2 ψ j for some constan ts c 1 and c 2 . Putting j = − 1 and 0 in the ab o v e expression and using the initial conditions in (4.4), we ﬁ n d f j = hφ j + f 0 ψ j . As the v ector with comp onen ts h and f 0 is an eig en vect or of the mono dromy matrix M ( z ) with eigen v alue h − 1 , we ﬁn d that φ p − 1 h + ψ p − 1 f 0 = 1 . S olving f or f 0 from this expression, we obtain the desired expression for f j .  W e next mak e the f ollo wing assumption. ( GA ) 2 α j 6 = 0 for j = 0 , · · · , p − 1 . Note that in particular, α p − 2 6 = 0 and so the degree of the p olynomial z p/ 2 ψ p − 1 ( z ) is p − 1 . The r o ots of this p olynomial will b e denoted by z k , k = 1 , · · · , p. (In general, the z k ’s are not necessarily all distinct.) Prop osition 4.5. F or e ach 0 ≤ j ≤ p − 2 , f j ( P ) is a single-value d mer omor phic function on the Riemann surfac e C . On the ﬁnite p art of C awa y f r om Q + , f j ( P ) has at worst p oles at the p oints P k = ( φ p − 1 ( z k ) , z k ) , wher e ψ p − 1 ( z k ) = 0 f or k = 1 , · · · , p − 1 . Mor e over, the z k ’s c oincide with the eigenvalues of the Dirichlet pr oblem E u = z u, u − 1 = 0 , u p − 1 = 0 . (4 . 2 8) Equivalently, the z k ’s ar e the zer os of the e quation det ( z [ ( g e ) ∗ − [ g o ( h )) = 0 , (4 . 29) wher e [ ( g e ) ∗ and [ g o ( h ) ar e ( p − 1) × ( p − 1) matric es obtaine d fr om ( g e ) ∗ and g o ( h ) by r emoving their last r ow and last c olumn. Pr o of. F or eac h j, it follo ws from (4.27) that f j ( P ) is meromorphic on C. Let us consider a p oint z k where ψ p − 1 ( z k ) = 0 . F r om (4.13), w e see that at such a p oint, the Flo quet m u ltipliers are giv en by φ p − 1 ( z k ) and φ p − 1 ( z k ) − 1 and hence ( φ p − 1 ( z k ) , z k ) and ( φ p − 1 ( z k ) − 1 , z k ) are p oin ts on C . Clearly , 1 − hφ p − 1 ( z ) v an- ishes at ( φ p − 1 ( z k ) − 1 , z k ). T h erefore, p ro vided that φ p − 1 ( z k ) 6 = ± 1 or equiv alen tly , ∆( z k ) 6 = ± 2 , f j ( P ) has a p ole at P k = ( φ p − 1 ( z k ) , z k ) . S ince ψ − 1 = 0 , w e see that 28 L.-C. LI, I. N E NCIU the solutions of ψ p − 1 ( z ) = 0 coincide with th e eigen v alues of the Diric hlet pr oblem in (4.28). On the other hand, observe that if ψ p − 1 ( z k ) = 0 , th en diag( − α p − 1 , θ 1 , · · · , θ p − 3 )    ψ 0 ( z k ) . . . ψ p − 2 ( z k )    = z k diag( θ ∗ 0 , · · · , θ ∗ p − 4 , α p − 2 )    ψ 0 ( z k ) . . . ψ p − 2 ( z k )    from the connection with (4.2 8) wh er e ψ 0 = 1 . As the matrix on the left hand side of the ab o ve form ula is [ g o ( h ) , wh ile the one on the r ight hand side is [ ( g e ) ∗ , the last assertion in the p rop osition follo ws.  R emark 4.6. The Diric hlet eigen v alues ab o ve should not b e confu sed w ith the Diric hlet d ata in Chapter 11 of Simon [S2]. While the latter quan tities alw a ys reside on the u n it circle and th e num b er of suc h quan tities is equal to p , it is n ot the case for the z k ’s, as is evident from the relation Q p − 1 k =1 z k = − α p − 1 α p − 2 . In wh at follo ws, we will d enote by h ± ( z ) (resp . f ± j ( z )) the v alues of the fun ction h ( P ) (resp . f j ( P )) on the ± sheets of the Riemann surf ace. Prop osition 4.7. F or j = 0 , · · · , p/ 2 − 1 , we have f − 2 j ( z ) ∼ −  Q p − 2 i =2 j ρ i  α p − 2 z − ( p/ 2 − j − 1) , f − 2 j +1 ( z ) ∼ α 2 j α p − 2   p − 2 Y i =2 j +1 ρ i   z − ( p/ 2 − j − 1) (4 . 30) as z → ∞ . Henc e f 2 j ( P ) and f 2 j +1 ( P ) have zer os of or der p/ 2 − j − 1 at P − . Pr o of. Consid er ﬁrst the ev en case. By using (4.25 ) and Pr op ositions 4.1 an d 4.2, w e hav e 1 − h − ( z ) φ p − 1 ( z ) ψ p − 1 ( z ) ψ 2 j ( z ) ∼ −  Q p − 2 i =2 j ρ i  α p − 2 . 1 z p/ 2 − j − 1 (4 . 31) as z → ∞ . Similary , h − ( z ) φ 2 j ( z ) ∼ − ¯ α 0 ρ p − 1  Q p − 1 i =2 j ρ i  z p/ 2 − j +1 (4 . 32) as z → ∞ . Therefore, on comparing (4.31) and (4.32), the assertion for the ev en case follo ws. W e will skip the details for th e o dd case as it pro ceeds in the same w a y .  THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 29 T o inv estigate the b eha viour of f + j ( z ) as z → ∞ , w e will ﬁrst esta blish an iden tit y for the pro d uct f + j ( z ) f − j ( z ) . T o this end, observ e that 1 − h ± φ p − 1 ( z ) ψ p − 1 ( z ) = 1 2 ψ p − 1 ( z ) h (2 − φ p − 1 ( z )( φ p − 1 ( z ) + ψ p ( z ))) + φ p − 1 ( z ) p ∆( z ) 2 − 4 i . (4 . 33) Therefore, by a d ir ect multiplica tion and usin g (4.31), w e ﬁnd f + j ( P ) f − j ( P ) = − ψ j ( φ j φ p − 1 + ψ j φ p ) + φ j ( ψ j ψ p + φ j ψ p − 1 ) ψ p − 1 = − ψ j φ j + p + φ j ψ j + p ψ p − 1 (4 . 34) where on the righ t hand side, w e ha ve omitted the v ariable z throughout. Note that in going from the ﬁrst line of (4.34) to the second line, w e h a ve used (4.12). Our next task is to in terpret th e numerator of the righ t hand sid e of (4.34), whic h is necessary b ecause p erform ing a direct asymptotic analysis of this qu an tit y b y using Pr op ositions 4.1 and 4.2 p ro v es to b e diﬃcult. Th at th is is so is due to the degeneracy of the tridiagonal matrices L and M . (Note that neither E n or its factors L and M satisfy the genericit y assumption in [MM].) F or this pur p ose, w e in tro duce for eac h 1 ≤ j ≤ p − 2 the shifted matrix E [ j ] whose ( k , l ) en try is giv en b y ( E [ j ] ) k l = E k + j,l + j . (4 . 35) On th e other hand, let ψ [ j ] denote the solution of E [ j ] u = z u, u − 1 = 0 , u 0 = 1 . (4 . 36) Prop osition 4.8. F or e ach 0 ≤ j ≤ p − 2 , f + j ( z ) f − j ( z ) = B j +1 ( z ) ψ [ j +1] p − 1 ( z ) ψ p − 1 ( z ) , (4 . 37) wher e B j +1 ( z ) = det  φ j ( z ) ψ j ( z ) φ j +1 ( z ) ψ j +1 ( z )  = ( ρ p − 1 ρ j , for j o dd − ρ p − 1 ρ j z , for j even . (4 . 38) Pr o of. F rom the deﬁn ition of E [ j ] and ψ [ j ] , it is clear that ψ [ j ] k ( z ) = c 1 ( z ) φ k + j ( z ) + c 2 ( z ) ψ k + j ( z ) (4 . 39) 30 L.-C. LI, I. N E NCIU for some c 1 ( z ) and c 2 ( z ) . By imp osing the initial conditions in (4.36) , w e ﬁnd that c 1 ( z ) = − ψ j − 1 ( z ) B j ( z ) , c 2 ( z ) = φ j − 1 ( z ) B j ( z ) . (4 . 40) Therefore, on substituting in to (4.39), we obtain B j ( z ) ψ [ j ] k ( z ) = − ψ j − 1 ( z ) φ k + j ( z ) + φ j − 1 ( z ) ψ k + j ( z ) . (4 . 41) Hence (4.37) follo ws from (4.34) if we replace j by j + 1 and let k = p − 1 in (4.41) . T o complete the pro of, it remains to calculate B j +1 ( z ) . Here w e mak e use of the quantit y W j ( z ) introd uced in the pro of of Pr op osition 4.3 whic h is related to B j +1 ( z ) b y the r elation W j ( z ) = ρ j B j +1 ( z ) . F r om the p ro of of Prop osition 4.3 (see relations (4.14 ) and (4.1 5)), we learn that W 2 j − 1 ( z ) is indep endent of the v alue of j, and the same holds tru e for W 2 j ( z ) . Consequent ly , we hav e ρ j B j +1 ( z ) = ρ j − 2 B j − 1 ( z ) . (4 . 42) F r om th is, we ﬁnd ρ j B j +1 ( z ) =  ρ − 1 B 0 ( z ) , for j o d d , ρ 0 B 1 ( z ) , for j even , =  ρ p − 1 , for j o dd , − ρ p − 1 z , for j ev en . (4 . 43) This completes the p ro of.  In our next resu lt, we will analyze ψ [ j ] ( z ) . Prop osition 4.9. (a) F or 0 ≤ j ≤ p/ 2 − 1 and 0 ≤ k ≤ p / 2 − 1 , ψ [2 j ] 2 k ( z ) = z k ρ 2 j · · · ρ 2 j +2 k − 1 + · · · + ¯ α 2 j +2 k − 1 α 2 j − 1 ρ 2 j · · · ρ 2 j +2 k − 1 1 z k , ψ [2 j ] 2 k +1 ( z ) = − α 2 j +2 k ρ 2 j · · · ρ 2 j +2 k z k − · · · − α 2 j − 1 ρ 2 j · · · ρ 2 j +2 k 1 z k +1 . (4 . 44) (b) F or 0 ≤ j ≤ p/ 2 − 1 and 0 ≤ k ≤ p/ 2 − 1 , ψ [2 j +1] 2 k ( z ) = ¯ α 2 j α 2 j +2 k ρ 2 j +1 · · · ρ 2 j +2 k z k + · · · + 1 ρ 2 j +1 · · · ρ 2 j +2 k 1 z k , ψ [2 j +1] 2 k +1 ( z ) = − ¯ α 2 j ρ 2 j +1 · · · ρ 2 j +2 k + 1 z k +1 − · · · − ¯ α 2 j +2 k + 1 ρ 2 j +1 · · · ρ 2 j +2 k + 1 1 z k . (4 . 45) In p articular, ψ [2 j ] p − 1 ( z ) = − α 2 j − 2 ρ 2 j − 1 Q p − 1 i =0 ρ i z p/ 2 − 1 − · · · − α 2 j − 1 ρ 2 j − 1 Q p − 1 i =0 ρ i 1 z p/ 2 (4 . 46) THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 31 and ψ [2 j +1] p − 1 ( z ) = − ¯ α 2 j ρ 2 j Q p − 1 i =0 ρ i z p/ 2 − · · · − ¯ α 2 j − 1 ρ 2 j Q p − 1 i =0 ρ i 1 z p/ 2 − 1 . (4 . 47) Pr o of. F or the equation E [ i ] u = z u asso ciated with the shifted matrix E [ i ] , the recurrence r elations in (4.3) ha v e to b e replaced b y ρ 2 k − 1 u 2 k − 1 − i − α 2 k − 1 u 2 k − i = z ( α 2 k u 2 k − i + ρ 2 k u 2 k +1 − i ) ¯ α 2 k +1 u 2 k +1 − i + ρ 2 k +1 u 2 k +2 − i = z ( ρ 2 k u 2 k − i − ¯ α 2 k u 2 k +1 − i ) . (4 . 48) F or i = 2 j, we can ther efore obtain (4.4 4) from (4.9) by s hifting the indices. F or i = 2 j + 1 , w e hav e to solv e (4.48) and an inductiv e argument leads to (4.45) .  Prop osition 4.10. F or j = 0 , · · · , p/ 2 − 1 , we have f + 2 j ( z ) ∼ − ¯ α 2 j  Q p − 2 i =2 j ρ i  z p/ 2 − j − 1 , f + 2 j +1 ( z ) ∼ 1  Q p − 2 i =2 j +1 ρ i  z p/ 2 − j − 1 (4 . 49) as z → ∞ . Henc e f 2 j ( P ) and f 2 j +1 ( P ) have p oles of or der p/ 2 − j − 1 at P + . Pr o of. F rom (4.46), (4,47), and Pr op osition 4.7, w e ha v e f + 2 j ( z ) f − 2 j ( z ) ∼ − ¯ α 2 j α p − 2 , f + 2 j +1 ( z ) f − 2 j +1 ( z ) ∼ α 2 j α p − 2 , (4 . 50) as z → ∞ . Therefore the assertion follo ws f rom (4.50) and (4.30).  W e next in vestig ate the b eha viour of f ± j ( z ) as z → 0 . Prop osition 4.11. F or j = 0 , · · · , p/ 2 − 1 , we have f − 2 j ( z ) ∼ − ¯ α 2 j − 1   p − 2 Y i =2 j ρ i   z p/ 2 − j , f − 2 j +1 ( z ) ∼   p − 2 Y i =2 j +1 ρ i   z p/ 2 − j − 1 (4 . 51) as z → 0 . Henc e at Q − , f 2 j ( P ) has a ze r o of or der p/ 2 − j, while f 2 j +1 ( P ) has a zer o of or der p/ 2 − j − 1 . Pr o of. By (4.25), Prop osition 4.1 and 4.2, w e ha v e 1 − h − ( z ) φ p − 1 ( z ) ψ p − 1 ( z ) ψ 2 j ( z ) ∼ −| α p − 1 | 2 ¯ α 2 j − 1   p − 2 Y i =2 j ρ i   z p/ 2 − j (4 . 52) 32 L.-C. LI, I. N E NCIU as z → 0 . On the other hand, h − ( z ) φ 2 j ( z ) ∼ − ¯ α 2 j − 1 ρ 2 p − 1   p − 2 Y i =2 j ρ i   z p/ 2 − j . (4 . 53) On com binin g (4.31) and (4.32) and simplify , we obtain th e ﬁrst relation in (4.50). The other relation in (4.51) f ollo ws in the same w ay .  Prop osition 4.12. F or j = 0 , · · · , p/ 2 − 1 , we have f + 2 j ( z ) ∼ 1 α p − 1  Q p − 2 i =2 j ρ i  z − ( p/ 2 − j ) , f + 2 j +1 ( z ) ∼ α 2 j +1 α p − 1  Q p − 2 i =2 j +1 ρ i  z − ( p/ 2 − j − 1) (4 . 54) as z → 0 . He nc e at Q + , f 2 j ( P ) has a p ole of or der p/ 2 − j while f 2 j +1 ( P ) has a p ole of or der p/ 2 − j − 1 . Pr o of. F rom (4.46), (4.47) and Pr op osition 4.7, w e ﬁn d f + 2 j ( z ) f − 2 j ( z ) ∼ − ¯ α 2 j − 1 α p − 1 , f + 2 j +1 ( z ) f − 2 j +1 ( z ) ∼ α 2 j +1 α p − 1 (4 . 55) as z → 0 . Th erefore the assertion follo ws from (4.55) and (4.51).  Com b ining Pr op ositons 4.5, 4.7, 4.10, 4.11 and 4.12, we obtain the main r esu lt of th e section. Theorem 4.13. F or 0 ≤ j ≤ p/ 2 − 1 , ( f 2 j ) ≥ − D −  p 2 − j − 1  P + +  p 2 − j − 1  P − −  p 2 − j  Q + +  p 2 − j  Q − , ( f 2 j +1 ) ≥ − D −  p 2 − j − 1  P + +  p 2 − j − 1  P − −  p 2 − j − 1  Q + +  p 2 − j − 1  Q − , (4 . 56) wher e D = P p − 1 k =1 P k . Corollary 4.14. F or 0 ≤ j ≤ p/ 2 − 1 , ( f 2 j + p ) ≥ − D + ( j + 1) P + − ( j + 1) P − + j Q + − j Q − , ( f 2 j +1+ p ) ≥ − D + ( j + 1) P + − ( j + 1) P − + ( j + 1) Q + − ( j + 1) Q − . (4 . 57) Pr o of. Th is follo ws fr om (4.51), the relation f k + p = h − 1 f k for all k and (4.26).  T o close, we presen t the follo wing result w hic h is essential in Section 6 b elo w. THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 33 Prop osition 4.15. F or e ach 0 ≤ j ≤ p/ 2 − 1 , the divisors U j 1 = D +  p 2 − j − 1  P + +  p 2 − j − 1  Q + −  p 2 − j − 1  P − −  p 2 − j − 1  Q − and U j 2 = D +  p 2 − j − 1  P + +  p 2 − j  Q + −  p 2 − j − 1  P − −  p 2 − j  Q − ar e gener al, i . e., dim L ( U j 1 ) = dim L ( U j 2 ) = 1 (4 . 58) wher e for a divisor U on C , L ( U ) = { mer omorphic function φ | ( φ ) ≥ −U } . Pr o of. W e will adapt an argument of [MM] to our situation. F or a divisor U on C, denote by Ω( U ) the set of meromorp hic 1-forms ω on C suc h that ( ω ) ≥ U . T ak e k and k ′ suc h that k + k ′ > g − 2 = p − 3 , then D + k P − + k ′ Q − has degree g + k + k ′ > 2 g − 2 . If ( ω ) ≥ D + k P − + k ′ Q − , then ω must b e 0 since a holomorphic 1-form can hav e at most 2 g − 2 zeros. Thus dim Ω( D + k P − + k ′ Q − ) = 0 . By Riemann-Ro c h, it follo ws that dim L ( D + k P − + k ′ Q − ) = k + k ′ + 1 > g − 1 . F or concreteness, tak e k = k ′ = p 2 , and we claim that L ( D + ( j − 1) P − + ( j − 1) Q − ) ( L ( D + j P − + ( j − 1) Q − ) ( L ( D + j P − + j Q − ) , 1 ≤ j ≤ p 2 . (4 . 59) T o establish th is claim, w e just ha ve to observ e that b y C orollary 4.14 ab o v e, w e ha ve f 2( j − 2)+1+ p ∈ L ( D + ( j − 1) P − + ( j − 1) Q − ) , bu t f 2( j − 2)+1+ p / ∈ L ( D + j P − + ( j − 1) Q − ) . Similarly , f 2( j − 1)+ p ∈ L ( D + j P − + ( j − 1) Q − ) , but f 2( j − 1)+ p / ∈ L ( D + j P − + j Q − ) . Thus it follo ws from dim L ( D + p 2 P − + p 2 Q − ) = p + 1 and the claim in (4.58 ) that dim L ( D ) = 1 . W e next show that dim L ( D + Q + − Q − ) = 1 . Here w e u se the fact that allo w ing an extra p ole increases dim L ( U ) by at most one. Thus w e hav e 1 ≤ dim L ( D + Q + − Q − ) ≤ dim L ( D − Q − ) + 1 . (4 . 60) But L ( D − Q − ) ( L ( D ) , as f p − 1 = 1 ∈ L ( D ) but f p − 1 / ∈ L ( D − Q − ) . Hence w e conclude from d im L ( D ) = 1 that dim L ( D − Q − ) = 0 . Consequen tly , it follo ws from (4.54) that dim L ( D + Q + − Q − ) = 1 . Based on th is, we can establish dim L ( D + Q + − Q − + P + − P − ) = 1 f rom the inequalit y 1 ≤ dim L ( D + Q + − Q − + P + − P − ) ≤ dim L ( D + Q + − Q − − P − ) + 1 (4 . 6 1) and the observ ation that L ( D + Q + − Q − − P − ) ( L ( D + Q + − Q − ). (This follo ws b ecause f p − 2 ∈ L ( D + Q + − Q − ) but f p − 2 / ∈ L ( D + Q + − Q − − P − ) . ) Pro ceed inductiv ely , we ha v e the assertion.  34 L.-C. LI, I. N E NCIU 5. Action-angle v aria ble s. As we sa w in Section 2 ab o ve, the p erio dic Ablo witz-Ladik equation can b e expressed in Lax pair form with L ax op erator giv en by E ( h ) . Therefore, the c har- acteristic p olynomial det( z I − E ( h )) is in v ariant under the Hamiltonian ﬂo w and pro vides us with a collect ion of conserved quantit ies. F rom [S2], we ha v e det( z I − E ( h )) =   p − 1 Y j =0 ρ j   z p 2 [∆( z ) − ( h + h − 1 )] = p 2 X − p 2 I j z j + p 2 − ( h + h − 1 ) z p 2   p − 1 Y j =0 ρ j   (5 . 1) where the functions I j as d eﬁned in the second line of (5.1) are suc h that I p/ 2 = I − p/ 2 = 1 , ¯ I j = I − j , j = 0 , · · · , p/ 2 − 1 . (5 . 2) Moreo v er, they are all p olynomials in the α j ’s, their conju gates, and P = Q p − 1 j =0 ρ j , j = 0 , · · · , p − 1 . By using the fact that the collection of p × p Flo q u et CMV matrices is a s ymplectic leaf of the Skly anin br ack et {· , ·} J ♯ , we b egin by repro vin g th e inv olution theorem in [N] and [S2]. Theorem 5.1. The func tions P , I 0 , R e I j , Im I j , j = 1 , · · · , p/ 2 − 1 pr ovide a c ol- le ction of p c onserve d quantities in i nv olution for the Ablowitz-L adik e quation. Pr o of. W rite det( z I − E ( h )) = p X r =0 E r ( E ( h )) z p − r . ( 5 . 3) Then u p to signs, the E r ’s are the elemen tary symmetric fu nctions. F rom (5.1) and (5.3), we ﬁnd th at I j ( E ) = I | h | =1 E p/ 2 − j ( E ( h )) dh 2 π ih , j = 0 , · · · , p/ 2 − 1 , P ( E ) = − I | h | =1 E p/ 2 ( E ( h )) dh 2 π i . (5 . 4) Hence the functions P, I 0 , Re I j , Im I j , j = 1 , · · · , p/ 2 − 1 are the pullbac ks of cen tral functions on ˜ G R w to the 2 p dimensional dressing orbit consisting of p × p Floquet CMV matrices. Consequently , the assertion follo ws from the abstract in v olution theorem in [STS2] and Theorem 3.6.  R emark 5.2. (a) F or readers who are not familiar with the abstract inv olution theorem in [S T S2], let us remark that its demonstration is a one line pro of making THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 35 use of the fact that for a cent ral f u nction, its left gradien t is the same as its righ t gradien t. (See the expression for th e P oisson brac ket {· , ·} J ♯ in (3.16) ab o ve for our case.) (b) Note that we are usin g the s y mb ol E as a sh orthand f or the unitary lo op E ( · ) in (5.4) ab ov e and w e will henceforth con tinue to use the sym b ol with this meaning. Since we will not b e usin g the extended CMV matrices in what follo w s, this should not cause an y confusion. Th us th e num b er of comm u ting in tegrals as pro v id ed b y the quan tities in the ab o v e theorem is exactly equal to one h alf the dimen s ion of the p hase s p ace. In the rest of the section, we will construct the v ariables (essen tially) conjugate to these actions. As in Section 4, we denote b y C the hyp erelliptic Riemann surface of gen us g = p − 1 corresp onding to th e aﬃne curv e I ( h, z ) = h det( z I − E ( h )) = 0 . On C, w e introdu ce the holomorphic 1-forms ξ k = z k − 1 ∂ I ∂ h dz , k = 1 , · · · , g = p − 1 . (5 . 5) W e also in tro duce the meromorphic 1-form ξ m = ( h + h − 1 ) z p/ 2 − 1 ∂ I ∂ h dz (5 . 6) with p oles at P ± and Q ± . Pic k a ﬁ x p oint P 0 on the ﬁn ite part of C an d put D 0 = g P 0 . Then for E ( h ) s atisfying the genericit y assumptions ( GA ) 1 and ( GA ) 2 , w e deﬁn e φ k ( E ) = Z D D 0 ξ k , k = 1 , · · · , g , ν ( E ) = Z D D 0 ξ m , (5 . 7) where D = P g k =1 P k is the divisor of p oles of f j , 0 ≤ j ≤ g − 1 in the ﬁn ite part of C. Note that in the deﬁnition of ν ( E ) , the paths of in tegratio n going from the p oints of D 0 to the p oin ts of D must a void the p oin ts P ± . T hese multi-v alued v ariables are w ell deﬁned b ecause the p oin ts in D 0 and D are in the ﬁnite part of C. O n the other hand, the m ulti-v aluedness can b e resolv ed in the standard wa y and we w ill not try to get into the details here. (See, for example, [DL T ] and [L2].) T o compute the P oisson br ac ket s b et ween the conserve d quant ities in Theorem 5.1 and the v ariables in (5.7), w e will m ak e use of a device in [DL T] (wh ic h has also pro v ed to b e successful in [L2]) whic h will allo w us to simplify the calculat ion. In the follo wing, we w ill deal with E ( h ) for h not necessarily on the un it circle. Note that in this general case, we ha v e the relatio n E ( h ) E ( ¯ h − 1 ) ∗ = E ( ¯ h − 1 ) ∗ E ( h ) = I , h ∈ C \ { 0 } (5 . 8) whic h can b e c hec ke d b y using the fact that E ( h ) is unitary for h ∈ ∂ D . F or our purp ose, we pic k a ﬁxed h 0 ∈ ( − 1 , 1) \ { 0 } , z 0 ∈ ∂ D suc h that ( h 0 , z 0 ) is n ot on C 36 L.-C. LI, I. N E NCIU and d eﬁne H h 0 ,z 0 ( E ) = Re log det ( z 0 I − E ( h 0 )) , J h 0 ,z 0 ( E ) = Im log det ( z 0 I − E ( h 0 )) . (5 . 9) As the reader will see in the calculation w hic h follo w, this c hoice of h 0 and z 0 is critical. Lemma 5.3. (a) The Hamiltonian e quation gener ate d by H h 0 ,z 0 ( E ) is given by the e quation ˙ E ( h ) = [ E ( h ) , B ( h ) ] (5 . 10 ) wher e B ( h ) =  i ( z 0 I − E ( h 0 )) − 1 E ( h 0 )  − + i Im  i ( z 0 I − E ( h 0 )) − 1 E ( h 0 )  0 − h  i ( z 0 − E ( h 0 )) − 1 E ( h 0 )  − i ∗ + iz 0 ( z 0 I − E ( h − 1 0 )) − 1 · 1 1 − hh 0 . (5 . 11) (b) The H amiltonian e quation gener ate d b y J h 0 ,z 0 ( E ) is g iven by the e quation ˙ E ( h ) = [ E ( h ) , C ( h ) ] (5 . 12) wher e C ( h ) =  ( z 0 I − E ( h 0 )) − 1 E ( h 0 )  − + i Im  ( z 0 I − E ( h 0 )) − 1 E ( h 0 )  0 − h  ( z 0 I − ( h 0 )) − 1 E ( h 0 )  − i ∗ − z 0 ( z 0 I − E ( h − 1 0 )) − 1 · 1 1 − hh 0 . (5 . 13) Pr o of. (a) If w e let φ ( g ) = Re log det( z 0 I − g ( h 0 )) for g ∈ e G R w , then fr om the second equation in (3.60) and a direct calculatio n, we ﬁnd th at the Hamiltonian equation generated by H h 0 ,z 0 is giv en b y (5.10), where B ( h ) =Π e b w i ( z 0 I − E ( h 0 )) − 1 E ( h 0 ) h h − h 0 =( i ( z 0 I − E ( h 0 )) − 1 E ( h 0 )) 0 + i Im  i ( z 0 I − E ( h 0 )) − 1 E ( h 0 )  0 − h  i ( z 0 I − E ( h 0 )) − 1 E ( h 0 )  − i ∗ +  i ( z 0 I − E ( h 0 )) − 1 E ( h 0 ) · h h − h 0  ∗ . (5 . 14) Note that in go ing fr om the ﬁr st line of (5.14) to th e second line, we h a ve used (3.12), together with the form u la f or Π b from [L1]. In the next step of the calculat ion, w e will try to rewrite the last term in the ab o ve expr ession in the d esired form, and it is here that th e c hoice of h 0 and z 0 is imp ortan t. T o wit, by using (5.8) , w e ha v e  i ( z 0 I − E ( h 0 )) − 1 E ( h 0 ) · h h − h 0  ∗ =  i ( z 0 E ( h − 1 0 ) ∗ − I ) − 1 · h h − h 0  ∗ = − i ( ¯ z 0 E ( h − 1 0 ) − I ) − 1 · ¯ h ¯ h − h 0 = iz 0 ( z 0 I − E ( h − 1 0 )) − 1 · 1 1 − hh 0 . (5 . 15) THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 37 Hence (5.11) follo ws from (5.14 ) and (5.1 5). (b) T he pro of is similar to (a) and s o w e will skip the d etails.  In our next t wo results, w e will d en ote the Po isson brac k et on the set of p × p Flo q u et C MV matrices induced from {· , ·} J ♯ simply by {· , ·} . Prop osition 5.4. F or j, k = 1 , · · · , p/ 2 − 1 , we have the fol lowing Poisson br acket r elations (a) n I 0 , − Im  φ p/ 2 2 o ( E ) = 1 , n I 0 , − Im  φ k + φ p − k 2 o ( E ) = 0 , n I 0 , − R e  φ k − φ p − k 2 o ( E ) = 0; (b) n R e I j , − Im  φ p/ 2 2 o ( E ) = 0 , n R e I j , − Im  φ k + φ p − k 2 o ( E ) = δ j,p/ 2 − k , n R e I j , − R e  φ k − φ p − k 2 o ( E ) = 0; (c) n Im I j , − Im  φ p/ 2 2 o ( E ) = 0 , n Im I j , − Im  φ k + φ p − k 2 o ( E ) = 0 , n Im I j , − R e  φ k − φ p − k 2 o ( E ) = δ j,p/ 2 − k ; (d) n P , − Im  φ p/ 2 2 o ( E ) = 0 , n P , − Im  φ k + φ p − k 2 o ( E ) = 0 , n P , − R e  φ k − φ p − k 2 o ( E ) = 0 . Pr o of. In order to compute { H h 0 ,z 0 , φ k } ( E ) , it suﬃces to ev aluate it on an op en dense set consisting of Flo quet CMV matrices E ( h ) for which (a) the p oints P j of th e divisor D are distinct, (b) s upp D ∩ { ∂ I ∂ h = 0 } = ∅ , (c) { Q j } p j =1 ∩ su p p D = ∅ , w here Q j = ( h − 1 0 , z j ( h 0 )) ∈ C, j = 1 , · · · , p, (d) { Q j } p j =1 ∩ { ∂ I ∂ h = 0 } = ∅ . So supp ose E ( h ) satisﬁes (a)- (d) ab ov e. Then in the neigh b orho o d of eac h P j , w e can tak e z to b e the lo cal co ord inate and express h in terms of z . Thus in particular, h j = h ( z j ) . Let E ( t ) b e the Hamiltonian ﬂo w generated b y H h 0 ,z 0 and let D ( t ) = P p − 1 j =1 P j ( t ) (where t is small) b e the divisor of p oles in the ﬁnite p art of C of the corresp onding eigen v ector with last comp onen t normalized to 1 , P j ( t ) = ( h j ( t ) , z j ( t )) . Th en { H h 0 ,z 0 , φ k } ( E ) = d dt    | t =0 φ k ( E ( t )) = p − 1 X j =1 z k − 1 j ∂ I ∂ h ( h j , z j ) · dz j ( t ) dt    | t =0 . (5 . 16) T o compute the rate of change of z j ( t ) at t = 0 , consider an eigen vecto r f ( z , t ) for z in a neigh b orho o d of z j suc h th at ( e p − 1 , f ( z j ( t ) , t )) = 0 for sm all v alues of t. Diﬀeren tiate this relation with resp ect to t at t = 0 , we obtain dz j ( t ) dt    t =0 = ( e p − 1 , B ( h j ) f ( z j , 0)) ( e p − 1 , ∂ f ∂ z ( z j , 0)) . 38 L.-C. LI, I. N E NCIU Therefore, on substituting this expression in to (5.16), we ﬁnd { H h 0 ,z 0 , φ k } ( E ) = p − 1 X j =1 z k − 1 j ∂ I ∂ h ( h j , z j ) · ( e p − 1 , B ( h j ) f ( z j , 0)) ( e p − 1 , ∂ f ∂ z ( z j , 0)) . (5 . 17) No w, on using the fact th at the last column of a strictly lo w er triangular matrix is the zero v ector, and ( e p − 1 , Re  i ( z 0 I − E ( h 0 )) − 1 E ( h 0 )) 0 f ( z j , 0)  = 0 , it follo w s from (5.11) th at ( e p − 1 , B ( h j ) f ( z j , 0)) =  e p − 1 ,  i  z 0 I − E ( h 0 )  − 1 E ( h 0 ) + iz 0  z 0 I − E ( h − 1 0 )  − 1 · 1 1 − h j h  f ( z j , 0)  . (5 . 18) Consequent ly , when w e sub stitute (5.18) into (5.17), the result is { H h 0 ,z 0 , φ k } ( E ) = i p − 1 X j =1 Res P j F k + i p − 1 X j =1 Res P j G k , (5 . 19) where F k =  e p − 1 , ( z 0 I − E ( h 0 )) − 1 E ( h 0 ) b v ( z , 0)  ξ k , G k =  e p − 1 , z 0 ( z 0 I − E ( h − 1 0 )) − 1 b v ( z , 0)  ξ k 1 − h 0 h (5 . 20) are meromorp hic 1-forms on C . By a similar calculation, w e also h a ve { J h 0 ,z 0 , φ k } ( E ) = p − 1 X j =1 Res P j F k − p − 1 X j =1 Res P j G k . (5 . 21) No w from [S2], w e kno w that for | z 0 | = 1 , ∆( z 0 ) ∈ R , and so th is implies J h 0 ,z 0 is a constan t indep endent of E ( h ) . Consequen tly , { J h 0 ,z 0 , φ k } ( E ) = 0 and hence it follo ws from (5.21) that P p − 1 j =1 Res P j F k = P p − 1 j =1 Res P j G k . Consequently , { H h 0 ,z 0 , φ k } ( E ) = 2 i p − 1 X j =1 Res P j G k . (5 . 22) Consider the meromorphic 1-form G k . Since ( ∂ I ∂ h ) ± ∼ ∓ 1 as z → 0 , it follo ws that ξ ± k ∼ ∓ z k − 1 dz as z → 0 . On the other hand, we hav e 1 1 − h 0 h + ∼ − P h 0 z p/ 2 , 1 1 − h 0 h − ∼ 1 as z → 0 . (5 . 23) Since the f j ’s are analytic at Q − , it follo ws that G k has no p oles at Q − . A t Q + , the most singular comp onent of b v ( z , 0) is f + 0 ( z ) ∼ 1 α p − 1 ( Q p − 2 i =0 ρ i ) z − p/ 2 . Hence it follo ws from the asymptotics of ξ + k , f + 0 ( z ) and (5.23) that G + k ∼ const. z k − 1 as z → 0 . (5 . 24) THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 39 So again G k has no p oles at Q + . Thus G k has p oles only at D and at Q j , j = 1 , · · · , p as we can similarly c hec k that there are no p oles at P ± . Hence by the residue theorem and (5.22), { H h 0 ,z 0 , φ k } ( E ) = − 2 i p X j =1 Res Q j G k . (5 . 25) T o sim p lify notation, let b ( k ) j = Res Q j G k and put b ( k ) = P p j =1 b ( k ) j . W e ﬁr s t calcu- late b ( k ) j . W e ha ve b ( k ) j = − 1 h 0 · z 0 z 0 − z j ( h 0 ) · z j ( h 0 ) k − 1 ∂ I ∂ h  h − 1 0 , z j ( h 0 )  = 1 h 0 · z 0 z 0 − z j ( h 0 ) · z j ( h 0 ) k − 1 ∂ I ∂ z  h − 1 0 , z j ( h 0 )  . (5 . 26) Therefore, b ( k ) = p X j =1 1 h 0 · z 0 z 0 − z j ( h 0 ) · z j ( h 0 ) k − 1 ∂ I ∂ z  h − 1 0 , z j ( h 0 )  = z 0 h 0 " lim R →∞ Z | z | = R z k − 1 ( z 0 − z ) · I ( h − 1 0 , z ) dz 2 π i + z k − 1 0 I ( h − 1 0 , z 0 ) # = z k 0 h 0 I ( h − 1 0 , z 0 ) = z k 0 det ( z 0 I − E ( h 0 )) (5 . 27) and so ﬁnally we conclude that { H h 0 ,z 0 , φ k } ( E ) = − 2 ib ( k ) = − 2 iz k 0 det ( z 0 I − E ( h 0 )) . (5 . 28) But on the other hand, it follo ws from { J h 0 ,z 0 , φ k } ( E ) = 0 and (5.1) that { H h 0 ,z 0 , φ k } ( E ) = { H h 0 ,z 0 + iJ h 0 ,z 0 , φ k } ( E ) = 1 det ( z 0 I − E ( h 0 ))   p/ 2 X j = − p/ 2 { I j , φ k } ( E ) z j + p/ 2 0 − z p/ 2 0 ( h 0 + h − 1 0 ) { P, φ k } ( E )   . (5 . 29) By equating (5.28) and (5.29) , we conclude that − 2 iz k 0 = p/ 2 X j = p/ 2 { I j , φ k } ( E ) z j + p/ 2 0 − z p/ 2 0 ( h 0 + h − 1 0 ) { P, φ k } ( E ) , 1 ≤ k ≤ p − 1 . (5 . 30) 40 L.-C. LI, I. N E NCIU W e now divide into three cases. Case 1: k ≥ p/ 2 + 1 In this case, all brac k ets are zero except for { I k − p/ 2 , φ k } ( E ) = − 2 i, i.e., { I j , φ k } ( E ) = − 2 iδ j,k − p/ 2 , { ¯ I j , φ k } ( E ) = 0 , 1 ≤ j ≤ p/ 2 − 1 , { P , φ k } ( E ) = 0 , { I 0 , φ k } ( E ) = 0 . (5 . 31) Hence { Re I j , φ k } ( E ) = − iδ j,k − p/ 2 , { Im I j , φ k } ( E ) = − δ j,k − p/ 2 , 1 ≤ j ≤ p/ 2 − 1 , { P , φ k } ( E ) = 0 , { I 0 , φ k } ( E ) = 0 . (5 . 32) Case 2: k ≤ p/ 2 − 1 In this case, we hav e { ¯ I j , φ k } ( E ) = − 2 iδ j,p/ 2 − k , { I j , φ k } ( E ) = 0 , 1 ≤ j ≤ p / 2 − 1 , { P , φ k } ( E ) = 0 , { I 0 , φ k } ( E ) = 0 , (5 . 33) whic h implies { Re I j , φ k } ( E ) = − iδ j,p/ 2 − k , { Im I j , φ k } ( E ) = δ j,p/ 2 − k , 1 ≤ j ≤ p/ 2 − 1 , { P , φ k } ( E ) = 0 , { I 0 , φ k } ( E ) = 0 . (5 . 34) Case 3: k = p/ 2 In this case, all brac k ets are zero except for − 2 i = { I 0 , φ p/ 2 } ( E ) − ( h 0 + h − 1 0 ) { P , φ p/ 2 } ( E ) . (5 . 35 ) Therefore, { Re I j , φ p/ 2 } ( E ) = 0 , { Im I j , φ p/ 2 } ( E ) = 0 , 1 ≤ j ≤ p/ 2 − 1 , { P , φ p/ 2 } ( E ) = 0 , { I 0 , φ p/ 2 } ( E ) = − 2 i. (5 . 36)  Prop osition 5.5. W ith ν deﬁne d as in r elation (5.7), we have the fol lowing Pois- son br acket r elation: n P , Im  ν 4 o ( E ) = 1 . Pr o of. It suﬃces to compute the Poi sson brack ets { H h 0 ,z 0 , ν } ( E ) and { J h 0 ,z 0 , ν } ( E ) on an op en dense set consisting of Floqu et CMV matrices E ( h ) wh ic h satisfy con- ditions (a)-(d) in Prop osition 5.4. Ind eed, by follo wing the same metho d of calcu- lation, we ﬁnd { H h 0 ,z 0 , ν } ( E ) = i p − 1 X j =1 Res P j F + i p − 1 X j =1 Res P j G, (5 . 37) THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 41 and { J h 0 ,z 0 , ν } ( E ) = p − 1 X j =1 Res P j F − p − 1 X j =1 Res P j G = 0 , (5 . 38) where F =  e p − 1 ,  z 0 I − E ( h 0 )  − 1 E ( h 0 ) b v ( z , 0)  ξ m , G =  e p − 1 , z 0  z 0 I − E ( h − 1 0 )  − 1 b v ( z , 0)  ξ m 1 − h 0 h . (5 . 39) Th us { H h 0 ,z 0 , ν } ( E ) = 2 i p − 1 X j =1 Res P j G. (5 . 40) Consider the meromorph ic 1-form G. As z → 0 , it follo ws from the asymp totics of h ± (see Section 4) an d ( ∂ I ∂ h ) ± that ( ξ m ) ± ∼ ∓ 1 P z dz . (5 . 41) Since the f j ’s are analytic at Q − , it follo ws from (5.23) and (5.41) that G has a simple p ole at Q − Indeed, it follo ws f rom Prop ositon 4.11 that Res Q − G = z 0   z 0 I − E ( h − 1 0 )  − 1  p − 1 ,p − 1 P . (5 . 42) In a similar w a y , it follo ws from (5.23), (5.41) and Prop osition 4.12 th at G also has a simp le p ole at Q + and Res Q + G = z 0   z 0 I − E ( h − 1 0 )  − 1  p − 1 , 0 P · ρ p − 1 h 0 α p − 1 . (5 . 43) No w consider the t wo p oint s P ± at inﬁnity and let u = 1 z b e the lo cal co ord inate. Then from (5.23) , (5.41) and Prop osition 4.10, we hav e f + 2 j ( z , 0) ( ξ m ) + 1 − h 0 h + ∼ ¯ α 2 j h 0 Q p − 2 i =2 j ρ i u j du as u → 0 , (5 . 44 ) while f + 2 j +1 ( z , 0) ( ξ m ) + 1 − h 0 h + ∼ − 1 h 0 Q p − 2 i =2 j +1 ρ i u j du as u → 0 . (5 . 4 5) F r om (5.44) and (5.45), w e conclude that G is analytic at P + . Similarly , b y making use of Prop osition 4.7 and (5.23), (5.41), w e ﬁnd that G has a simple p ole at P − with Res P − G = z 0   z 0 I − E ( h − 1 0 )  − 1  p − 1 ,p − 2 P · ρ p − 2 α p − 2 − z 0   z 0 I − E ( h − 1 0 )  − 1  p − 1 ,p − 1 P . (5 . 46) 42 L.-C. LI, I. N E NCIU Since G obvi ously has p oles at the p oin ts of D and at the p oints Q j , j = 1 , · · · , p, it follo ws b y the r esidue theorem, (5.40), (5.42), (5.43) and (5.46) that { H h 0 ,z 0 , ν } ( E ) = − 2 i p X j =1 Res Q j G − 2 iz 0 P h  z 0 I − E ( h − 1 0 )  − 1  p − 1 , 0 · ρ p − 1 h 0 α p − 1 +   z 0 I − E ( h − 1 0 )  − 1  p − 1 ,p − 2 · ρ p − 2 α p − 2 i . (5 . 47) T o compute the s econd term on the righ t-hand side of (5.47), w e in tro duce the follo wing ad-ho c notation: if A is a p × p m atrix and 0 ≤ j 1 ≤ · · · ≤ j l ≤ p − 1, 0 ≤ k 1 ≤ · · · ≤ k l ≤ p − 1 are t wo sets of indices, with l ≥ 1, then we denote b y A [ j 1 , . . . , j l ; k 1 , . . . , k l ] the submatrix obtained from A by deleting ro w s j 1 , ..., j l and columns k 1 , . . . , k l . F or simplicit y of notation, let M = z 0 I − E ( h − 1 0 ) . Then the terms app earing on the right-hand side of (5.47) can b e iden tiﬁed as  M − 1  p − 1 , 0 = det( M [0 , p − 1]) det( M ) and  M − 1  p − 1 ,p − 2 = det( M [ p − 2 , p − 1]) det( M ) . T o pro ceed, w e expand b oth det( M [0 , p − 1]) an d d et( M [ p − 2 , p − 1]) along their 0th columns, which mak es all the minors app earing the calculation h 0 -indep end en t, and so allo ws us to separate the h 0 -dep end ent terms fr om the h 0 -indep end en t ones. This leads to ρ p − 1 h 0 α p − 1 ·  M − 1  p − 1 , 0 + ρ p − 2 α p − 2 ·  M − 1  p − 1 ,p − 2 = m 1 + m 2 , (5 . 48) where the h 0 -dep end ent term is m 1 = 1 det( M )  1 h 0 · ρ 0 ρ p − 1 det  M [0 , 1; 0 , p − 1]  + h 0 · ρ p − 2 ρ p − 1 det  M [ p − 2 , p − 1; 0 , p − 1]   (5 . 49) and m 2 is h 0 -indep end en t. While the minors app earing in m 2 ha ve a structur e whic h cannot b e ea sily simpliﬁed, the minors app earing in m 1 can b e computed explicitely . Indeed, f or eac h 0 ≤ j ≤ p 2 − 1, consider the 2 × 2 blo c k s A j =  − ρ 2 j − 1 ¯ α 2 j z 0 + α 2 j − 1 ¯ α 2 j − ρ 2 j − 1 ρ 2 j α 2 j − 1 ρ 2 j  and ˜ A j =  − ρ 2 j ¯ α 2 j +1 − ρ 2 j ρ 2 j +1 z 0 + α 2 j ¯ α 2 j +1 α 2 j ρ 2 j +1  . Note A j and ˜ A j are, resp ectiv ely , the left and righ t “halv es” of th e 2 × 4 blo c ks app earing in the extended matrix z 0 I − E . In particular, d ir ect inv estigatio n sho ws THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 43 that M [0 , 1; 0 , p − 1] is a blo c k bi-diagonal m atrix, having the blo c ks A 1 , . . . , A p 2 − 1 on the diagonal and ˜ A 1 , . . . , ˜ A p 2 − 2 on the upp er diagonal, while M [ p − 2 , p − 1; 0 , p − 1] has ˜ A 0 , . . . , ˜ A p 2 − 2 on the diagonal and A 1 , . . . , A p 2 − 2 on the lo w er d iagonal. This implies that det  M [0 , 1; 0 , p − 1]  = p 2 − 1 Y j =1 det  A j  = p 2 − 1 Y j =1 z 0 ρ 2 j − 1 ρ 2 j = z p 2 − 1 0 p − 2 Y j =1 ρ j , and det  M [ p − 2 , p − 1; 0 , p − 1]  = p 2 − 2 Y j =0 det  ˜ A j  = p 2 − 2 Y j =0 z 0 ρ 2 j ρ 2 j +1 = z p 2 − 1 0 p − 3 Y j =0 ρ j . Plugging these t wo exp r essions in (5.49) leads to a v ery simple exp ression: m 1 = z p 2 − 1 0 · P det  z 0 I − E ( h − 1 0 )  ·  h 0 + 1 h 0  . (5 . 50) On the other h and, pro ceeding exactly as in the pr o of of Prop osition 5.4, we ﬁnd that p X j =1 Res Q j G = z p/ 2 0  h 0 + 1 h 0  det  z 0 I − E ( h 0 )  . (5 . 51) By com b ining (5.48), (5.50 ), and (5.51) into (5.47), and using (5.38), w e conclud e that { H h 0 ,z 0 + iJ h 0 ,z 0 , ν } ( E ) = 1 det  z 0 I − E ( h 0 )  h − 4 iz p/ 2 0  h 0 + 1 h 0  − 2 i z 0 P · m 2 i . By th e analogue for ν of relation (5.29), together with the fact that m 2 is h 0 - indep en den t, we conclude that { P , ν } ( E ) = 4 i , whic h leads directly to our claim.  R emark 5.6. (a) In the Prop osition ab o ve, w e were unable to obtain the Poi sson brac kets b et ween the I j ’s and ν. Nev ertheless, th e f u nctional in d ep end ence of the in tegrals follo ws by combining Prop osition 5.4 and Prop osition 5.5. (b) Prop osition 5.4 sh ows that the Hamiltonian ﬂo w generated by P do es not giv e rise to non tr ivial motio n on the Jacobi v ariet y of C, this is the r eason for th e in- tro duction of th e algebro-ge ometric v ariable ν. (See also Prop osition 6.2 b elo w in this connection.) Note that in p articular, th is means that the p erio dic defo cusing Ablo witz-Ladik equation is n ot an algebraica lly completely integrable system in the sense of Adler and v an Mo erb eke [AMV]. F or other examples of integ rable systems in v olving sp ectral cur ves which are not algebraically integrable, we refer the reader to [DL T] and [L2]. 44 L.-C. LI, I. N E NCIU 6. Solving the equations via factorization problems. In this section, we will s olve the Hamiltonian equations of m otion generated by the commuting inte grals in Section 5 via factoriza tion problems. W e b egin b y wr iting d own the Hamiltonian equations of motion b y u sing P r op o- sition 3.9, Theorem 3.6 and (5.4). T o d o so, for a map F : g l ( p, C ) − → C , we let ∇ F ( M ) = ( ∂ F /∂ m ij ) denote its gradien t. In wh at follo ws, w e will regard Re I j , Im I j and P as functions of g e , g o , where E ( h ) = g e g o ( h ) , and ˜ E ( h ) = g o ( h ) g e . W e will use the n otation of Prop osition 3.9 (b). As an example, if H = Im I p/ 2 − j , then the corresp onding central fun ction ϕ is giv en by ϕ ( X ) = Im H | h | =1 E j ( X ( h )) dh 2 π ih . Prop osition 6.1. The Hamiltonian e quations of mo tion gener ate d by H = Im I p/ 2 − j , R e I p/ 2 − j , and P ar e given by ˙ g e = (Π ˜ b w ( D ϕ ( E ( h )))) g e − g e (Π ˜ k w ( D ϕ ( ˜ E ( h )))) , ˙ g o ( h ) = (Π ˜ b w ( D ϕ ( ˜ E ( h )))) g o ( h ) − g o ( h )(Π ˜ b w ( D ϕ ( E ( h )))) , (6 . 1) wher e D ϕ ( E ( h )) =      E ( h ) ∇ T E j ( E ( h )) , for H = Im I p/ 2 − j i E ( h ) ∇ T E j ( E ( h )) , for H = R e I p/ 2 − j − ih E ( h ) ∇ T E p/ 2 ( E ( h )) , for H = P , (6 . 2) and similarly f or D ϕ ( ˜ E ( h )) . In (6.1), the parameter h is on the unit circle, ho w ever, we will r emov e this restriction later on. (S ee (6.29 ) b elo w .) Before s olving these equations, let us sp ell out ∇ T E j ( x ) more explicitly . T o do so, observe that ∇ T E j ( x ) (see (5.3) ab ov e) ob ey the recursion relations ∇ T E j +1 ( x ) = x ∇ T E j ( x ) − E j ( x ) I , 0 ≤ j ≤ p, (6 . 3) where by con ven tion ∇ T E p +1 ( x ) ≡ 0 . Sin ce ∇ T E 0 ( x ) = 0 , by solving the recur sion relations backw ards, w e obtain ∇ T E j ( x ) = − j − 1 X i =0 E j − 1 − i ( x ) x i , j = 1 , · · · , p. (6 . 4 ) The equations of motion generated by P are the simplest to solve. Although w e could easily wr ite do wn the solutions of these equations without recourse to Prop osition 6.1 ab o ve, h o wev er, we will d o it by th e prop osition in order to ac hiev e uniformity in our treatment. Prop osition 6.2. The H amiltonian e quations of motion gener ate d by P simplify to ˙ g e = g e Λ 1 − Λ 2 g e ˙ g o ( h ) = g o ( h ) Λ 2 − Λ 1 g o ( h ) (6 . 5) THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 45 wher e Λ 1 = i R e  A 1  p 2  0 = diag (0 , iP , 0 , iP , · · · , 0 , iP ) , (6 . 6) and Λ 2 = i R e  A − 1  p 2  0 = diag ( iP , 0 , iP , 0 , · · · , iP , 0) . (6 . 7) The solutions of (6.5) ar e ther efor e g iven by g e ( t ) = e − t Λ 2 (0) g e (0) e t Λ 1 (0) , g o ( h, t ) = e − t Λ 1 (0) g o ( h, 0) e t Λ 2 (0) . (6 . 8 ) Henc e α j ( t ) = α j (0) e itP (0) , ρ j ( t ) = ρ j (0) , j = 0 , · · · , p − 1 . (6 . 9) Pr o of. According to (6.2) and (6.3), D ϕ ( E ( h )) = ih p 2 − 1 X j =0 E j ( E ( h )) E ( h ) p 2 − j . (6 . 10) F or j = 1 , · · · , p 2 − 1 , it follo ws from (2.11) that Π ˜ b w  ihE j ( E ( h )) E ( h ) p 2 − j  = ihE j ( E ( h )) E ( h ) p 2 − j . (6 . 11) Similarly , by u s ing the f act that A − 1  p 2  is low er triangular, we ﬁnd Π ˜ b w  ihE 0 ( E ( h )) E ( h ) p 2  = ih E ( h ) p 2 − i Im  iA − 1  p 2  0 (6 . 12) as E 0 ≡ 1 . Therefore, on using (6.11) and (6.12), w e obtain Π ˜ b w D ϕ ( E ( h )) = D ϕ ( E ( h )) − i Re  A − 1  p 2  0 . (6 , 13) By a similar calculation, w e ﬁn d Π ˜ b w D ϕ ( e E ( h )) = D ϕ ( e E ( h )) − i Re  A 1  p 2  0 . (6 . 14) Hence, on substituting (6.13) and (6.14) into (6.1), an d usin g the obvio us facts that E ( h ) p 2 − j g e = g e e E ( h ) p 2 − j , g o ( h ) E ( h ) p 2 − j = e E ( h ) p 2 − j g o ( h ) , we obtain th e equations in (6.5). Th e form ulas in (6.6) and (6.7) for Λ 1 and Λ 2 in terms of P then follo w from Lemma 2.2 and (2.19). As P is a conserved q u an tit y , it is easy to ve rify that the expr essions in (6.8) giv e solutions to the equations in (6.5). Finally the s olution form ulas in (6.9) are obtained from (6.8) by multiplying ou t.  T o s olve the Hamiltonian equations generated by H = Im I p/ 2 − j , Re I p/ 2 − j , we will mak e use of Pr op osition 3.9 (c), w hic h means w e h a ve to solv e explicitly for eac h t > 0 the follo win g factorizat ion problems e tDϕ ( E ( h, 0)) = k 1 ( h, t ) b 1 ( h, t ) − 1 , e tDϕ ( ˜ E ( h, 0)) = k 2 ( h, t ) b 2 ( h, t ) − 1 (6 . 15) 46 L.-C. LI, I. N E NCIU for k i ( · , t ) ∈ ˜ K w , b i ( · , t ) ∈ ˜ B w , i = 1 , 2 . Ho w ever, from the d eﬁnition of ˜ K w , it is easy to sho w that it is enough to solv e e − tDϕ ( E ( h, 0)) e − tDϕ ( E ( h, 0)) ∗ = b 1 ( h, t ) b 1 ( h, t ) ∗ , e − tDϕ ( ˜ E ( h, 0)) e − tDϕ ( ˜ E ( h, 0)) ∗ = b 2 ( h, t ) b 2 ( h, t ) ∗ (6 . 16) for b 1 ( · , t ) , b 2 ( · , h ) ∈ ˜ B w . Note that for i = 1 , 2 , b i ( · , t ) (resp. b i ( · , t ) ∗ ) can b e ex- tended analytically in the in terior (resp. exterior) of the unit circle | h | = 1 . S o (6.16) is a Riemann-Hilb ert p roblem. I n order to solv e this problem explicitly , we will ﬁrs t transform th e pro duct on the left hand side of (6.1 6) in to a f orm which mak es the problem more tractable. T o this end, note th at it f ollo ws f r om (6.4) th at ∇ T E j ( E ( h, 0)) = − j − 1 X i =0 E j − 1 − i ( E ( h, 0)) E ( h, 0) i , j = 1 , · · · , p. (6 . 17) As E ( h, 0) comm utes with E ( h, 0) ∗ for h ∈ ∂ D , it follo ws from (6.17) and (6.2) that D ϕ ( E ( h, 0)) commutes with D ϕ ( E ( h, 0)) ∗ for h ∈ ∂ D . In a similar w a y , w e see that D ϕ ( ˜ E ( h, 0)) comm utes with D ϕ ( ˜ E ( h, 0)) ∗ . C onsequen tly , we can rewr ite (6.16) as e − t ( Dϕ ( E ( h, 0))+ D ϕ ( E ( h, 0)) ∗ ) = b 1 ( h, t ) b 1 ( h, t ) ∗ , e − t ( Dϕ ( ˜ E ( h, 0))+ D ϕ ( ˜ E ( h, 0)) ∗ ) = b 2 ( h, t ) b 2 ( h, t ) ∗ , h ∈ ∂ D . (6 . 18) No w if we compute D ϕ ( E ( h, 0)) ∗ (resp. D ϕ ( ˜ E ( h, 0))) more carefully , w e ﬁnd D ϕ ( E ( h, 0)) ∗ = ± D ϕ ( E ( h, 0) − 1 ) , D ϕ ( ˜ E ( h, 0)) ∗ = ± D ϕ ( ˜ E ( h, 0) − 1 ) , h ∈ ∂ D , (6 . 19) where w e pic k the + sign for H = Im I p/ 2 − j and the − sign for H = Re I p/ 2 − j . Hence (6.18) b ecomes e − t ( Dϕ ( E ( h, 0)) ± D ϕ ( E ( h, 0) − 1 )) = b 1 ( h, t ) b 1 ( h, t ) ∗ , e − t ( Dϕ ( ˜ E ( h, 0)) ± Dϕ ( ˜ E ( h, 0) − 1 )) = b 2 ( h, t ) b 2 ( h, t ) ∗ , h ∈ ∂ D , (6 . 20) where the c hoice of sign is describ ed in the p revious sentence. As the explicit solution of the factorization problem in (6.20) will inv olv e con- structing b i ( h, t ) , i = 1 , 2 , for v alues of h not on unit circle, it is necessary to in tro duce some Lie algebras and pro j ection op erators w h ic h complements those in Section 3. F or this pu rp ose, let A b e the ring of Laurent p olynomials in the v ari- able h an d let g l p ( A ) b e the Lie algebra of p × p matrix fun ctions with entries in A equipp ed w ith the p oin t wise Lie brac ket. W e will consider the follo wing Lie subalgebras of g l p ( A ): e b =    n X j =0 X j h j ∈ g l p ( C [ h ]) | X 0 ∈ b    , e k =  X ∈ g l p ( A ) | X ( h ) + X ( ¯ h − 1 ) ∗ = 0  . (6 . 21) THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 47 Then analogous to (3.8), we hav e the sp litting g l p ( A ) = e k ⊕ e b . (6 . 22) F or X ∈ g l p ( A ) , d eﬁne (Π + X )( h ) = X j > 0 X j h j , (Π − X )( h ) = X j < 0 X j h j , (6 . 23) then the pr o jection op erator onto e b associated with the splitting in (6.22) is give n b y (Π e b X )( h ) = (Π + X )( h ) + Π b X 0 + ((Π − X )( ¯ h − 1 )) ∗ . (6 . 2 4) Theorem 6.3. F or H = R e I p/ 2 − j , Im I p/ 2 − j , ther e exist unique ho lomorp hic matrix- value d functions b 1 ( · , t ) : CP 1 \ {∞} − → GL ( p, R ) , b 2 ( · , t ) : CP 1 \ {∞} − → GL ( p, R ) (6 . 25) which ar e smo oth in t , solve the factorizatio n pr oblems e − t ( Dϕ ( E ( h, 0)) ± D ϕ ( E ( h, 0) − 1 )) = b 1 ( h, t ) b 1 ( ¯ h − 1 , t ) ∗ , e − t ( Dϕ ( ˜ E ( h, 0)) ± D ϕ ( ˜ E ( h, 0) − 1 )) = b 2 ( h, t ) b 2 ( ¯ h − 1 , t ) ∗ , h ∈ CP 1 \ { 0 , ∞} (6 . 26) (wher e the + sign c orr esp onds to H = Im I p/ 2 − j and the − sign c orr esp onds to H = R e I p/ 2 − j ) and satisfy b i (0 , t ) ∈ B , b i ( h, t ) − 1 ˙ b i ( h, t ) ∈ Im Π e b i = 1 , 2 . (6 . 27) Mor e over, for h ∈ CP 1 \ { 0 , ∞} , the formulas g e ( t ) = b 1 ( h, t ) − 1 g e (0) b 2 ( h, t ) = b 1 ( ¯ h − 1 , t ) ∗ g e (0)( b 2 ( ¯ h − 1 , t ) ∗ ) − 1 , g o ( h, t ) = b 2 ( h, t ) − 1 g o ( h, 0) b 1 ( h, t ) = b 2 ( ¯ h − 1 , t ) ∗ g o ( h, 0)( b 1 ( ¯ h − 1 , t ) ∗ ) − 1 (6 . 28) give solutions of the e quations ˙ g e = (Π e b D ϕ ( E ( · )))( h ) g e − g e (Π e b D ϕ ( e E ( · )))( h ) , ˙ g o ( h ) = (Π e b D ϕ ( e E ( · )))( h ) g o ( h ) − g o ( h )(Π e b D ϕ ( E ( · )))( h ) . (6 . 29) Final ly, for generic initial data g e (0) and g o ( h, 0) , b 1 ( h, t ) and b 2 ( h, t ) c an b e c on- structe d by me ans of theta functions asso ciate d with the Riemann surfac e of the sp e ctr al curve C = { ( h, z ) | det ( z I − g e (0) g o ( h, 0)) = 0 } for v alues of t > 0 for which α j ( t ) 6 = 0 , j = 0 , · · · , p − 1 . 48 L.-C. LI, I. N E NCIU Pr o of. W e will prov e the r esult f or H = Im I p/ 2 − j . Th e argum en t for the other case is similar. W e start with un iqueness of the facto rs b i ( h, t ) , i = 1 , 2 . T o pro ve this, supp ose b 0 i ( h, t ) , i = 1 , 2 is a second pair of solutions of the factorizat ion problem. Then from b i ( h, t ) b i ( ¯ h − 1 , t ) ∗ = b 0 i ( h, t ) b 0 i ( ¯ h − 1 , t ) ∗ , we ha ve g i ( h, t ) ≡ b 0 i ( h, t ) − 1 b i ( h, t ) = b 0 i ( ¯ h − 1 , t ) ∗ ( b i ( ¯ h − 1 , t ) ∗ ) − 1 , h ∈ CP 1 \ { 0 , ∞} . (6 . 30) Clearly the fun ction g i ( · , t ) deﬁ ned in (6.30) ab o ve can b e extended to an analytic function ev erywhere, hence by Liouville’s theorem, g i ( h, t ) = c i ( t ) . T o determine c i ( t ) , note that g i (0 , t ) = b 0 i (0 , t ) − 1 b i (0 , t ) ∈ b . On the other h and, g i ( ∞ , t ) = ( g i (0 , t ) − 1 ) ∗ is u pp er triangular with p ositiv e diagonal entries on the diagonal. Hence c i ( t ) ≡ I . T o establish the existence of b i ( h, t ) , i = 1 , 2 , note that E ( h, t ) (resp. e E ( h, t )) exists for h ∈ CP \ { 0 , ∞} since it exists f or h ∈ ∂ D . Hence w e can obtain b i ( h, t ) , i = 1 , 2 as solutions of the equations ˙ b 1 ( h, t ) = − b 1 ( h, t )(Π b D ϕ ( E ( · , t )))( h ) , (6 . 31) and ˙ b 2 ( h, t ) = − b 2 ( h, t )(Π b D ϕ ( e E ( h, t )))( h ) . (6 . 32) Clearly , the analyticit y pr op erties and (6.27) are satisﬁed by deﬁnition. W e next consider the pro d uct b 1 ( h, t ) b 1 ( ¯ h − 1 , t ) ∗ . By diﬀeren tiating, we ha ve d dt ( b 1 ( h, t ) b 1 ( ¯ h − 1 , t ) ∗ ) = − b 1 ( h, t )  (Π e b D ϕ ( E ( · , t )))( h ) + ((Π e b D ϕ ( E ( · , t )))( ¯ h − 1 )) ∗  b 1 ( ¯ h − 1 , t ) ∗ . (6 . 33) No w, from th e d eﬁnition of Π e b , it is straightfo rw ard to c hec k that (Π e b X )( h ) + ((Π e b X )( ¯ h − 1 )) ∗ = X ( h ) + ( X ( ¯ h − 1 )) ∗ . (6 . 3 4) Consequent ly , we obtain (Π e b D ϕ ( E ( · , t )))( h ) + ((Π e b D ϕ ( E ( · , t )))( ¯ h − 1 )) ∗ = D ϕ ( E ( h, t )) + ( D ϕ ( E ( ¯ h − 1 , t ))) ∗ = D ϕ ( E ( h, t )) + D ϕ ( E ( h, t ) − 1 ) . (6 . 35) Substitution of (6.35) in to (6.33 ) therefore giv es the r elation d dt ( b 1 ( h, t ) b 1 ( ¯ h − 1 , t ) ∗ ) = − b 1 ( h, t )( Dϕ ( E ( h, t )) + D ϕ ( E ( h, t ) − 1 )) b 1 ( ¯ h − 1 , t ) ∗ = − b 1 ( h, t ) b 1 ( ¯ h − 1 , t ) ∗  D ϕ ( F ( h, t )) + D ϕ ( F ( h, t ) − 1 )  (6 . 36) THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 49 where we hav e used the f act that ϕ is a cen tral fun ction, and wher e F ( h, t ) = ( b 1 ( ¯ h − 1 , t ) ∗ ) − 1 E ( h, t ) b 1 ( ¯ h − 1 , t ) ∗ . No w, by d irect diﬀerentia tion, u sing (6.31), the equation ˙ E ( h, t ) = (Π e b D ϕ ( E ( · , t )))( h ) E ( h, t ) − E ( h, t )(Π e b D ϕ ( E ( · , t )))( h ) , (6 . 37) and (6.35), w e ﬁnd that d dt F ( h, t ) = ( b 1 ( ¯ h − 1 , t ) ∗ ) − 1 ( D ϕ ( E ( h, t )) + D ϕ ( E ( h, t ) − 1 )) E ( h, t ) b 1 ( ¯ h − 1 , t ) ∗ − ( b 1 ( ¯ h − 1 , t ) ∗ ) − 1 E ( h, t )( D ϕ ( E ( h, t )) + D ϕ ( E ( h, t ) − 1 )) b 1 ( ¯ h − 1 , t ) ∗ =0 . (6 . 38) Therefore, F ( h, t ) = E ( h, 0) and so (6.36) b ecomes d dt ( b 1 ( h, t ) b 1 ( ¯ h − 1 , t ) ∗ ) = − b 1 ( h, t ) b 1 ( ¯ h − 1 , t ) ∗  D ϕ ( E ( h, 0)) + D ϕ ( E ( h, 0 − 1 ))  . (6 . 39) This shows b 1 ( h, t ) satisﬁes the ﬁ rst relation in (6.26). In a similar fashion, w e can sho w that b 2 ( h, t ) satisﬁes the second relation in (6.26). Finally , we w ill sho w that g e ( t ) and g o ( h, t ) as deﬁn ed in (6.28) satisfy (6.29). First, note that by u sing the relatio n e E ( h, 0) = g e (0) − 1 E ( h, 0) g e (0) and (6.26) , w e ha ve b 1 ( h, t ) − 1 g e (0) b 2 ( h, t ) = b 1 ( ¯ h − 1 , t ) ∗ g e (0)( b 2 ( ¯ h − 1 , t ) ∗ ) − 1 , b 2 ( h, t ) − 1 g o ( h, 0) b 1 ( h, t ) = b 2 ( ¯ h − 1 , t ) ∗ g o ( h, 0)( b 1 ( ¯ h − 1 , t ) ∗ ) − 1 . (6 . 40) Diﬀeren tiate g e ( t ) = b 1 ( h, t ) − 1 g e ( o ) b 2 ( h, t ) and g o ( h, t ) = b 2 ( h, t ) − 1 g o ( h, 0) b 1 ( h, t ) with resp ect to t, we ﬁnd ˙ g e ( t ) = − b 1 ( h, t ) − 1 ˙ b 1 ( h, t ) g e ( t ) + g e ( t ) b 2 ( h, t ) − 1 ˙ b 2 ( h, t ) , ˙ g o ( h, t ) = − b 2 ( h, t ) − 1 ˙ b 2 ( h, t ) g o ( h, t ) + g o ( h, t ) b 1 ( h, t ) − 1 ˙ b 1 ( h, t ) . (6 . 41) On the other h and, by d iﬀeren tiating the ﬁrst relation in (6.26 ) with r esp ect to t, and m u ltiply the r esulting expression on the left by b 1 ( h, t ) − 1 and on th e r igh t by ( b 1 ( ¯ h − 1 , t ) ∗ ) − 1 , we obtain − b 1 ( ¯ h − 1 , t ) ∗ ( D ϕ ( E ( h, 0)) + D ϕ ( E ( h, 0) − 1 ))( b 1 ( ¯ h − 1 , t ) ∗ ) − 1 = b 1 ( h, t ) − 1 ˙ b 1 ( h, t ) + ˙ b 1 ( ¯ h − 1 , t ) ∗ ( b 1 ( ¯ h − 1 , t ) ∗ ) − 1 . (6 . 42) But f rom the fact that ϕ is a cen tr al function and (6.40) , we see that b 1 ( ¯ h − 1 , t ) ∗ ( D ϕ ( E ( h, 0)) + D ϕ ( E ( h, 0) − 1 ))( b 1 ( ¯ h − 1 , t ) ∗ ) − 1 = D ϕ ( E ( h, t )) + D ϕ ( E ( h, t ) − 1 ) (6 . 43) 50 L.-C. LI, I. N E NCIU where E ( h, t ) = g e ( t ) g o ( h, t ) . Th u s (6.42) b ecomes − ( D ϕ ( E ( h, t )) + D ϕ ( E ( h, t ) − 1 )) = b 1 ( h, t ) − 1 ˙ b 1 ( h, t ) + ( b 1 ( ¯ h − 1 , t ) − 1 ˙ b 1 ( ¯ h − 1 , t )) ∗ . (6 . 44) No w let X ( h, t ) = b 1 ( h, t ) − 1 ˙ b 1 ( h, t ) − ( b 1 ( ¯ h − 1 , t ) − 1 ˙ b 1 ( ¯ h − 1 , t )) ∗ , Y ( h, t ) = D ϕ ( E ( h, t )) − D ϕ ( E ( ¯ h − 1 , t ) − 1 ) . (6 . 45) Clearly , X ( · , t ) ∈ e k , th us Π e b X ( · , t ) = 0 . On the other hand, as Dϕ ( E ( h, t ) − 1 ) = D ϕ ( E ( ¯ h − 1 , t )) ∗ , we also hav e Y ( · , t ) ∈ e k and Π e b Y ( · , t ) = 0 . Consequen tly , when we apply Π e b to b oth sides of (6.44), w e obtain b 1 ( h, t ) − 1 ˙ b 1 ( h, t ) = − (Π e b D ϕ ( E ( · , t )))( h ) . (6 . 46) Similarly , f rom the second relation in (6.26), we can show th at b 2 ( h, t ) − 1 ˙ b 2 ( h, t ) = − (Π e b D ϕ ( e E ( · , t )))( h ) . (6 . 47 ) Finally , substituting (6.46) and (6.47) in to (6.41) gives (6.2 9). This completes the pro of of the theo rem m o dulo the assertion o n the construction of b 1 ( h, t ) and b 2 ( h, t ) via Riemann theta fun ctions.  W e no w turn to th e construction of b 1 ( h, t ) and b 2 ( h, t ) via theta functions. Again, w e will giv e deta ils for H = Im I p/ 2 − j , lea ving the other case to the in terested reader. T h e follo win g prop osition sho ws we can construct b 2 ( h, t ) f r om b 1 ( h, t ) and the s olution of a ﬁnite dimen sional factorization problem. Prop osition 6.4. L et l ( t ) ∈ B , u ( t ) ∈ K b e the solution of the factorization pr oblem b 1 (0 , t ) − 1 g e (0) = u ( t ) l ( t ) − 1 . (6 . 48) Then b 2 ( h, t ) = g e (0) ∗ b 1 ( h, t ) u ( t ) . (6 . 49) Pr o of. Sin ce e E ( h, 0) = g e (0) − 1 E ( h, 0) g e (0) , the factorizatio n pr oblem for b 2 ( h, t ) in (6.26) can b e r ewritten as e − t ( Dϕ ( E ( h, 0))+ D ϕ ( E ( h, 0) − 1 )) = g e (0) b 2 ( h, t ) b 2 ( ¯ h − 1 , t ) ∗ ( g e (0)) − 1 . (6 . 50) Therefore, wh en we compare this with th e ﬁrst relation in (6.26), we obtain b 2 ( h, t ) b 2 ( ¯ h − 1 , t ) ∗ = g e (0) ∗ b 1 ( h, t )( g e (0) ∗ b 1 ( ¯ h − 1 , t )) ∗ . ( 6 . 51) No w let l ( t ) ∈ B , u ( t ) ∈ K b e the solution of the factorizatio n pr oblem in (6.48). Then g e (0) ∗ b 1 ( h, t ) = ( l ( t ) + O ( h )) u ( t ) ∗ (6 . 52) THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 51 and g e (0) ∗ b 1 ( ¯ h − 1 , t ) = ( l ( t ) + O ( ¯ h − 1 )) u ( t ) ∗ . (6 . 5 3) Substitute (6.52) and (6.53) int o (6.51), w e obtain b 2 ( h, t ) b 2 ( ¯ h − 1 , t ) = ( l ( t ) + O ( h ))( l ( t ) ∗ + O ( h − 1 )) . (6 . 54) Therefore, from the un iqueness of solution of the factoriza tion problem (Theorem 6.3), we conclude that b 2 ( h, t ) = l ( t ) + O ( h ) = g e (0) ∗ b 1 ( h, t ) u ( t ) .  T o constru ct b 1 ( h, t ) , w e inv ok e the formula in (6.2) and (6.17), according to whic h we ha ve ( D ϕ ( E ( h ( P ) , 0)) + D ϕ ( E ( h ( P ) , 0) − 1 )) b v ( P ) = µ j ( P ) b v ( P ) (6 . 55) where µ j is meromorphic on the h yp erelliptic Riemann surface C . F rom the ﬁrst relation in (6.26), e − tµ j ( P ) b 1 ( h ( P ) , t ) − 1 b v ( P ) = b 1 ( h ( P ) , t ) − 1 e − t ( Dϕ ( E ( h ( P ) , 0))+ Dϕ ( E ( h ( P ) , 0) − 1 )) b v ( P ) = b 1 ( h ( P ) − 1 , t ) ∗ b v ( P ) (6 . 56) for h ( P ) ∈ C P 1 \ { 0 , ∞} . Since E ( h, t ) = b 1 ( h, t ) − 1 E ( h, 0) b 1 ( h, t ) , w e ha ve E ( h ( P ) , t ) b 1 ( h ( P ) , t ) − 1 b v ( P ) = z ( P ) b 1 ( h ( P ) , t ) − 1 b v ( P ) . (6 . 57) On th e other hand, as E ( h ( P ) , t ) =( E ( h ( P ) − 1 , t ) ∗ ) − 1 = b 1 ( h ( P , t ) ∗ E ( h ( P ) , 0)( b 1 ( h ( P ) − 1 , t ) ∗ ) − 1 , (6 . 58) w e also ha ve E ( h ( P ) , t ) b 1 ( h ( P ) − 1 , t ) ∗ b v ( P ) = z ( P ) b 1 ( h ( P ) − 1 , t ) ∗ b v ( P ) . (6 . 59) Th us if w e let v t + ( P ) = b 1 ( h ( P ) , t ) − 1 b v ( P ) , v t − ( P ) = b 1 ( h ( P ) − 1 , t ) ∗ b v ( P ) , (6 . 60) then (6.56), (6.57) and (6.59) give e − tµ j ( P ) v t + ( P ) = v t − ( P ) , h ( P ) ∈ CP 1 \ { 0 , ∞} E ( h ( P ) , t ) v t ± ( P ) = z ( P ) v t ± ( P ) . (6 . 61) 52 L.-C. LI, I. N E NCIU In this w a y , we are led to scalar factorization problems. Note that b ecause b 1 (0 , t ) is lo w er triangular, it follo ws from (4.50) and (6.60) that (( v t + ) 2 j ) ≥ − D (0) +  p 2 − j − 1  P − +  p 2 − j  Q − (( v t + ) 2 j +1 ) ≥ − D (0) +  p 2 − j − 1  P − +  p 2 − j − 1  Q − (6 . 62) on C \ { h ( P ) = ∞} . Sim ilarly , b ecause b 1 (0 , t ) ∗ is u pp er triangular, we ﬁnd that (( v t − ) 2 j ) ≥ − D (0) −  p 2 − j − 1  P + −  p 2 − j  Q + (( v t − ) 2 j +1 ) ≥ − D (0) −  p 2 − j − 1  P + −  p 2 − j − 1  Q + (6 . 63) on C \ { h ( P ) = 0 } . W e will ﬁrst solve the follo wing scalar factorization problem (cf. [RST S], [DL]) e − tµ j ( P ) ω t + ( P ) = ω t − ( P ) , P ∈ C \ { h ( P ) = 0 , ∞} , ( ω t + ) ≥ − D (0) , on C \ { h ( P ) = ∞} , ( ω t − ) ≥ − D (0) , on C \ { h ( P ) = 0 } . (6 . 64) T o do so, w e ﬁx a canonical homology b asis { a j , b k } 1 ≤ j,k ≤ g of the Riemann surface asso ciated with th e sp ectral curve, and let { ω i } 1 ≤ i ≤ g b e a cohomology b asis dual to { a j , b k } , i.e., Z a j ω i = δ ij , Z b j ω i = Ω ij , (6 . 65) where (Ω ij ) is the Riemann matrix. Let θ ( z ) = θ ( z , Ω ) = X m ∈ Z g exp { 2 π i ( m, z ) + π i ( m, Ω m ) } (6 . 66) b e th e Riemann theta function asso ciated with the matrix Ω . Let ω = ( ω 1 , · · · , ω g ) . Cho ose a nonsin gular e ∈ C g in the theta divisor, i.e. θ ( e ) = 0 , the prim e form E e ( x, y ) ≡ θ  e + R y x ω  6≡ 0 with the additional p rop erty that E e ( P ± , P ) are not iden tically zero in P . (See Lemma 3.3 of [M].) Let P 0 b e a ﬁxed p oin t on th e ﬁnite part of the Riemann su rface, then by Corollary 3.6 of [M], there exists an eﬀectiv e divisor D g − 1 of degree g − 1 su c h that e = ∆ − R D g − 1 ( g − 1) P 0 ω , where ∆ is the v ector of Riemann constants. Also note that θ e + Z D g − 1 + P D (0) ω ! = θ ∆ + Z ( g − 1) P 0 + P D (0) ω ! = 0 ⇐ ⇒ P ∈ D (0) . (6 . 67) No w let dφ + b e the un ique meromorphic diﬀeren tial of th e second kind with v an- ishing a -p erio d s with p oles only at { h ( P ) = ∞} such that ( dφ + − dµ j )( P ) is regular in C \ { h ( P ) = 0 } . Set ω t + ( P ) = exp ( t φ + ( P )) θ  e + R D g − 1 + P D (0) ω + t Φ +  θ  e + R D g − 1 + P D (0) ω  , (6 . 68 ) THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 53 where φ + ( P ) = R P P 0 dφ + and Φ + is the v ector of b -p erio ds of dφ + , i.e., (Φ + ) j = 1 2 π i Z b j dφ + , j = 1 , · · · , g . (6 . 69) By u sing (6.69), th e b asic p r op erty θ ( z + γ ′ + Ω γ ) = θ ( z ) exp  2 π i ( − ( γ , z ) − 1 2 ( γ , Ω γ ))  , and (6.67), it is s tr aigh tforward to chec k that ω t + is single-v alued and meromorp hic in C \ { h ( P ) = ∞} with ( ω t + ) ≥ − D (0) there. Moreo v er, e − tµ j ( P ) ω t + ( P ) is mero- morphic in C \ { h ( P ) = 0 } with ( e − tµ j ω t + ) ≥ − D (0) b ecause dφ + ( P ) − dµ j ( P ) is regular in C \{ h ( P ) = 0 } . T h us ω t + and ω t − ≡ e − tµ j ω t + solv es the scalar factorization problem (6.64). Next, w e compare (6.64) and (6.61), this give s ω t + ( P ) − 1 v t + ( P ) = ω t − ( P ) − 1 v t − ( P ) , C \ { h ( P ) = 0 , ∞} . ( 6 . 70) Therefore, e v t ( P ) ≡  ω t + ( P ) − 1 v t + ( P ) , P ∈ C \ { h ( P ) = ∞} ω t − ( P ) − 1 v t − ( P ) , P ∈ C \ { h ( P ) = 0 } (6 . 71) is meromorphic on C. Moreo ver, it follo ws from (6.62), (6.63) and th e expressions for ω t ± that ( e v t 2 j ) ≥ − e D ( t ) −  p 2 − j − 1  P + +  p 2 − j − 1  P − −  p 2 − j  Q + +  p 2 − j  Q − , ( e v t 2 j +1 ) ≥ − e D ( t ) −  p 2 − j − 1  P + +  p 2 − j − 1  P − −  p 2 − j − 1  Q + +  p 2 − j − 1  Q − , (6 . 72) where e D ( t ) is the divisor of zeros of the function θ e + Z D g − 1 + P D (0) ω + t Φ + ! . (6 . 73) Since E ( h ( P ) , t ) v t ± ( P ) = z ( P ) v t ± ( P ) , it follo ws from the d eﬁ nition of e v t ( P ) that E ( h ( P ) , t ) e v t ( P ) = z ( P ) e v t ( P ) . (6 . 74) But on the other hand, E ( h ( P ) , t ) b v ( t, P ) = z ( P ) b v ( t, P ) , (6 . 75 ) 54 L.-C. LI, I. N E NCIU where the last comp onen t of b v ( t, P ) is equal to 1 and ( b v 2 j ( t, · )) ≥ − D ( t ) −  p 2 − j − 1  P + +  p 2 − j − 1  P − −  p 2 − j  Q + +  p 2 − j  Q − , ( b v 2 j +1 ( t, · )) ≥ − D ( t ) −  p 2 − j − 1  P + +  p 2 − j − 1  P − −  p 2 − j − 1  Q + +  p 2 − j − 1  Q − . (6 . 76) Clearly , e v t ( P ) = e v t p − 1 ( P ) b v ( t, P ) . Let e D 0 ( t ) b e the divisor of zeros of e v t p − 1 ( P ) so that ( e v t p − 1 ) = e D 0 ( t ) − e D ( t ) . (6 . 77) Then it follo ws from th e relation connecting e v t ( P ) and b v ( t, P ) ab o ve, (6.72) and (6.77) th at ( b v 2 j ( t, · )) = ( e v t 2 j ) − ( e v t p − 1 ) ≥ − e D 0 ( t ) −  p 2 − j − 1  P + +  p 2 − j − 1  P − −  p 2 − j  Q + +  p 2 − j  Q − . (6 . 78) Similarly , ( b v 2 j +1 ( t, · )) ≥ − e D 0 ( t ) −  p 2 − j − 1  P + +  p 2 − j − 1  P − −  p 2 − j − 1  Q + +  p 2 − j − 1  Q − . (6 . 79) Clearly , w e m ust ha ve e D 0 ( t ) ≥ D ( t ) . Sin ce deg D ( t ) = deg e D 0 ( t ) = g , w e m u st ha ve e D 0 ( t ) = D ( t ) . But D ( t ) is a general divisor b y Prop osition 4.12, so from ( e v t p − 1 ) = e D 0 ( t ) − e D ( t ) = D ( t ) − e D ( t ) , we must ha ve e v t p − 1 = constan t and this implies e D ( t ) = D ( t ) . Thus we can solv e for e v t ( P ) and hen ce v t ± ( P ) u p to multi ples b ecause of P rop osition 4.12. Indeed, by making use of the prime form, w e can write do w n the explicit exp r ession e v t 2 j ( P ) = c 2 j ( t )  E e ( P − , P ) E e ( P + , P )  p 2 − j − 1 ·  E e ( Q − , P ) E e ( Q + , P )  p 2 − j × θ  e + R D g − 1 + P + ( p 2 − j − 1 ) P − + ( p 2 − j ) Q − D ( t )+ ( p 2 − j − 1 ) P + + ( p 2 − j ) Q + ω  θ  e + R D g − 1 + P D ( t ) ω  , (6 . 80) where c 2 j ( t ) h as to b e determined . Similarly , w e hav e e v t 2 j +1 ( P ) = c 2 j +1 ( t )  E e ( P − , P ) E e ( P + , P )  p 2 − j − 1 ·  E e ( Q − , P ) E e ( Q + , P )  p 2 − j − 1 × θ  e + R D g − 1 + P + ( p 2 − j − 1 ) P − + ( p 2 − j − 1 ) Q − D ( t )+ ( p 2 − j − 1 ) P + + ( p 2 − j − 1 ) Q + ω  θ  e + R D g − 1 + P D ( t ) ω  , (6 . 81) THE P ERIODIC AL EQUA TION AND FLOQUET CMV MA T R ICES 55 where c 2 j +1 ( t ) is also as y et un d etermined. W e are now ready to construct b 1 ( h, t ) an d in the p ro cess, we will also determine c j ( t ) . F or give n h ∈ C P 1 whic h is n ot a b ranc h p oin t of the coord inate function h ( P ) , there exist p p oints P 0 ( h ) , · · · , P p − 1 ( h ) of the Riemann surface C lying o ver h. Th erefore we can d eﬁne the matrices V ± ( h, t ) = ( v t ± ( P 0 ( h )) · · · v t ± ( P p − 1 ( h ))) b V θ ( h ) = ( b v ( P 0 ( h )) · · · b v ( P p − 1 ( h ))) (6 . 82) where b v ( P ) can b e obtained f rom the formula for e v t ( P ) by setting t = 0 (since ω t =0 ± ( P ) ≡ 1 and b 1 ( h, t = 0) = I ) and so can b e computed in terms of theta functions. Of course, v t ± ( P ) are also in terms of theta functions. With these matrices, it follo ws that b 1 ( h, t ) = b V θ ( h ) V + ( h, t ) − 1 . (6 . 83) Of course, v t ± ( P ) = ω t ± ( P ) e v t ( P ) are determined only up to the c j ( t )’s. W rite v t ± ( P ) = c ( t ) v θ ± ( t, P ) (6 . 84) where v θ ± ( t, P ) are kno wn and c ( t ) = d iag ( c 0 ( t ) , · · · , c p − 1 ( t )) is to b e determined. Then V ± ( h, t ) = c ( t ) V θ ± ( h, t ) ( 6 . 85) where V θ ± ( h, t ) = ( v θ ± ( t, P 0 ( h )) · · · v θ ± ( t, P p − 1 ( h ))) . With these d eﬁnitions, b 1 ( h, t ) = b V θ ( h ) V θ + ( h, t ) − 1 c ( t ) − 1 . (6 . 86) As b 1 ( h, 0) = I , the ab o ve relation determines c (0) via the f orm ula c (0) = b V θ ( h ) V θ + ( h, 0) − 1 . (6 . 87) T o d etermine c ( t ) for t > 0 , w e br ing in k 1 ( h, t ) = e tDϕ ( E ( h, 0)) b 1 ( h, t ) = e tDϕ ( E ( h, 0)) b V θ ( h ) V θ + ( h, t ) − 1 c ( t ) − 1 (6 . 88) whic h is required to b e unitary for h ∈ ∂ D . Therefore, if we equate the expression for k 1 (1 , t ) from (6.88) w ith the corresp on d ing one for ( k 1 (1 , t ) ∗ ) − 1 , we obtain | c ( t ) | 2 = c ( t ) ∗ c ( t ) . Explicitly , | c ( t ) | 2 = ( b V θ (1) V θ + (1 , t ) − 1 ) ∗ e t ( Dϕ ( E (1 , 0))+ Dϕ ( E (1 , 0) − 1 )) b V θ V θ + (1 , t ) − 1 . (6 . 89) W r ite c ( t ) = | c ( t ) | e iη ( t ) , wh ere e iη ( t ) = d iag ( e iη 0 ( t ) , · · · , e iη p − 1 ( t ) ) . It remains to determine e iη ( t ) . 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Nenciu, Dep ar t ment of Ma thema tics, University of Illinois a t Chicago, Chicago, IL, USA and Instit ute of Ma thema tics “Simon Stoilow” of the R omanian A c ademy, Bucharest, Romania E-mail addr ess : nenciu@ uic.e du

The periodic defocusing Ablowitz-Ladik equation and the geometry of Floquet CMV matrices

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