Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies

Blo c k T o eplitz determinants, constrained KP and Gelfand-Dick ey hierarc hi es. Ma ttia Caf asso (SISSA - T rieste) Abstract W e prop ose a metho d for computing an y Gelfand-Dick ey τ function living in Segal- Wilson Gr assmannian as the asymptotics of blo ck T oeplitz determinant asso cia ted to a certain class of symbols W ( t ; z ). Also truncated blo ck T o eplitz determinants asso ciated to the same sym b ols are shown to be τ function for rational r eductions of KP. Connection with Riemann- Hilber t pro blems is inv estigated b oth from the p oint of view of in tegr able s y stems and block T o eplitz op erator theory . Examples of applications to algebro-g eometric solutions are giv en. In tro duc tion This pap er deals with the app licatio n s of blo c k T o eplitz determinan ts and their asymptotics to the stud y of in tegrable h ierarc hies. Asymptotics of b lo c k T o eplitz determinants and their applications to physics is a dev eloping fi eld of researc h; in recen t y ears it has b een shown ho w to compute some physically relev an t quan tities (e.g. correlation fun ctions) studying asymptotics of some b lo c k T o eplitz determinants (see [25],[26],[27]). In p articular in [25] and [26] authors, for the fir st time, sh o we d effectiv e computations for the case of blo ck T o eplitz determinan ts with sym b ols that do not ha ve half truncated F our ier series. Th is is of particular in terest for us as, with our app roac h, w e w ill b e able to do the same for certa in blo c k T o eplitz determinan ts asso ciated to algebro-geometric solutions of Gelfand Dic ke y h ierarc hies. Let us m en tion some theoretical r esults ab out (blo c k) T oep litz determinan ts w e will use in this pap er. Given a function γ ( z ) on the circle we denote T N ( γ ) the T o eplitz matrix with symb ol γ giv en b y T N ( γ ) :=           γ (0) . . . . . . γ ( − N ) γ (1) . . . . . . γ ( − N +1) . . . . . . . . . . . . γ ( N ) . . . . . . γ (0)           where γ ( k ) are the F our ier co efficien ts γ ( z ) = P k γ ( k ) z k . W e u se the term blo ck T o eplitz for the case of m atrix-v alued symbol γ ( z ). In that case th e en tries γ ( j − i ) of the ab o ve matrix are n × n matrices themselv es. W e denote D N ( γ ) := det T N ( γ ) and we use the notati on T ( γ ) for the N × N matrix obtained letting N go to infin it y . The main goal of the theory of T o eplitz determinants is to compute D N ( γ ) as N go es to infi nit y 1 and find expressions for D N ( γ ) as w ell as f or its limit in terms of F redholm determinant s. First result is due to Szeg¨ o that in 1952 ga ve a formula for asymptotics of D N ( γ ) in the s calar case [5]. This r esult has b een generalized b y H. Widom in the 70’s ([6],[7] and [8]) for the matrix case; namely he pro ved that u nder s uitable analytical assu mptions it exists the limit D ∞ ( γ ) := lim N → ∞ D N ( γ ) G ( γ ) N = det( T ( γ ) T ( γ − 1 )) where G ( γ ) is a normalizing constant and the op erator T ( γ ) T ( γ − 1 ) is suc h that its d etermi- nan t is well defined as a F redh olm determinan t (see section 2 for the p r ecise statemen t). Once the asymptotics had b een computed the next quite natural question was to fi nd an expr essions relating directly D N ( γ ), and not ju st its asymptotics, to certain F redh olm d eterminan ts. Th e problem w as solv ed man y ye ars later by Borod in and Oko un k o v in [9] f or the scalar case and generalized, in the same y ear, for matrix case by E. Basor and H.Widom in [10]. F or matrix v alued case Boro din -Ok ounko v f orm ula reads D N ( γ ) = D ∞ ( γ ) det( I − K γ ,N ) (here w e assu me G ( γ ) = 1). The op er ator ( I − K γ ,N ) can b e wr itten exp licitely in co ordi- nates knowing certain Riemann-Hilb ert factorizations of γ . Its F redh olm determinant is w ell defined (see section 2 for details). Now man y pro ofs of Boro din -Ok ounko v form ula are known (for instance [11] con tains another pr o of of the same formula, see also the earlier pap er [12]). In th is pap er w e apply blo ck-T oeplitz d etermin ants to the compu tation of τ function of an (almost) arb itrary solution of Gelfand-Dic k ey hierarch y ∂ L ∂ t j = [( L j n ) + , L ] . ( L different ial op erator of order n , j 6 = nk ). More precisely to a giv en p oin t W = W ( z ) H ( n ) + in the b ig cell of Segal-Wilson ve ctor-v alued Grassmannian w e asso ciate a n × n matrix-v alued sym b ol W ( t ; z ) obtained deforming W ( z ) (see formula (10 )). In this w ay w e defin e a sequence of N -truncated blo c k T oep litz determinan ts { τ W,N ( t ) } N > 0 whic h are sho wn to b e s olutions of certain rational reductions of KP; this is our First result: E very symb ol W ( t ; z ) defines thr ough its trunc ate d determinants a se que nc e { τ W,N ( t ) } N > 0 of solutions for KP such that τ W,N ( t ) ∈ cKP 1 ,nN ∩ cKP n,n ∀ N > 0 . Here we used the notation from [20]; gi ven a τ function for KP with corresp on d ing Lax pseudo d ifferen tial op erator L w e sa y that τ ∈ cKP m,n iff L m can b e written as the ratio of t w o differen tial op er ators of order m + n and n resp ectiv ely . This s equ ence adm its a stable limit which is shown to b e equal to the Gelfand-Dic k ey τ function τ W ( t ) asso ciated to W ; this quan tit y can b e compu ted usin g S zeg¨ o-Widom’s theorem. This will giv e us the remark able iden tit y τ W ( ˜ t ) = det h P W ( ˜ t ; z ) i (1) 2 where P W ( ˜ t ; z ) is the F redholm op erator app earing in Szeg¨ o-Widom’s theorem (h ere w e put ˜ t instead of t to remem b er that, when w orking with W ∈ Gr (n) , times t nj m ultiple of n must b e set to 0). Next step is the s tudy of Riemann-Hilb ert (also called Wiener-Hopf ) factorization of symbol W ( t ; z ) giv en by W ( t ; z ) = T − ( t ; z ) T + ( t ; z ) (2) with T − and T + analytical in z outside and insid e S 1 resp ectiv ely and normalized as T − ( ∞ ) = I . Here w e assume that the s y mb ol can b e extend ed to an analytic function in a n eigh b orho o d of S 1 . Using Plemelj’s w ork [13] we show that T − ( ˜ t ; z ) m ust satify the in tegral equation P T W ( ˜ t ; z ) T − ( ˜ t ; z ) = I (3) and w e wr ite a solution of (2) in terms of w a ve fu nction ψ W ( ˜ t ; z ) corresp ond in g to W . In this w a y w e arrive to our second r esult: Second result: T ake W ∈ Gr ( n ) in the big c el l and its c orr esp onding τ function τ W ( ˜ t ) . τ W ( ˜ t ) i s e qual to the F r e dholm determinant of the homo ge ne ous inte gr al e quation asso ciate d to (3) which is r elate d to Riemann-Hilb ert pr oblem (2). The solution of this Riemann-Hilb ert pr oblem is unique for every value of p ar ameters ˜ t that makes τ W ( ˜ t ) 6 = 0 and c an b e c ompute d by me ans of r e late d wave f u nction ψ W ( ˜ t ; z ) . A t the end of th e p ap er we consider a particular class of symbols W ( t ; z ) corresp onding to algebro-geometric solutions of Gelfand-Dic key hierarc hies. W e formulate an alternativ e Riemann-Hilb ert pr oblem equiv alen t to (2 ) and explain ho w to s olve it u sing θ -fun ctions. In this w ay w e give concrete form ulas for a wide class of s ym b ols that do not hav e half truncated F ourier series. W e think this is quite remark able since concrete results for non half truncated sym b ols were a v ailable, till now, just for the concrete cases pr esented in [25] an d [26 ]. The pap er is organized as follo ws: • First section states some results ab out Segal-Wilson Grassmannian and related lo op groups w e w ill need in the sequ el; p ro ofs can b e f ound in [1 ] and [2 ]. • Second section states Szeg¨ o-Widom’s theorem and r elated results obtained b y Widom in [6],[7] and [8] and the Boro din-Okounk o v formula for blo c k T o eplitz determinant [10]. • In the third section we in tro d uce and stud y the sequence of truncated determinants { τ W,N ( t ) } N > 0 and its stable limit τ W ( t ). W e w ant to remark that the p rop erty of stabilit y w as stated for the first time in [15] (see also [16]) and our sequ en ce is actually a subsequence of the s tabilizing c hain studied in [17]; nev ertheless, to our b est kno wledge, this is the fir st time th at blo c k T o eplitz determinants en ter the game and also the observ ation that τ W,N ∈ cKP n,n seems to b e something new. • F ourth section is dev oted to establishing the connectio n b et w een int egral equations for- m ulated b y Plemelj in [13] and F redh olm op erator app earing in Szeg¨ o-Widom’s theorem. • In the fifth secti on w e show how to write Riemann-Hilbert factorization of W ( ˜ t ; z ) in terms of wa ve fu nction ψ W ( ˜ t ; z ). Of course r elation b et ween Gelfand-Dic k ey hierarc hy 3 and factorization problem is something kno wn ; our exp osition her e is closely r elated to [14]. Moreo ver, kn o wing Riemann-Hilb ert factorization of W ( ˜ t ; z ), we can apply Boro din-Oko un k o v formula to giv e an exp r ession of any τ W,N ( ˜ t ) as F red holm d etermi- nan t and a recurs ion relation to go fr om τ W,N ( ˜ t ) to τ W,N +1 ( ˜ t ). • Last section give s explicit formulas for sym b ols and τ fu n ctions asso ciated to algebro- geometric rank one solutions of Gel fand -Dic k ey hierarc hies. Also we formulate an alter- nativ e Riemann-Hilb ert pr ob lem equiv alen t to (2 ) in analogy with what h as b een done in [25] and [26]. W e explain how to solve it using θ -fu nctions. Ac knowle dgemen ts I am grateful to A.Its for a fru itful discus sion in w hic h he p ointed out that our Riemann- Hilb ert pr oblem can b e r ed uced to a pr oblem with constan t jump as in [25],[26] and ga ve me some in teresting hin ts ab ou t p ossible develo pm en ts of th is work. J. v an de L eur s u ggested me to v erify if ve ctor-costrained reductions of KP had some relatio ns with blo c k T o eplitz determinant, I am grateful to h im for this really u s eful s u ggestion. Moreo ver I wish to thank my advisor B.Dubro vin for h is constan t supp ort and suggestio ns he gav e me dur ing many hours sp en t together discussing the preparation of this pap er. This wo r k is partially supp orted b y the Eu rop ean Science F oun dation Programme “Meth- o ds of In tegrable Systems, Geometry , Applied Mathematics” (MISGAM), the Marie Curie R TN “Europ ean Net w ork in Geometry , Mathemati cal Ph ysics and Applications” (ENIGMA), and b y the Italian Ministry of Univ ersities and Researc hes (MUR) researc h grant PRIN 2006 “Geometric metho ds in the theory of nonlinear wa ve s and their applications”. 1 Segal-Wilson G rassmannian and related lo op groups Here w e recall some defi n itions and r esults f rom [1 ] and [2 ] that will b e useful in the sequel. Definition 1.1. L et H ( n ) := L 2 ( S 1 , C n ) b e the sp ac e of c omplex ve ctor-value d squar e- inte gr able functions. We cho ose a orthono rmal b asis given by { e α,k := (0 , . . . , z k , . . . , 0) T : α = 1 . . . n, k ∈ Z } and the p olarization H ( n ) = H ( n ) + ⊕ H ( n ) − wher e H ( n ) + and H ( n ) − ar e the close d subsp ac es sp anne d by elements { e α,k } with k ≥ 0 and k < 0 r esp e ctively. In the s equel in order to a v oid cumbersome notations we will write H instead of H (1) . Definition 1.2 ([2]) . The Gr assmannian Gr( H ( n ) ) mo dele d on H ( n ) c onsists of the subset of close d subsp ac es W ⊆ H ( n ) such that: • the ortho gonal pr oje ction pr + : W → H ( n ) + is a F r e dholm op er ator. • the ortho gonal pr oje ction pr − : W → H ( n ) − is a Hilb e rt-Schmidt op er ator. 4 Mor e over we wil l denote Gr ( n ) the subset of Gr( H ( n ) ) give n by subsp ac es W such that z W ⊆ W . It’s wel l kno wn [1 ] that through Segal-Wilson theory w e can associate a solution of n th Gelfand-Dic key hierarc hy to ev ery elemen t of Gr ( n ) ; this is the reason why we are in terested in them. Lemma 1.3 ([1]) . The map Ξ : H ( n ) − → H ( f 0 ( z ) , . . . , f n − 1 ( z )) T 7− → ˜ f ( z ) := f 0 ( z n ) + . . . + z n − 1 f n − 1 ( z n ) is an isometry. Its inv e rse is gi ven by f k ( z ) = 1 n X ζ n = z ζ − k ˜ f ( ζ ) wher e the sum runs over the n th r o ots of z . Prop osition 1.4. Under the isometry Ξ we c an identify Gr ( n ) with the subset { W ∈ Gr( H ) : z n W ⊆ W } It is obvious that lo op groups act on Hilb ert s p aces defined ab o v e by m u ltiplications. W e w ant to define a certain loop group L 1 2 Gl( n, C ) with go o d analytical prop erties acting transitiv ely on Gr ( n ) ; in s uc h a wa y w e can obtain any W ∈ Gr ( n ) just acting on the reference p oint H ( n ) + with this group. Go o d analit ycal prop erties w ill b e necessary as w e wan t to construct sym b ols of some T o eplitz op erators out of elemen ts of this group and then apply Widom’s results (see b elo w). Giv en a m atrix g we denote with k g k its Hilb ert-Schmidt norm k g k 2 = n X i,j =1 k g i,j k 2 Definition 1.5. Given a me asur able matrix-value d lo op γ we define two norms k γ k ∞ and k γ k 2 , 1 2 as k γ k ∞ := ess sup k z k =1 k γ ( z ) k k γ k 2 , 1 2 := X k  | k | k γ ( k ) k 2  1 2 wher e we have F ourier exp ansion γ ( z ) = ∞ X k = −∞ γ ( k ) z k . Definition 1.6. L 1 2 Gl( n, C ) is define d as the lo op gr oup of invertible me asur able lo ops γ such that k γ k ∞ + k γ k 2 , 1 2 < ∞ . Prop osition 1.7 ([2]) . L 1 2 Gl( n, C ) acts tr ansitively on Gr ( n ) and the isotr opy gr oup of H ( n ) + is the gr oup of c onstant lo ops Gl ( n , C ) . 5 Pro of can b e foun d in [2], h ere w e ju st m ention the principal steps necessary to arrive to this r esult. • W e defin e a subgroup Gl r es ( H ( n ) ) of in vertible linear maps g : H ( n ) → H ( n ) acting on Gr( H ( n ) ) (the restricted general linear group). • W e pr o v e that ev ery element of Gl r es ( H ( n ) ) comm uting with multiplic ation b y z m ust b elong to L 1 2 Gl( n, C ). • W e tak e an element W ∈ Gr ( n ) and a basis { w 1 , ..., w n } of the orth ogonal complemen t of z W in W . • Out of this b asis, p utting vec tors side by side, w e construct W and easily c heck that W = W ( z ) H ( n ) + . • W e verify that multiplica tion b y W b elongs to Gl r es ( H ( n ) ); since it ob viously comm utes with m ultiplication b y z we conclude that W ( z ) ∈ L 1 2 Gl( n, C ). 2 Szeg¨ o-Widom theorem for blo c k T oeplitz determinan ts. In his w ork ([6],[7] and [8]) H. Widom expressed the limit, for the size go ing to infi nit y , of certain blo c k T o eplitz determinant s as F redholm determinant s of an op erator P acting on H ( n ) + . Also h e ga v e t wo different co rollaries that allo w u s to compute this determinant in some p articular cases. In this section we recall, without pr o ofs, these results. Moreo v er w e state Boro din-Okounk o v formula as presente d in [10] f or matrix case. W e b egin with some notations; giv en a lo op γ ∈ L 1 2 Gl( n, C ) w e denote with T N ( γ ) the blo ck T o eplitz matrix giv en by T N ( γ ) :=           γ (0) . . . . . . γ ( − N ) γ (1) . . . . . . γ ( − N +1) . . . . . . . . . . . . γ ( N ) . . . . . . γ (0)           where we h a v e the F ourier expansion γ ( z ) = P k γ ( k ) z k .W e denote D N ( γ ) its determinant. W e use the notation T ( γ ) for the N × N matrix obtained letting N go to infi nit y . R emark 2.1 . It’s easy to see that, in the base w e hav e c hosen ab ov e for H ( n ) , T ( γ ) is nothing but the matrix representa tion of pr + ◦ γ : H ( n ) + − → H ( n ) + Theorem 2.2 (S zeg¨ o-Wi d om theorem,[8]) . Supp ose γ ∈ L 1 2 Gl( n, C ) and ∆ 0 ≤ θ ≤ 2 π arg  det  γ ( e iθ )   = 0 6 Then it exists the limit D ∞ ( γ ) := lim N → ∞ D N ( γ ) G ( γ ) N = det( T ( γ ) T ( γ − 1 )) wher e G ( γ ) = exp  1 2 π Z 2 π 0 log  det γ ( e iθ )  dθ  The p ro of of the theorem is contai ned in [8]; instead of rewriting it we s im p ly consider the op erator T ( γ ) T ( γ − 1 ) and explain the meaning of ”det” in this case. Lemma 2.3. Consider γ 1 , γ 2 ∈ L 1 2 Gl n ( n, C ) ; we have T ( γ 1 γ 2 ) − T ( γ 1 ) T ( γ 2 ) = h X k ≥ 1 γ ( i + k ) 1 γ ( − j − k ) 2 i i,j ≥ 0 . Pr oo f T he ( i, j )-entry of left hand side r eads ∞ X k = −∞ γ ( i − k ) 1 γ ( k − j ) 2 − ∞ X k =0 γ ( i − k ) 1 γ ( k − j ) 2 = − 1 X k = ∞ γ ( i − k ) 1 γ ( k − j ) 2 = ∞ X k =0 γ ( i + k + 1) 1 γ ( − k − j − 1) 2 .  In particular c ho osing γ 1 = γ and γ 2 = γ − 1 w e obtain I − T ( γ ) T ( γ − 1 ) = h X k ≥ 1 γ ( i + k ) ( γ − 1 ) ( − j − k ) i i,j ≥ 0 Definition 2.4. P γ := T ( γ ) T ( γ − 1 ) = " δ j i −  X k ≥ 1 γ ( i + k ) ( γ − 1 ) ( − j − k )  # i,j ≥ 0 (4) Thanks to the fact th at X i ≥ 0 X k ≥ 1 k γ ( i + k ) k 2 = X k ≥ 1 k k γ ( k ) k 2 < ∞ the pro du ct we h a v e written on the righ t of (4) is a pro duct of t wo Hilb ert-Schmidt op erators. So P γ differs from the iden tit y by a nucle ar op erator. Hence its determinant is well defi ned (see f or ins tance [28]). In our n otation w e obtained the equalit y D ∞ ( γ ) = det( P γ ) (5) W e will call P γ Plemelj’s op erator as it is related in a clear wa y with a Riemann-Hilb er t factorizat ion p roblem (see s ection 4) already considered by Josip Plemelj in 1964 [13]. Unfortunately , in concrete cases, det ( P γ ) turns out to b e really hard to compute; nev ertheless w e can u se some shortcuts also p ro vided by Widom in his wo r k s ([6],[7] and [8]). 7 Prop osition 2.5 ([6]) . Supp ose that γ satisfies c onditions imp ose d in Sze g¨ o-Widom the or em and, mor e over, γ ( i ) = 0 for i ≥ j + 1 or γ ( i ) = 0 for i ≤ j + 1 . Then D ∞ ( γ ) = D j ( γ − 1 ) G ( γ ) j (6) Prop osition 2.6 ([8]) . Supp ose we have a symb ol γ satisfying c onditions imp ose d i n Sze g¨ o- Widom the or em. Supp ose mor e over that γ dep ends on a p ar ameter x in such a way that the function x → γ ( x ) is differ e ntiable. If γ − 1 admits two Riema nn-H ilb ert factorizations γ − 1 ( z ) = t + ( z ) t − ( z ) = s − ( z ) s + ( z ) such that t + ( z ) := X k ≥ 0 t ( k ) + z k s + ( z ) := X k ≥ 0 s ( k ) + z k t − ( z ) := X k ≤ 0 t ( k ) − z k s − ( z ) := X k ≤ 0 s ( k ) − z k Then d dx log( D ∞ ( γ )) = i 2 π I trace "  ( ∂ z t + ) t − − ( ∂ z s − ) s +  ∂ x γ # dz . (7) Also D N ( γ ) can b e expressed as a F redholm determinan t as p oin ted out for the scalar case in [9 ] and generalized f or matrix case in [10]. Theorem 2.7 (Boro din -Ok ounko v formula, [10]) . Supp ose that our symb ol γ ( z ) satisfying c onditions of Sze g¨ o-Widom’s the or em admits two Rieman n-H ilb ert factorizations γ ( z ) = γ + ( z ) γ − ( z ) = θ − ( z ) θ + ( z ) such that γ + ( z ) := X k ≥ 0 γ ( k ) + z k θ + ( z ) := X k ≥ 0 θ ( k ) + z k γ − ( z ) := X k ≤ 0 γ ( k ) − z k θ − ( z ) := X k ≤ 0 θ ( k ) − z k and G ( γ ) = 1 . Then f or eve ry N D N ( γ ) = D ∞ ( γ ) det( I − K γ ,N ) (8) wher e, in c o or dinates, we have ( K γ ,N ) ij =      0 if min { i, j } < N P ∞ k =1 ( γ − θ − 1 + ) ( i + k ) ( θ − 1 − γ + ) ( − j − k ) otherwise. R emark 2.8 . O n e can easily ve rify that θ − 1 − γ + is the inv ers e of γ − θ − 1 + so that, again, we deal with op erators of t yp e T ( φ ) T ( φ − 1 ) with φ = γ − θ − 1 + . Also we w an t to p oint out that the assumption G ( γ ) = 1 is not necessary . The formula for G ( γ ) 6 = 1 is written in [11]; since in our case we will alwa ys hav e G ( γ ) = 1 we wrote the formula as it was giv en in [10]. 8 3 τ functions for constrained KP and G elfand-Dic k ey hierarc h ies as b lo c k T o eplitz determinan ts. It is w ell known that giv en a p oint W ∈ Gr its time ev olution W ( t ) := exp( t 1 z + t 2 z 2 + t 3 z 3 + . . . ) W is nothing b ut K P flo w. Moreo v er p oin ts W ∈ Gr ( n ) corresp ond to solutions of Gelfand-Dic key hierarc hies (this is the celebrated Grassmann ian form ulation of KP hierarch y d u e to M.Sato, see for instance [3]) . In their pap er [1] Segal and Wilso n ga v e a form ula for the corresp ondin g τ f unction τ W as d eterminan t (F redholm d eterminan t) of th e pro jection of W ( t ) onto H + . Here we tak e a sligh tly different appr oac h that generalizes w hat has b een done by Itzykson and Zu b er in the stu dy of Witten-Kon tsevic h τ fun ction in [15] (see also [16] and [18]). Th is approac h all ows u s to d efine not ju s t τ W but also a sequence of { τ W,N } N > 0 appro ximating τ W and b eing themselve s τ functions for some redu ctions of K P. Supp ose w e ha ve an elemen t W ∈ Gr ( n ) ; thanks to results stated in Section 1 w e can r epresen t this element as W =     w 11 . . . . . . w n 1 . . . . . . . . . . . . . . . . . . . . . . . . w 1 n . . . . . . w nn     H ( n ) + = W ( z ) H ( n ) + with W ( z ) ∈ L 1 2 Gl( n, C ). Also w e assum e that the m atrix W ( z ) = { w ij ( z ) } i,j =1 ..n satisfies                w ii = 1 + O ( 1 z ) w ij = z ( O ( 1 z )) , i > j w ij = O ( 1 z ) , i < j This means th at we restrict to the big cell, i.e. we assum e W ∼ = H ( n ) + . In the sequel w e will alw a ys assu me that W b elongs to the big cell. Obvio u s ly w e ha ve a base for W ∈ Gr ( n ) giv en b y { z s w j : s ∈ N , j = 1 . . . n } where w j is the column ve ctor ( w 1 j ...w nj ) T . U sin g the isomorphim Ξ : H ( n ) → H the corresp ondin g base for W ∈ Gr is giv en b y { ω ns + j = z ns Ξ( w j ) : s ∈ N , j = 1 . . . n } and, as in Section 1, w e h a v e [Ξ( w j )]( z ) = n X i =1 z i − 1 w j i ( z n ) Note that thanks to the big-cell assumption we ha ve ω ns + j ( z ) = z ns + j − 1 (1 + O ( 1 z )). F or th ese p oin ts W ∈ Gr ( n ) and vect ors sp anning them w e define the standard time ev olution 9 (KP fl o w) giv en b y ω ns + j ( t ; z ) := exp X i> 0 t i z i ! ω ns + j ( z ) = exp( ξ ( t, z )) ω ns + j ( z ) No w w e wan t to define the τ function asso ciated to W as limit for N → ∞ of some blo ck T o eplitz d eterminan ts τ W,N . Definition 3.1. T ake M = N n a multiple of n . τ W,N ( t ) := det " I z − i ω j ( t ; z ) dz # 1 ≤ i,j ≤ M = N n (9) Fist of all we wan t to prov e that τ W,N is a blo ck T o eplitz determinan t and write explicitely the symbol. Lemma 3.2. F or every j = 1 . . . n we have w j ( t ; z ) := Ξ − 1 ( ω j ( t, z )) = exp( ξ ( t, Λ)) w j ( z ) wher e we denote Λ :=           0 . . . . . . . . . z 1 0 . . . . . . 0 0 1 . . . . . . 0 . . . . . . . . . . . . . . . 0 . . . 0 1 0           Pr oo f W e simply v erify th at multiplicatio n by z on Gr corresp ond s to multiplicati on by Λ on Gr ( n ) through the isomorphism Ξ − 1 .  Prop osition 3.3. τ W,N is the N-trunc ate d ( n × n ) − blo ck T o eplitz determinant with symb ol W ( t ; z ) := exp( ξ ( t, Λ)) W ( z ) (10) Pr oo f T ake i, j ≤ n and s, v ≤ N ; the ( i + sn, j + v n )-en try of th e matrix in the righ t hand side of (9) is giv en by I z − i − sn ω j + vn ( t ; z ) dz = I z − i − sn z vn ω j ( t ; z ) dz = I z − i +( v − s ) n X k ∈ Z ,l =1 ..n w j l ( t ) ( k ) z nk + l − 1 dz = w j i ( t ) ( s − v ) so that the r igh t h and sid e of (9) is the trans p osed of the N -truncated n × n blo c k T o eplitz matrix with symbol W ( t ; z ).  In the s equel of th is pap er w e w ill call su c h symbols Gelfand-Dic k ey (GD) symb ols. No w generaliz ing what has b een done by Itzykson and Z ub er in [15] w e expand τ W,N ( t ) in c haracters. In this wa y , assigning d egree m to t m , it’s p ossible to state a certain prop ert y of stabilit y for τ W,N ( t ); namely every term of degree less or equal to Q will b e ind ep endent of N for Q ≤ N . 10 Prop osition 3.4. τ W,N ( t ) = X l 1 ,...,l nN ≥ 0  Y i ω ( − l i + i − 1) i  χ l 1 ,...,l nN ( X ) wher e X = diag ( x 1 , ..., x nN ) is r elate d to times { t i } thr ough M iwa’s p ar ametrization t k := trace X k k ! and χ l 1 ,...,l nN ( X ) := det   x l 1 + nN − 1 1 x l 2 + nN − 2 1 . . . x l nN 1 . . . . . . . . . . . . x l 1 + nN − 1 nN x l 2 + nN − 2 nN . . . x l nN nN   det   x nN − 1 1 x nN − 2 1 . . . 1 . . . . . . . . . . . . x nN − 1 nN x nN − 2 nN . . . 1   Pr oo f W e start from determinant representa tion (9 ). T he ( i, j )-entry of th e matrix w ill b e X n ω ( − n ) j p n + i − 1 ( t ) where for ev ery n ≥ 0 p n ( t ) := 1 2 π i I exp( ξ ( t, z )) z n +1 dz are the classical Sch u r p olynomials and p n ( t ) = 0 for ev ery negativ e n . Then resumming ev erything we obtain τ W,N ( t ) = X k 1 ,...,k nN  Y j ω ( − k j ) j  det[ p k j + i − 1 ( t )] i,j =1 ...nN with k j ≥ 1 − j . Equiv alentl y we write τ W,N ( t ) = X l 1 ,...,l nN ≥ 0  Y j ω ( − l j + j − 1) j  det[ p l j − j + i ( t )] i,j =1 ...nN . On the other hand it’s well known th at u nder Miw a’s parametrization this last d eterminan t can b e w ritten as χ l 1 ,...,l nN ( X ) (see for instance [15],[16]); this completes the p ro of.  No w it’s easy to see that assigning degree 1 to ev ery x i (whic h is equiv alen t to assigning degree m to t m ) we obtain deg( χ l 1 ...l nN ) = M X i =1 l i F rom this easily verified prop ert y w e obtain the follo wing 11 Lemma 3.5. Supp ose d eg( χ l 1 ,...,l nN ) = Q ≤ nN . Then, if the char acter is differ ent fr om zer o, we have χ l 1 ...l nN = χ l 1 ,...,l Q , 0 ,..., 0 . Pr oo f S upp ose l j 6 = 0, j > Q and l i = 0 ∀ i > j . The j th column of the matrix [ p l j − j + i ( t )] has p ositiv e subscripts l 1 + j − 1 , l 2 + j − 2 , . . . , l j . On the other hand P l i = Q ; hence the sum of these subs cripts is Q + j − 1 X r =0 r ≤ j X r =0 r hence t wo subs cripts must b e equal, then t wo lines of the matrix are equal.  F rom this corollary it follo ws directly the follo wing result. Prop osition 3.6. Up to de gr e e Q the function τ W,N ( t ) do es not dep end on N with N ≥ Q . This prop osition allo ws us to defi ne in a r igorous sens e τ W ( t ) := lim N → ∞ τ W,N ( t ) (11) where the limit is defined as limit of formal graded series in t i ; this means that lim N → ∞ deg( τ W,N ( t ) − τ W ( t )) = ∞ W e will sa y that τ W ( t ) is th e stable limit of τ W,N ( t ) for N → ∞ . On the other hand, in the sequel, we will p ro v e that the symb ol W ( t ) satisfies Szeg¨ o-Widom’s condition for eve ry v alues of t i so that the limit in (11) exist p oin t wise in time parameters and can b e wr itten as a F r edholm determinant. No w, follo wing again [15], we write a differen tial op erator ∆ W,N ( t ) associated to the function τ W,N ( t ). In the sequel we will alwa ys write D for the partial deriv ativ e with resp ect to t 1 . W e will pr ov e that for ev ery N the pseudo-differenti al op erator ∆ W,N ( t ) D − nN satisfies S ato’s equations for the dressing (see [3]) and we reco v er the u sual relation b etw een τ and w av e functions. Lemma 3.7. Define f s,N ( t ) := X k >s ω ( − k ) s p k + nN − 1 ( t ) Then we have τ W,N ( t ) = W r( f 1 ,N ( t ) , . . . , f nN ,N ( t )) := det[ D nN − j f i,N ( t )] 1 ≤ i,j ≤ nN Pr oo f F r om definition 3.1 the ( i, j )-en try of matrix d efining τ W,N ( t ) is X k >j ω ( − k ) j p k + i − 1 ( t ) On the other hand w e h a v e D nN − j f i,N ( t ) = D nN − j X k >i ω ( − k ) i p k + nN − 1 ( t ) ! = X k >i ω ( − k ) i p k + j − 1 ( t ) (using the equation D s ( p m ( t )) = p m − s ( t )). Hence w e obtained the p ro of.  12 Definition 3.8. We define the differ ential op er ator ∆ W,N ( t ) of or der N in D as ∆ W,N ( f ) := W r( f , f 1 ,N ( t ) , . . . , f nN ,N ( t )) W r( f 1 ,N ( t ) , . . . , f nN ,N ( t )) wher e f ∈ H ( n ) dep ends in a differ entiable way on { t i } i ≥ 1 . Prop osition 3.9. The fol lowing e quations for time-derivatves of ∆ W,N ( t ) holds: ∂ ∂ t i (∆ W,N ( t )) = (∆ W,N ( t ) D i ∆ − 1 W,N ( t )) + − ∆ W,N ( t ) D i (12) Pr oo f It is enough to prov e the equalit y of the t wo differen tial op erators w hen acting on f 1 ,N ( t ) , ...f nN ,N ( t ) wh ic h are nN indip enden t solutions of the equation (∆ W,N ( t ))( f ( t )) = 0 But this amoun ts to pro ving " ∂ ∂ t i (∆ W,N )( t ) # f j,N ( t ) + ∆ W,N ( t ) ∂ i ∂ t i 1 f j,N ( t ) = 0 ∀ j whic h is tr u e iff ∂ ∂ t i (∆ W,N ( t ) f j,N ( t )) = 0 ∀ j. This equalit y is obviously satisfied.  Multiplying ∆ W,N from the right with D − nN w e found a p seudo differential op erator that, in fact, giv es a solution of KP equations. Definition 3.10. S W,N ( t ) := ∆ W,N ( t ) D − nN Prop osition 3.11. S W,N ( t ) is a monic pseudo-differ ential op er ator of or der 0 satisying Sato’s e quation (se e for instanc e [3]) ∂ ∂ t i ( S W,N ( t )) = ( S W,N ( t ) D i S − 1 W,N ( t )) + − S W,N ( t ) D i (13) Henc e the monic pseudo-differ ential op er ator of or der 1 L W,N ( t ) := ( S W,N ( t ) D S − 1 W,N ( t )) (14) satisfies the usual L ax system for KP ∂ L W,N ∂ t k = [( L k W,N ) + , L W,N ] (15) Pr oo f It is ob vious that S W,N is a monic pseud o-differen tial op erator of order 0 since ∆ W,N , whic h is of order nN , is normalized so that the leading term is equal to 1. Equ ation (13 ) follo w s directly from (12). T he deriv ation of Lax system from Sato’s equations is wel l known: one has just to derive the r elation L W,N S W,N = S W,N D 13 for t k and use the obvious r elation [ L W,N , L k W,N ] = 0  It remains to p r o v e that τ W,N ( t ) is r eally the τ fu nction for these solutions L W,N ( t ) of KP equations. W e r ecall the usual relations b et ween the d ressing S , the wa ve f unction ψ and τ function giv en by ψ ( t ; z ) = S ( t )(exp( ξ ( t, z ))) = exp( ξ ( t, z )) τ ( t − 1 [ z ] ) τ ( t ) (w e recall th at th e notation t − 1 [ z ] stands for the ve ctor with i th comp onent equ al to t i − 1 iz i ) All w e hav e to prov e is the follo win g Prop osition 3.12. ψ W,N ( t ; z ) := S W,N ( t )(exp( ξ ( t, z ))) = exp( ξ ( t, z )) τ W,N ( t − 1 [ z ] ) τ W,N ( t ) (16) Pr oo f E quiv alent ly w e pr ov e th at (∆ W,N ( t )) exp( ξ ( t, z )) = exp( ξ ( t, z )) z nN τ W,N ( t − 1 [ z ] ) τ W,N ( t ) Since we h a v e p n ( t − 1 [ z ] ) = p n ( t ) − z − 1 p n − 1 ( t ) the righ t h and sid e of the equalit y ab ov e can b e wr itten as z nN e ξ ( x,t ) det   D nN − 1 f 1 − z − 1 D nN f 1 . . . f 1 − z − 1 D f 1 . . . . . . . . . D nN − 1 f nN − z − 1 D nN f nN . . . f nN − z − 1 D f nN   W r( f 1 , . . . , f nN ) (here deriv ativ e is with resp ect to t 1 , we don’t write dep endence on f i on t to a vo id hea vy notation) The left hand side can b e written as det     z nN e ξ ( t,z ) . . . . . . e ξ ( t,z ) D nN f 1 . . . . . . f 1 . . . . . . . . . . . . D nN f nN . . . . . . f nN     W r( f 1 , . . . , f nN ) . It is easy to chec k that these tw o exp ressions are equal.  W e wan t no w to study th e stru cture of L W,N with more atten tion; our inv estigation will lead us to disco v er that, actually , we are dealing with rational reductions ([19],[18]) of KP. First of all we recall a useful lemma (pro of can b e foun d for instance in [29]). Lemma 3.13. L et { g 1 , ..., g m } b e a b asis of line arly indep endent solutions of a differ ential op er ator K of or der m . Then one c an factorize K as K = ( D + T m )( D + T m − 1 ) · · · ( D + T 1 ) with T j = W r( g 1 , . . . , g m − 1 ) W r( g 1 , . . . , g m ) . 14 W e will state now pr op erties of symm etry for f s,N that will b e usefu l in the sequel. Prop osition 3.14. The fol lowing e qualities hold: f s,N +1 ( t ) = f s − n,N ( t ) (17) f s,N ( t ) = D n f s + n,N ( t ) (18) Pr oo f W e will use the equalit y ω ( − k ) s = ω ( − k + n ) s + n whic h follo ws from the v ery definition of th ese coefficients. Then f or (17) we hav e f s,N +1 ( t ) = X k ω ( − k ) s p k + n + nN − 1 ( t ) = X k ω ( − k − n ) s − n p k + n + nN − 1 ( t ) = X k ω ( − k ) s − n p k + nN − 1 ( t ) = f s − n,N ( t ) F or (18) we ha ve D n ( f s + n,N ( t )) = D n X k ω ( − k ) s + n p k + nN − 1 ( t ) = X k ω ( − k ) s + n p k + nN − 1 − n ( t ) = X k ω ( k − n ) s p k − n + nN − 1 ( t ) = f s,N ( t )  Theorem 3.15. F or every N > 0 the pseudo diffe r ential op er ator L W,N and its n th p ower L W,N = L n W,N c an b e factorize d as • L W,N = L 1 ,W,N ( L 2 ,W,N ) − 1 • L W,N = M 1 ,W,N ( M 2 ,W,N ) − 1 wher e al l the factors ar e differ ential op e r ators and • ord( L 1 ,W,N ) = nN + 1 , ord( L 2 ,W,N ) = nN • ord( M 1 ,W,N ) = 2 n, ord( M 2 ,W,N ) = n Pr oo f T he fi rst factorization comes d irectly f rom th e fact that L W,N ( t ) = ∆ W,N ( t ) D (∆ W,N ( t )) − 1 F or the second factorization we note that we hav e the factorizat ion L W,N ( t ) = ∆ W,N ( t ) D n (∆ W,N ( t )) − 1 where the fi rst op erator ∆ W,N ( t ) D n has order M + n while th e second (i.e. ∆ W,N ) has order nN . Moreo v er as follo ws from (18) we ha v e 15 • ∆ W,N f i,N = 0 ∀ i = 1 , . . . , nN • ∆ W,N D n f i,N = 0 ∀ i = n + 1 , . . . , nN hence using lemma 3.13 on e can simplify factorization ab o ve as L W,N = M 1 ,W,N ( M 2 ,W,N ) − 1 where M 2 ,W,N is give n explicitely by the formula M 2 ,W,N = ( D + K n,N )( D + K n − 1 ,N ) ... ( D + K 1 ,N ) with K j,N = D " log W r( f n +1 ,N , . . . , f nN ,N , f 1 ,N , . . . , f j − 1 ,N ) W r( f n +1 ,N , . . . , f nN ,N , f 1 ,N , . . . , f j,N ) !#  Theorem 3.16. The se q u enc e {L W,N } N ≥ 1 satisfies r e cu rsion r elation L W,N +1 = T N L W,N ( T N ) − 1 (19) with T N = ( D + T n,N )( D + T n − 1 ,N ) ... ( D + T 1 ,N ) T j = D log W r( f 1 ,N , ..., f nN ,N , f 1 ,N +1 , ...f j − 1 ,N +1 ) W r( f 1 ,N , ..., f nN ,N , f 1 ,N +1 , ...f j,N +1 ) ! . Pr oo f W e observ e that thanks to (17 ) • ∆ W,N f i,N = 0 ∀ i = 1 , . . . , nN • ∆ W,N +1 f i,N = 0 ∀ i = n + 1 , . . . , nN • ∆ W,N +1 f i,N +1 = 0 ∀ i = 1 , . . . , n Hence u sing again (3.13) w e obtain th e recurs ion relation ∆ W,N +1 = T N ∆ W,N and from this last equation w e r eco ver the recursion relation for th e Lax op erator.  Using the notation of [20] w e s ay that, for eve ry N , τ W,N ∈ cKP 1 ,nN ∩ cKP n,n . This means that giv en a τ fun ction for KP with corresp ond ing Lax p s eudo differentia l op erator L w e sa y that τ ∈ cKP m,n iff L m can b e written as the ratio of t wo d ifferential operators of order m + n and n resp ectiv ely . Th ese sp ecial reductions of KP b egun to b e studied in 1995 by Dic k ey and K ric hev er ([18],[19 ]); a geometric in terpr etation of corresp ond ing p oints in the Gr assm annian h as b een give n in [21] and [22 ]. Going b ack to theorem 3.16 we p oint out that the fir st decomp osition as well as the recursion formula are already known and, as p ointe d out in [20 ], come simply fr om the fact that we hav e a truncated dressin g. Actually our sequence of { τ W,N } N ≥ 1 is a p art of a sequence already stud ied by Dic key in [17] und er the name of stabilizing chain; in that article Dic k ey alrea dy pro vided the recursion formula written ab o ve as well as s ome differen tial equations for co efficien ts of T N . Nev erth eless, to our 16 b est kno wledge, connectio n w ith blo ck T o eplitz determinan ts never app eared b efore. Also th e fact that τ W,N ∈ cKP n , n is something new. It could b e in teresting to fi nd recursion relations as w ell as differential equations for M 1 ,W,N and M 2 ,W,N ; w e plan to do it in a sub sequent w ork. Till no w all w e can do is to inf er from recur s ion f orm ula for Lax op erator the f ollo w ing form ula ( M 1 ,W,N +1 ) − 1 T N M 1 ,W,N = ( M 2 ,W,N +1 ) − 1 T N M 2 ,W,N . (20) No w we wa nt to go one step furth er and see what happ ens for N → ∞ . Ob viously thanks to the prop ert y of stabilizati on stated in p rop osition 3.6 w e can define a p seudo differentia l op erator L W and a wa ve fu nction ψ W related to τ W in the same w ay as for finite N and w e will obtain a solution of KP as w ell. Actually a stronger statemen t holds. Prop osition 3.17. Given W ∈ Gr ( n ) the functions τ W , ψ W and L W := ( L W ) n ar e r e sp e c- tively the τ function, the wave function and the differ ential op er ator of or der n c orr esp onding to a solution of n th Gelfand-Dickey hier ar chy. Pr oo f It is kno wn [1] that s u bspaces satisfying z n W ⊆ W corresp ond to solutions of n th Gelfand-Dic key hierarch y . What we hav e to pro ve is that L W ( t ) = ( L W ( t )) + . In the s equel w e will omit dep enden ce on times of L W and S W if n o ambiguit y arises. F rom the usual relation ∂ ψ W ∂ t n ( t ; z ) = ( L W ( t )) + ψ W ( t ; z ) (21) w e obtain imm ed iately ∂ S W ∂ t n + S W D n = ( L W ) + S W so that w e hav e to pro ve that ∂ S W ∂ t n = 0 On the other hand ψ W ( t ; z ) = exp( ξ ( t, z ))(1 + ∞ X 1=1 s i ( t ) z − i ) where S W ( t ) = 1 + ∞ X 1=1 s i ( t ) D − i Using this explicit expression for the wa ve function and su bstituting in (21) w e obtain ( L W ( t )) + ψ W ( t ; z ) − z n ψ W ( t ; z ) = exp ( ξ ( t, z )) ∞ X i =1 ∂ s i ( t ) ∂ t n z − i The left h an d side of this equation lies on W ( t ) = exp( ξ ( t, z )) W for every t so that multiplying b oth terms for exp( − ξ ( t, z )) one obtains th at they b elong to subsp aces transv erse one to the other ( W and H − ), hence b oth of them v anish. This means that ∂ s i ∂ t n = 0 for ev ery i .  In virtue of this prop osition, wh en computing τ W asso ciated to W ∈ Gr ( n ) , we will alw ays omit times t j n m ultiple of n . Setting { t j n = 0 , j ∈ N } will b e imp ortan t in order to b e able to ap p ly S zeg¨ o-Widom’s theorem; in this case we will write ˜ t instead of t . 17 Prop osition 3.18. T ake any W ∈ Gr ( n ) in the big c el l of Gr ( n ) and a c orr esp onding GD symb ol W ( t ; z ) . Then τ W ( ˜ t ) = det( P W ( ˜ t ; z ) ) . (22 ) Pr oo f All w e ha ve to p r o v e is that conditions of Szeg¨ o-Widom’s theorem are s atisfied and G ( W ( ˜ t ; z )) = 1. W e observ e that W ( ˜ t ; z ) ∈ L 1 2 Gl( n, C ) ∀ ˜ t since we can alw a ys fin d W ( z ) ∈ L 1 2 Gl( n, C ) such th at W = W ( z ) H ( n ) + and exp( ξ ( ˜ t, Λ)) is con tin uous ly differen tiable (ob viously wh en r estricted to a finite n umb er of times). Moreo ver det[exp( ξ ( ˜ t, Λ))] = 1 since w e d eleted times m ultiple of n and det( W ( z )) = 1 + O ( z − 1 ) by b ig cell assu mption. This implies that w e h a v e ∆ 0 ≤ θ ≤ 2 π det  W ( ˜ t ; e iθ )  = 0 and G ( W ( ˜ t ; z )) = 1 .  W e are now in the p osition to state th e main result of this pap er, pr o of follo ws from r esults obtained ab o ve. Theorem 3.19. Giv e n any p oint W ∈ Gr ( n ) and c orr esp onding GD symb ol W ( t ; z ) the fol lowing facts hold true: • { τ W,N ( t ) := D N ( W ( t ; z )) } 0 ≤ N < ∞ is a se que nc e of τ functions for KP asso ciate d to wave function ψ W,N ( t ; z ) and pseudo differ ential op er ators L W,N ( t ) given r esp e ctively b y (16) and (14). • F or e v ery N > 0 we have τ W,N ∈ cKP 1 ,nN ∩ cKP n,n . Explicit factorizations of L ax op er ator and its n th p ower ar e given in the or em 3.15. • The se que nc e admits stable limit τ W ( ˜ t ) . The stable limit is a solution of n th Gelfand-Dickey hier ar chy. It is e qual to the F r e dholm determinant det P W ( ˜ t ; z ) . R emark 3.20 . Also all the τ W,N ( ˜ t ) can b e expressed as F redh olm determinan ts; in order to giv e explicit exp r essions we need a certain Riemann-Hilb ert factorizatio n of sym b ol W ( ˜ t ; z ). This factorization will b e obtained in section 5 and it will b e exploited to exp ress τ W,N ( ˜ t ) as a F redholm determinant. 18 4 Riemann-Hilb ert problem and Plemelj’s in tegral form ula. It is evident from pr op osition 2.6 that R iemann -Hilb ert decomp ositions of symbol γ for a blo c k T o eplitz op erator plays an imp ortan t role in computing D ∞ ( γ ). Here w e will sho w that actually Plemelj’s op erator itself enters in a integ ral equation (see [13]) giving solutions of Riemann-Hilb er t pr oblem ϕ + ( z ) = γ T ( z ) ϕ − ( z ) . (23) Here ϕ + ( z ) and ϕ − ( z ) are resp ectiv ely analytical functions d efined inside and outside th e circle. In this s ection we consider a smaller class of lo ops; γ ( z ) will b e a matrix-v alued function that extends analytically on a neigh b orho o d of S 1 . F or con v enience of the reader we recall here main steps to arrive to Plemelj’s inte gral form ula [13]. Lemma 4.1. Supp ose that f + ( z ) , f − ( z ) ar e functions on S 1 satisfying | f ( ζ 2 ) − f ( ζ 1 ) | < | ζ 2 − ζ 1 | µ C for some p ositive c onstants µ, C and for eve ry ζ 1 , ζ 2 ∈ S 1 . N e c essary and sufficient c onditions for f + ( z ) and f − ( z ) to b e b oundary values of analytic functions r e g ular inside or outside S 1 ⊆ C and with value c at infinity ar e r esp e ctiv e ly 1 2 π i I f + ( ζ ) − f + ( z ) ζ − z dζ = 0 (24) 1 2 π i I f − ( ζ ) − f − ( z ) ζ − z dζ + f − ( z ) − c = 0 (25) W e ha ve to p oint out that h ere b oth ζ and z lies on S 1 so that one has to b e careful and define (24) and (25 ) as appropriate limits. Namely on e p ro v es that taking ζ slight ly inside or outside S 1 along the normal and making it app r oac h to the circle we obtain the s ame result whic h will b e, by d efinition, the v alue of our integral. No w supp ose w e w an t to fi nd solutions of (23); w e normalize the problem r equiring ϕ − taking v alue C at in finit y . T aking an appropriate linear com bination of (24) and (25) and usin g (23) we fin d that ϕ − ( z ) m ust satisfy the equation C = ϕ ( z ) − 1 2 π i I ( γ T ) − 1 ( z ) γ T ( ζ ) − I ζ − z ϕ ( ζ ) dζ (26) Note th at here w e do not ha ve to tak e any limit since the in tegrand is we ll defined for ev ery p oin t of S 1 . W e also wan t to consid er the asso ciate homogeneous equation 0 = ϕ ( z ) − 1 2 π i I ( γ T ) − 1 ( z ) γ T ( ζ ) − I ζ − z ϕ ( ζ ) dζ (27) as w ell as its adjoint 0 = ψ ( z ) + 1 2 π i I γ ( z ) γ − 1 ( ζ ) − I ζ − z ψ ( ζ ) dζ (28) Ob viously , as usu al in F redh olm’s theory , the equ ations (27) and (28) either hav e only trivial solution or they ha ve the same num b er of linearly indep end en t solutions. 19 Lemma 4.2. Consider two adjoint RH pr oblems ϕ + ( z ) = γ ( z ) T ϕ − ( z ) (29) ψ + ( z ) = γ ( z ) ψ − ( z ) (30) normalize d as ψ − ( ∞ ) = ϕ − ( ∞ ) = 0 . Any solution ϕ − of (29) i s a solution of (27) as wel l as any solution ψ + of (30) i s a solution of (28). Pr oo f W e just rep eat computations made for non-homogeneous case.  No w w e introdu ce a new in tegrable op erator acting on H ( n ) + and pro ve that it is actually equal to th e Plemelj’s op erator. Definition 4.3. F or every f ∈ H ( n ) + we define [ ˜ P γ ψ ]( z ) := pr + ψ ( z ) + 1 2 π i I γ ( z ) γ − 1 ( ζ ) − I ζ − z ψ ( ζ ) dζ ! (31) wher e p r + denote the pr oje ction onto H ( n ) + . Prop osition 4.4. ˜ P γ = P γ . Pr oo f W e write ˜ P γ in co ordinates and verify we obtain the same as in (4). T o do so as in the definition of in tegrals (24) and (25) w e compute (31) imp osing | ζ | < | z | ; th e form u la w ill hold when ζ appr oac h to S 1 in the same wa y as in (24) and (25). F or a consistency chec k we will prov e we obtain the same resu lt imp osing | ζ | > | z | . Let’s start with | ζ | < | z | ; we h a v e ψ ( z ) + 1 2 π i I γ ( z ) γ − 1 ( ζ ) − I ζ − z ψ ( ζ ) dζ = ψ ( z ) + 1 2 π i I X k ≥ 1 ζ k z k  I − X p,q ∈ Z γ ( p ) ( γ − 1 ) ( q ) z p ζ q  X s ≥ 0 ψ ( s ) ζ s dζ ζ Imp osing k + q + s = 0 w e get that this is equ al to ψ ( z ) + X p ∈ Z X k ≥ 1 X s ≥ 0 γ ( p ) ( γ − 1 ) ( − k − s ) ψ ( s ) z p − k = ψ ( z ) + X t ∈ Z X k ≥ 1 X s ≥ 0 γ ( t + k ) ( γ − 1 ) ( − k − s ) ψ ( s ) z t T aking the pr o jection on H ( n ) + w e obtain exactly f orm ula (4). No w for | ζ | > | z | we hav e ψ ( z ) + 1 2 π i I γ ( z ) γ − 1 ( ζ ) − I ζ − z ψ ( ζ ) dζ = ψ ( z ) + 1 2 π i I X k ≥ 0 ζ k z k  X p,q ∈ Z γ ( p ) ( γ − 1 ) ( q ) z p ζ q − I  X s ≥ 0 ψ ( s ) ζ s dζ ζ 20 Imp osing q + s = k w e arrive to X k ,s ≥ 0 X p ∈ Z γ ( p ) ( γ − 1 ) ( k − s ) ψ ( s ) z k + p = X k ,s ≥ 0 X t ∈ Z γ ( t − k ) ( γ − 1 ) ( k − s ) ψ ( s ) z t T aking the pro jection on H ( n ) + w e obtain th at this is equal to T ( γ ) T ( γ − 1 ) so that the t wo computations for | ζ | < | z | and for | ζ | > | z | coincide in virtue of lemma 2.3  Theorem 4.5. Supp ose we ar e given a symb ol γ ( z ) analytic in a neig hb orho o d of S 1 and such that D ∞ ( γ ) 6 = 0 Then the R iemann-Hilb ert pr oblem ϕ + ( z ) = γ ( z ) T ϕ − ( z ) normalize d as ϕ − ( ∞ ) = C admits (if existing) a uni q ue solution. Pr oo f Supp ose w e ha ve t w o distinct s olutions ( ϕ 1 − , ϕ 1+ ) and ( ϕ 2 − , ϕ 2+ ); taking the d if- ference we obtain a non-trivial solution of (29). Th en also (30) adm its non trivial solutions and the same holds for (28). But this means that we ha ve a n on zero ψ ( z ) ∈ H ( n ) + suc h that [ P γ ψ ]( z ) = 0 which is im p ossible s ince det( P γ ) = D ∞ ( γ ) 6 = 0  Existence of factorization w ill b e treated in the next section for th e sp ecific case of Gelfand- Dic k ey symbols. F or a general treatmen t of the problem of existence s ee [13]. 5 F actorization for Gelfand-Dic k ey sym b ols Here we will p ro ve th at for Gelfand-Dic k ey symbols we can write the unique solution of factorizat ion (23) in terms of data L W ( ˜ t ) , ψ W ( ˜ t ; z ). W e recall that L W ( ˜ t ) and ψ W ( ˜ t ; z ) are the stable limits of L W,N ( ˜ t ) and ψ W,N ( ˜ t ; z ). They repr esen t the differenti al op erator and th e w a ve fun ction associated to the solution τ W ( ˜ t ). Our exp osition here is closely related to [14]. A t the end of th e s ection we will u se the factorization obtained to expr ess an y τ W,N ( ˜ t ) as a F redholm determinant. As we ha ve written b efore in the pr o of of p rop osition 3.17 we ha ve the relation L W ( ˜ t ) ψ W ( ˜ t ; z ) = z n ψ W ( ˜ t ; z ) (32) where ψ W ( ˜ t ; z ) admits asymptotic expansion ψ W ( ˜ t ; z ) = exp( ξ ( ˜ t, z ))(1 + O ( z − 1 )) No w out of ψ W w e construct n time-dep endent functions ψ W,i ( ˜ t ; z ) := D i ( ψ W ( ˜ t ; z )) : i = 0 , . . . , n − 1 b elonging to the subsp ace W ∈ Gr. 21 Definition 5.1. Ψ W ( ˜ t ; z ) :=           1 ζ 1 . . . ζ n − 1 1 1 ζ 2 . . . ζ n − 1 2 . . . . . . . . . . . . 1 ζ n . . . ζ n − 1 n           − 1           ψ W, 0 ( ˜ t ; ζ 1 ) ψ W, 1 ( ˜ t ; ζ 1 ) . . . ψ W,n − 1 ( ˜ t ; ζ 1 ) ψ W, 0 ( ˜ t ; ζ 2 ) ψ W, 1 ( ˜ t ; ζ 2 ) . . . ψ W,n − 1 ( ˜ t ; ζ 2 ) . . . . . . . . . . . . ψ W, 0 ( ˜ t ; ζ n ) ψ W, 1 ( ˜ t ; ζ n ) . . . ψ W,n − 1 ( ˜ t ; ζ n )           wher e ζ i is the i th r o ot of z . Prop osition 5.2. The matrix Ψ W ( ˜ t ; z ) admits asymp otic exp ansion Ψ W ( ˜ t ; Λ) = exp( ξ ( ˜ t ; Λ))( I + O ( z − 1 )) Mor e over under the isomorphism Ξ − 1 : H → H ( n ) we c an write W ∈ Gr ( n ) as W = Ψ W (0 , z ) H ( n ) + (33) Pr oo f On e has to note that the i th column of matrix Ψ W ( ˜ t ; z ) is nothing b ut Ξ − 1 ( ψ W,i ( ˜ t, z )) so that asymptotic expansion follo ws easily . E quation (33) corresp onds to the f act that { z ns ψ W,i (0 , z ) : s ∈ Z } is a b asis for W .  Observe that, since w e also hav e W = W ( z ) H ( n ) + w e obtain Ψ W (0 , z ) = W ( z )( I + O ( z − 1 ) . F rom this equation and from lemma 2.3 it follo ws that for ev ery N > 0 we ha ve T N  W ( ˜ t ; z )  I + O ( z − 1 )   = T N ( W ( ˜ t ; z )) T N ( I + O ( z − 1 )) . No w since for eve ry N det( T N ( I + O ( z − 1 ))) = 1 w e will assu me, without loss of generalit y , that Ψ W (0 , z ) = W ( z ) since this is tr ue mo d ulo an irrelev an t term that do es not affe ct v alues of determinan ts we w ant to compute. W e no w wan t to define a matrix Φ W ( ˜ t ; z ) analytic in z near 0 and with similar prop erties as Ψ W ( ˜ t ; z ) . Definition 5.3. L et φ W ( ˜ t ; z ) b e the uniq u e solution of L W ( ˜ t ) φ W ( ˜ t ; z ) = z n φ W ( ˜ t ; z ) analytic in z = 0 and such that ( D i φ )(0 , z ) = z i : i = 0 , . . . , n − 1 22 We define Φ W ( ˜ t ; z ) :=           1 ζ 1 . . . ζ n − 1 1 1 ζ 2 . . . ζ n − 1 2 . . . . . . . . . . . . 1 ζ n . . . ζ n − 1 n           − 1           φ W, 0 ( ˜ t ; ζ 1 ) φ W, 1 ( ˜ t ; ζ 1 ) . . . φ W,n − 1 ( ˜ t ; ζ 1 ) φ W, 0 ( ˜ t ; ζ 2 ) φ W, 1 ( ˜ t ; ζ 2 ) . . . φ W,n − 1 ( ˜ t ; ζ 2 ) . . . . . . . . . . . . φ W, 0 ( ˜ t ; ζ n ) φ W, 1 ( ˜ t ; ζ n ) . . . φ W,n − 1 ( ˜ t ; ζ n )           wher e as b efor e ζ i is the i th r o ot of z and φ W,i ( ˜ t ) := D i ( φ W ( ˜ t ; z )) : i = 0 , . . . , n − 1 . R emark 5.4 . Φ W ( ˜ t ; z ) adm its regular expansion in z = 0 and Cauc hy initial v alues we imp osed on φ W imply Φ W (0; z ) = I . Prop osition 5.5. Ψ W ( ˜ t ; z )Φ − 1 W ( ˜ t ; z ) do es not dep end on t i for any i . Pr oo f I t is well known that equations ∂ ∂ t i f = ( L i n W ) + f satisfied by φ W and ψ W can b e translated into matrix equations ∂ ∂ t i F = F M satisfied by Ψ W ( ˜ t ; z ) and Φ W ( ˜ t ; z ) (one can write explicitely M in terms of co efficien ts of ( L i n W ) + ). Hence we h a v e ∂ ∂ t i (Ψ W ( ˜ t ; z )Φ − 1 W ( ˜ t ; z )) = Ψ W ( ˜ t ; z ) M Φ − 1 W ( ˜ t ; z ) − Ψ W ( ˜ t ; z )Φ − 1 W ( ˜ t ; z )Φ W ( ˜ t ; z ) M Φ − 1 W ( ˜ t ; z ) = 0  Theorem 5.6. Given a Gelfand-Dickey symb ol W ( ˜ t ; z ) = exp  ξ ( ˜ t, Λ)  W ( z ) one c an factorize it as W ( ˜ t ; z ) = " exp  ξ ( ˜ t, Λ)  Ψ W ( − ˜ t, z ) # Φ − 1 W ( − ˜ t ; z ) wher e the term i nside the squar e br acket is analytic ar ound z = ∞ and the other is analytic ar ound z = 0 . F or assigne d values of ˜ t for which τ W ( ˜ t ) 6 = 0 this is the uni q ue solution of the factorization pr oblem (23) normalize d at infinity to the identity. 23 Pr oo f Usin g the p revious prop osition we ha ve W ( ˜ t ; z ) = exp ξ ( ˜ t, Λ) W ( z ) = exp( ξ ( ˜ t, Λ))Ψ W (0 , z ) = exp( ξ ( ˜ t, Λ))Ψ( − ˜ t ; z )Φ − 1 W ( − ˜ t ; z )Φ W (0; z ) = exp( ξ ( ˜ t, Λ))Ψ( − ˜ t ; z )Φ − 1 W ( − ˜ t ; z ) Unicit y of the f actorization follo w s from section 4.  Corollary 5.7. F or every N > 0 τ W,N ( ˜ t ) = τ W ( ˜ t ) det( I − K W ( ˜ t ; z ) ,N ) with ( K W ( ˜ t ; z ) ,N ) ij =      0 if min { i, j } < N P ∞ k =1 (Ψ W ( − ˜ t ; z )) ( i + k ) (Ψ W ( − ˜ t ; z ) − 1 ) ( − j − k ) otherwise. Pr oo f It is enough to apply Boro din -Ok ounko v formula using factorization obtained ab o ve.  Corollary 5.8. F or every N > 0 τ W,N ( ˜ t ) τ W,N +1 ( ˜ t ) = det  T  Ψ W ( − ˜ t ; z )  T  Ψ W ( − ˜ t ; z ) − 1   N ,N (observe that the right hand side of this e quation is an or dinary n × n determinant, not a F r e dholm determinant). Pr oo f τ W,N ( ˜ t ) τ W,N +1 ( ˜ t ) = det( I − K W ( ˜ t ; z ) ,N ) det( I − K W ( ˜ t ; z ) ,N +1 ) . On the other hand the op erator ( I − K W ( ˜ t ; z ) ,N +1 ) − 1 ( I − K W ( ˜ t ; z ) ,N ) can b e w ritten as a b lo c k matrix obtained taking the iden tit y matrix and r eplacing the N th blo c k column by the N th blo c k column of the matrix w ith ( i, j )-entry equal to ∞ X k =1 (Ψ W ( − ˜ t ; z )) ( i + k ) (Ψ W ( − ˜ t ; z ) − 1 ) ( − j − k ) . Hence p ro of is obtained applying lemma 2.3  6 Rank one stationary reductions and corresp onding Gelfand-Dic k ey sym b ols W e w ant to describ e, more explicitely , GD symb ols corresp onding to solutions of Gelfand- Dic k ey hierarc hies obtained b y rank one stationary reductions. In order to emphasize that w e are d ealing with r ank-one generic case instead of the standard expr ession Krichever lo cus w e will sp eak ab out Bur chnal l- Chaundy lo cus . 24 Definition 6.1. Giv e n a p oint W ∈ Gr ( n ) we say that W stays in Bur chnal l-Chaundy lo cus iff the L ax op er ator L W of the c orr esp onding solutio n satisfies [ L W , M W ] = 0 for some differ ential op er ator M W of or der m c oprime with n . Without loss of g ener ality we also assume m > n . The name w e use is d ue to the fact th at, already in 1923, Burchnall and Chaun dy were the fi r st to stu d y algebras of comm u tin g differentia l op erators in [23] where they stated this imp ortant prop osition we will u se in the sequel. Prop osition 6.2 ([23]) . Given a p air of c ommuting differ ential op er ator L, M with r elatively prime or ders it exists an irr e ducible p olynomial F ( x, y ) such that F ( x, y ) = x m + ... ± y n and F ( L, M ) = 0 . This prop osition in particular allo ws us to asso ciate to ev ery Burchnall-Chaundy solu- tion a sp ectral cur v e defined b y p olynomial relation existing b etw een the pair of comm uting differen tial op erators. F rom the Grassmann ian p oin t of view one can defi n e an action A of pseudo d ifferen tial op erators in v ariable t 1 on H by A : ΨDO × H − → H  ( t 1 ) m ∂ n ∂ t n 1 , ϕ ( z )  7− →  ∂ n ∂ z n  z n  ϕ ( z ) and, using this action, prov e the f ollo w ing pr op ostition Prop osition 6.3 ([4]) . Given a p oint W in the Bur chnal l-Chaundy lo cus one has z n W ⊆ W (34) b ( z ) ⊆ W (35) wher e L W and M W ar e of or der n and m r esp e ctively and b ( z ) is a series in z wh ose le ading term is z m . Conversely, if W satisfies ab ove pr op erties, it stays in the Bur chnal l-Chaundy lo cus. Pr oo f W e just ske tc h the p ro of and mak e r eferen ce to Mulase’s article [4]. Sup p ose w e are giv en L W and M W ; u n der conju gation with the d ressing S W ( ˜ t ) w e h a v e S − 1 W ( ˜ t ) L W ( ˜ t ) S W ( ˜ t ) = ∂ n ∂ t n 1 Under the action A this giv es inv ariance of W with r esp ect to z n while in v ariance with resp ect to b ( z ) is obtained acting with S − 1 W ( ˜ t ) M W ( ˜ t ) S W ( ˜ t ) Vicev er s a given W we reconstruct th e dressing S W ( ˜ t ); u sing it we defin e L W ( ˜ t ) and M W ( ˜ t ) conjugating pseud o different ial op erators corresp onding to z n and b ( z ). In particular observ e that also z n and b ( z ) w ill satisfy the same p olynomial relation as L W ( ˜ t ) and M W ( ˜ t ).  25 R emark 6.4 . Without loss of generalit y we can assu m e 1 2 π i I b ( z ) z ns +1 dz = 0 ∀ s ∈ Z . (36) No w supp ose we are giv en an element W = W ( z ) H ( n ) + ∈ Gr ( n ) in the Burchnall-Chaundy lo cus. Using the explicit isomorphism Ξ we can construct a matrix B ( z ) := b (Λ) su c h that B ( z ) W ⊆ W. (37) Prop osition 6.5. C ( z ) := W − 1 ( z ) B ( z ) W ( z ) has the fol lowing pr op erties: • C ( z ) is p olynomial in z . • trace( C ( z )) = 0 • m = max i ( j − i + n d eg C ij ( z )) ∀ j = 1 . . . n • The char acteristic p olynomial p C ( z ) ( λ ) of C ( z ) defines the sp e ctr al curve of the solution. Pr oo f Equation (37) can b e equiv alen tly w r itten as W − 1 ( z ) B ( z ) W ( z ) H ( n ) + ⊆ H ( n ) + and this means precisely that C ( z ) can’t hav e terms in z − k for any k > 0. The other prop erties are satisfied if and only if they are equally satisfied b y B ( z ) so that w e will p ro ve them for B ( z ) instead of C ( z ). B ( z ) is traceless thanks to equation (36) and thanks to the fact that trace(Λ k ) = 0 ∀ k 6 = sn The third prop erties is satisfied as B ( z ) = b (Λ) represen ts in H m ultiplication by a series whose leading term is equal to m . F or the last prop ert y w e obs er ve that if F ( x, y ) is the p olynomial defin ing the sp ectral curve , i.e. F ( L W , M W ) = 0, then w e w ill ha v e F (diag ( z , z , . . . , z ) , B ( z )) = 0 as w ell; on the other h and thank s to C ayley- Hamilton theorem we hav e p B ( z ) ( B ( z )) = 0 . Since F is ir reducible and p B ( z ) ( λ ) has the same f orm p B ( z ) ( λ ) = λ n + . . . ± z m w e conclude th at th ey are equal.  Observe that since W ( z ) is defin ed m o dulo multiplica tion on the left by inv ertible trian- gular matrices also C ( z ) is d efined m o dulo conju gation by elemen ts of th e group ∆ of u pp er trianguar in vertible m atrices. It was a remark able observ ation of Sc hw arz [24] that actually Burc hn all-Chaun dy lo cu s can b e describ ed by means of matrices with p rop erties as in prop o- sition 6.5 mo dulo the actio n of ∆. Here we ad ap t the results of [24] to our s itu ation. Namely w e explain how, giv en C ( z ), one can reco ve r W ( z ) and the corresp onding sp ectral curve. 26 Prop osition 6.6. Given a matrix C ( z ) such that: • C ( z ) is p olynomial in z . • trace( C ( z )) = 0 • m = max i ( j − i + n d eg C ij ( z )) ∀ j = 1 . . . n it exists a unique W = W ( z ) H ( n ) + in Bur chnal l- Chaundy lo cus such that i ts sp e ctr al curve is define d by p C ( z ) ( λ ) . In order to prov e this pr op osition we n eed tw o lemmas. Lemma 6.7. Given a p olynomial matrix C ( z ) such that m = max i ( j − i + n d eg C ij ( z )) ∀ j = 1 . . . n (with m and n c oprime) c o efficients of char acteristic p olynomial p C ( z ) ( λ ) := λ n + c 1 ( z ) λ n − 1 + . . . + c n ( z ) satisfy n deg c s ≤ ms ∀ s = 1 , . . . , n − 1 deg c n = m Pr oo f F r om n deg C i,j ≤ m − j + i and definition of determinan t f ollo ws immed iately that n deg c s ≤ ms ∀ s = 1 , . . . , n. Strict inequalit y for s < n follo ws from the fact that m and n are copr im e. F or the equalit y deg c n = deg  det( C ( z ))  = m w e ob s erv e that in ev ery line there is a un iqu e elemen t C ij ( z ) suc h th at m = j − i + n deg C ij ( z ); taking this uniqu e elemen t for every lin e and multiplying them w e will obtain the leading term of determinant whic h w ill b e of ord er m .  Lemma 6.8. The e quation λ n + c 1 ( z ) λ n − 1 + . . . + c n ( z ) = 0 (38) with n deg c s ≤ ms ∀ s = 1 , . . . , n − 1 deg c n = m and n, m c oprime has n distinct solutions { λ i = b ( ζ i ) , i = 1 . . . n } with b ( ζ ) = ζ m (1 + O ( ζ − 1 )) (as u sual ζ i is the i th r o ot of z ). 27 Pr oo f I mp osing λ i = ζ m i w e hav e a s olution of the equation ( ζ m i ) n + c 1 ( ζ n i )( ζ m i ) n − 1 + . . . + c n ( ζ n i ) = 0 at the leading ord er mn . Then imp osing λ i = ζ m i (1 + l 1 ζ − 1 i ) and plugging it into the equation (38) one obtains ( ζ m i + l 1 ζ m − 1 i ) n + c 1 ( ζ n i )( ζ m i + l 1 ζ m − 1 i ) n − 1 + . . . + c n ( ζ n i ) = O ( ζ mn i ) l 1 can b e found so that terms of order nm − 1 in the equation v anish ; going on solving the equation term by term we obtain λ i = ζ m i  1 + X j < 0 l j ζ − j i  Clearly co efficients l j do not dep end on th e choice of the ro ot ζ i so th at it exists b ( λ ) with stated prop erties.  No w we can pr o v e pr op osition 6.6. Pr oo f W e s tart computing the c h aracteristic p olynomial p C ( z ) ( λ ); thanks to lemmas 6.7 and 6.8 we fin d n d istin ct r o ots b ( ζ 1 ) , . . . , b ( ζ n ) with prop erties stated ab ov e. The aim is to fi nd W ( z ) suc h that W ( z ) b (Λ) W − 1 ( z ) = C ( z ) Since we h a v e n distinct solutions { b ( ζ i ) , i = 1 , . . . , n } of the equation p C ( z ) ( λ ) = 0 it exists a matrix Υ( ζ 1 , . . . , ζ n ) suc h that Υ( ζ i ) C ( z )Υ − 1 ( ζ i ) =      b ( ζ 1 ) 0 . . . 0 0 b ( ζ 2 ) . . . 0 0 . . . . . . 0 0 . . . . . . b ( ζ n )      On the other hand it’s easy to observe that the matrix Λ can b e diagonalized as Λ =     1 ζ 1 . . . ζ n − 1 1 1 ζ 2 . . . ζ n − 1 2 . . . . . . . . . . . . 1 ζ n . . . ζ n − 1 n     − 1      ζ 1 0 . . . 0 0 ζ 2 . . . 0 0 . . . . . . 0 0 . . . . . . ζ n          1 ζ 1 . . . ζ n − 1 1 1 ζ 2 . . . ζ n − 1 2 . . . . . . . . . . . . 1 ζ n . . . ζ n − 1 n     and this means that m ultiplication by b ( z ) can b e wr itten in H ( n ) + as m ultiplication by     1 ζ 1 . . . ζ n − 1 1 1 ζ 2 . . . ζ n − 1 2 . . . . . . . . . . . . 1 ζ n . . . ζ n − 1 n     − 1      b ( ζ 1 ) 0 . . . 0 0 b ( ζ 2 ) . . . 0 0 . . . . . . 0 0 . . . . . . b ( ζ n )          1 ζ 1 . . . ζ n − 1 1 1 ζ 2 . . . ζ n − 1 2 . . . . . . . . . . . . 1 ζ n . . . ζ n − 1 n     28 Hence we h a v e W ( z ) = Υ − 1 ( ζ i )     1 ζ 1 . . . ζ n − 1 1 1 ζ 2 . . . ζ n − 1 2 . . . . . . . . . . . . 1 ζ n . . . ζ n − 1 n     Note th at W ( z ) is defined mod ulo the action of ∆ so th at, b y construction, C ( z ) corresp onds to a unique W ∈ Gr ( n ) suc h that W = W ( z ) H ( n ) + .  R emark 6.9 . As it wa s p oin ted out by Sch w arz [24], matrices C ( z ) with prop erties stat ed ab o ve can b e used to describ e p oints in the Grassmannian describing string solutions of Gelfand-Dic key hierarchies, i.e. solutions asso ciated to reduction of t yp e [ L, M ] = 1 This class of solutions has n ot b een treated in this article since they d o not liv e in S egal-Wi lson Grassmannian but jus t on Sato’s Grassman n ian constru cted on the s pace of formal series; this means that we cannot use any more Szeg¨ o-Wi dom theorem as the analytical requir ements are not satisfied. Nev ertheless some results obtained in section 3 still hold since the p r op ert y of stabilit y for { τ W,N ( t ) } do es not dep end on analytical prop erties of the symbol W ( z ). Hence one can try to apply the approac h used in this article to the study of these (m uc h less studied) string solutions; p erhaps results ob tained by Okounk o v and Boro din in [9] f or formal series and a generalizatio n to blo ck case can play in th e s etting of formal theory the same role p la y ed b y Szeg¨ o-Widom theorem in this pap er. Example 6.10 (Sym metric n -co verings) . T ake a sym metric n -co vering C of P 1 giv en by equa- tion λ n = P ( z ) = nk +1 Y j =1 ( z − a j ) (39) F or this p articular t y p e of cur ves, c ho osing in a appr opriate w a y the divisor on the cur v e, we can write explicitely W ( z ) , B ( z ) and C ( z ). W e start to observ e th at for an y W corresp onding to th is sp ectral cu rv e we hav e b ( z ) W ⊆ W with b ( z ) = P ( z n ) 1 n Then it’s easy to prov e that th e corresp onding B ( z ) = b (Λ) can b e written as B ( z ) =            0 0 . . . 0 z n − 1 n P ( z ) 1 n z − 1 n P ( z ) 1 n 0 . . . 0 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . . . 0 z − 1 n P ( z ) 1 n 0            29 No w we defin e n fu nctions w i ( z ) :=  P ( z ) z  i − 1 n 1 Q ( i − 1) k j =1 ( z − a j ) , i = 1 , . . . , n. W e tak e W := diag( w 1 ( z ) , ..., w n ( z )) It is easy to verify that the m atrix C ( z ) = W − 1 ( z ) B ( z ) W ( z ) =             0 0 . . . 0 z n − 1 n P ( z ) 1 n w n ( z ) w 1 ( z ) z − 1 n P ( z ) 1 n w 1 ( z ) w 2 ( z ) 0 . . . 0 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . . . 0 z − 1 n P ( z ) 1 n w n − 1 ( z ) w n ( z ) 0             is p olynomial in z . It is worth noticing that this example already giv es all p ossible d ou b le co v erings; hence for any (p ossibly singular) h yp erelliptc surface w e found (assigning a par- ticular d ivisor) the GD symbol of the corresp ond ing algebro-geo metric rank one solution of KdV. Example 6.11 (Rational solutions) . As p oin ted out b y Segal and Wilson [1], subspace of Burc hn all-Chaun dy locu s corresp on d ing to rational curves are giv en by W = W ( z ) H ( n ) + with W ( z ) rational in z . In p articular the corresp onding Gelfand-Dic key symb ol w ill satisfy h y- p othesis giv en in p r op osition 2.5 so that we reco ver the follo wing (kno wn) resu lt. Pr op osition 6.12 . E very r ational solution of Gelfand-Dicke y hier ar chies c an b e written as a finite-size determinant. F or instance, for n = 2, taking W ( z ) =  1 − d 2 z − 1 0 0 1 − c 2 z − 1  H 2 + the in verse of Gelfand-Dic k ey sym b ol is equal to W − 1 ( ˜ t ; z ) =     cosh( z 1 2  P i ≥ 0 t 2 i +1 z 2 i )  − z 1 2 sinh( z 1 2  P i ≥ 0 t 2 i +1 z 2 i )  − z − 1 2 sinh( z 1 2  P i ≥ 0 t 2 i +1 z 2 i )  cosh( z 1 2  P i ≥ 0 t 2 i +1 z 2 i )          z z − d 2 0 0 z z − c 2     Simply taking the residue one obtains that the corresp onding τ fu nction will b e equal to τ W ( t 1 , t 3 , ... ) = det     cosh  P i ≥ 0 t 2 i +1 d 2 i +1  − d sinh  P i ≥ 0 t 2 i +1 d 2 i +1  − c − 1 sinh  P i ≥ 0 t 2 i +1 c 2 i +1  cosh  P i ≥ 0 t 2 i +1 c 2 i +1      and reco v er 2-solitons solution for KdV . 30 W e wa nt to p oint out that, for algebro geometric solutions treated in this section, th e problem of factorizat ion for Gelfand Dic k ey sym b ol can b e easily translated in to a Riemann - Hilb ert p roblem on some cuts on the plane with constan t jumps. F or simplicit y w e reduce to the case n = 2; the pr o cedure us ed here is equiv alen t to the one u sed by Its, Jin and Korepin in [25] and generalized by I ts, Mezzadri and Mo in [26]. Supp ose w e w ant to solv e the factorizatio n problem W ( ˜ t ; z ) := exp  ξ ( ˜ t, Λ)  W ( z ) = T − ( ˜ t ; z ) T + ( ˜ t ; z ) for our GD sym b ol w ith W ( z ) = diag( w 1 ( z ) , w 2 ( z )) as in example 6.10; since it will app ear man y times we denote A the matrix A :=  1 √ z 1 − √ z  Also w e imp ose P ( z ) := 2 g +1 Y j =1 ( z − a j ) with all a j ha ving mo d ulo less then 1 and k a 1 k < k a 2 k < . . . < k a 2 g +1 k W e d en ote l 1 , ...l g +1 the orient ed in terv als ( a 1 , a 2 ) , ( a 3 , a 4 ) , ... ( a 2 g +1 , ∞ ). Instead of lo oking for T − ( ˜ t ; z ) and T + ( ˜ t ; z ) w e d efine a new matrix S ( ˜ t ; z ) imp osing      S ( ˜ t ; z ) := A exp  − ξ ( ˜ t, Λ)  T − ( ˜ t ; z ) z ≥ 1 S ( ˜ t ; z ) := A W ( z ) T − 1 + ( ˜ t ; z ) z ≤ 1 Prop osition 6.13. S ( ˜ t ; z ) has the fol lowing pr op erties: • It has no jumps on S 1 • It has jumps on intervals l j ; pr e cisely c al ling S L ( ˜ t ; z ) and S R ( ˜ t ; z ) the values of S ( ˜ t ; z ) appr o aching fr om the lef t and appr o ching fr om the right the interval we have S L ( ˜ t ; z ) :=  0 1 1 0  S R ( ˜ t ; z ) • It is invertible in any p oints but a j ; ther e it has singular b ehaviour of typ e S ( ˜ t ; z ) ∼  1 1 1 − 1  ( z − a j ) 0 @ 1 0 0 ± 1 2 1 A S j ( ˜ t ; z ) with S j ( ˜ t ; z ) invertible in a j ; minus is for a 1 , . . . , a g , plus for the others. 31 • At infinity it b e haves as S ( ˜ t ; z ) ∼   exp  − √ z ( t 1 z + t 3 z + . . . )  √ z exp  − √ z ( t 1 z + t 3 z + . . . )  exp  √ z ( t 1 z + t 3 z + . . . )  − √ z exp  √ z ( t 1 z + t 3 z + . . . )    Pr oo f Let’s call S + ( ˜ t ; z ) and S − ( ˜ t ; z ) th e limiting v alues of S ( ˜ t ; z ) app roac hing the u nit circle from inside and outside; we ha v e S + ( ˜ t ; z ) S − 1 − ( ˜ t ; z ) = A exp( − ξ ( ˜ t, z )) T − ( ˜ t ; z ) T + ( ˜ t ; z ) W − 1 ( z ) A − 1 = A exp ( − ξ ( ˜ t, z )) exp( ξ ( ˜ t, z )) W ( z ) W − 1 ( z ) A − 1 = I and this pro ves we ha ven’t an y jump s on S 1 . W riting explicitely S ( ˜ t ; z ) as                                      S ( ˜ t ; z ) =     exp  − √ z ( t 1 z + t 3 z + . . . )  √ z exp  − √ z ( t 1 z + t 3 z + . . . )  exp  √ z ( t 1 z + t 3 z + . . . )  − √ z exp  √ z ( t 1 z + t 3 z + . . . )      T − ( ˜ t ; z ) z ≥ 1 S ( ˜ t ; z ) =         1 ( P ( z )) 1 2 Q g j =0 ( z − a j ) 1 − ( P ( z )) 1 2 Q g j =0 ( z − a j )         T − 1 + ( ˜ t ; z ) z ≤ 1 w e obtain almost immediately the other p oin ts of the pr op osition; the only thing we ha ve to observ e is that b oth T + ( ˜ t ; z ) and T − ( ˜ t ; z ) are inv ertible inside and outside the circle resp ec- tiv ely . This is b ecause w e hav e det W ( ˜ t ; z ) = ( P ( z )) 1 2 Q g j =0 ( z − a j ) = det( T + ( ˜ t ; z )) det ( T − ( ˜ t ; z )) This condition com bined w ith lim z →∞ det( T − ( ˜ t ; z )) = 1 giv es det( T + ( ˜ t ; z )) = 1 det( T − ( ˜ t ; z )) = ( P ( z )) 1 2 Q g j =0 ( z − a j ) .  The Riemann-Hilb ert pr ob lem giv en by prop ostion 6.13 is equ iv alen t to the one pr op osed in section 5. What can b e done is to wr ite explicitely the solution S ( ˜ t ; z ) u sing θ functions 32 asso ciated to the cur ve; this is what has b een d one in [25] and [26]. Actually comparin g previous prop osition with results obtained in section 5 we immediately r ealize that S ( ˜ t ; z ) =   ψ W, 0 ( ˜ t ; √ z ) ψ W, 1 ( ˜ t ; √ z ) ψ W, 0 ( ˜ t ; − √ z ) ψ W, 1 ( ˜ t ; − √ z )   so that all w e hav e to d o in our case is to write do wn Bak er-Akhiezer function in terms of sp ecial functions. W e can carry on the same pro cedure for n arbitary; the only difference will b e that th e ju mp matrices will remain constan t bu t more complicated; in any case the solution of th is Riemann-Hilb ert problem with constant ju mps will b e S ( ˜ t ; z ) =           ψ W, 0 ( ˜ t ; ζ 1 ) ψ W, 1 ( ˜ t ; ζ 1 ) . . . ψ W,n − 1 ( ˜ t ; ζ 1 ) ψ W, 0 ( ˜ t ; ζ 2 ) ψ W, 1 ( ˜ t ; ζ 2 ) . . . ψ W,n − 1 ( ˜ t ; ζ 2 ) . . . . . . . . . . . . ψ W, 0 ( ˜ t ; ζ n ) ψ W, 1 ( ˜ t ; ζ n ) . . . ψ W,n − 1 ( ˜ t ; ζ n )           Explicit form u las inv olving sp ecial fu nctions can b e u sed here to app ly p rop osition 2.6 to our case. F or instance taking the elliptic cu r v e C give n b y equation w 2 = 4 z 3 − g 2 z − g 3 with uniformization giv en by the W eierstass ℘ function ( z , w ) = ( ℘ ( u ) , ℘ ′ ( u )) one can w rite w a ve f unction as ψ ( x, t, u ) := σ ( u − c − x ) σ ( c ) σ ( u − c ) σ ( x + c ) exp  xζ ( u ) − 1 2 t℘ ′ ( u )  (here ζ and σ are W eierstrass ζ and σ fu n ction resp ectiv ely , x and t corresp ond to the first and the thir d time). With some tedious computations, making th e c hange of v ariables u = u ( z ), the right h and side of equation (7) can b e obtained. 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