Fredholm determinants and pole-free solutions to the noncommutative Painleve II equation

F redholm determinan ts and p ole-free solutions to the noncomm utativ e P ainlev ´ e I I equation M. Bertola †‡ 1 2 , M. Cafasso †‡ 3 † Centr e de r e cher ches math´ ematiques, Universit´ e de Montr ´ eal C. P. 6128, suc c. c entr e vil le, Montr ´ eal, Qu´ eb e c, Canada H3C 3J7 ‡ Dep artment of Mathematics and Statistics, Conc or dia University 1455 de Maisonneuve W., Montr ´ eal, Qu´ eb e c, Canada H3G 1M8 Abstract W e extend the formalism of in tegrable op erators ` a la Its-I zergin-Korepin-Slavno v to matrix- v alued con volution o p erators o n a semi–inﬁnite interv al and to matrix integral op erators with a kernel of the form E T 1 ( λ ) E 2 ( µ ) λ + µ thus pro ving that their resolv ent op erators can b e expressed in terms of solutions of some sp eciﬁc Riemann-Hilb ert problems. W e also describe some ap- plications, mainly to a noncommutativ e vers ion of Painlev ´ e I I (recently introdu ced by Retakh and Ru btsov), a related n oncomm utative eq uation of P ainlev ´ e typ e. W e construct a particular family of solutions of the noncommutativ e P ainlev´ e II t hat are p ole-free (for real v alues of the v ariables) and hence analo gous to th e H astings-McLeod solution of (commutativ e) P ainlev´ e I I. Such a solution plays the same role as its commutativ e counterpart relativ e to the T racy– Widom theorem, b ut for the computation of the F red h olm determinant of a m atrix versi on of the Airy kernel. 1 W ork supported in part b y the Natural Sciences and Engineering Researc h Council of Canada (NSERC) 2 bertola@crm. umon treal.ca 3 cafasso@crm.umont real.ca Con ten ts 1 In tro du ctio n and resul ts 1 2 Matrix con volution op erators on a semi–i n ﬁni te in terv al. 5 3 Riemann-Hi lb ert problems wi th di ﬀeren t asymptotics and their m utual relation- ship 9 4 T a u functio n s a nd F redholm determ i nan ts 19 5 Applications: F redholm dete rminan ts and noncommuta tiv e P ai nl ev´ e I I, XXXIV 27 5.1 Noncommutativ e Painlev ´ e II a nd its p o le -free solutions . . . . . . . . . . . . . . . . . 30 5.1.1 The g eneral Sto kes’ data/Riemann– Hilber t pr o blem fo r Ψ . . . . . . . . . . . 3 1 5.2 Pole-free solutions of noncommutativ e Painlev´ e I I and F redholm determinants . . . . 37 5.3 Noncommutativ e PXXXIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 0 1 In tro duction and results The pap er aims a t extending the gener a l theory of in tegrable op era tors of Its-Izer gin-Kor epin- Slavno v (I IKS for short) [ 11 ] to op erato rs o f “Hankel” form (see b elow). Leaving aside for the time being any analytical consideration, the issue is the study of integral op erator s on L 2 ( γ + , C r ) with a kernel of the following form K ( λ, µ ) = E T 1 ( λ ) E 2 ( µ ) λ + µ , λ, µ ∈ γ + (1.1) where γ + is a contour contained in a half-plane o f C (so that the denominator do es not v a nish) 4 and the matrices E j : γ + → Mat( p × r, C ) are suitable (analytic) functions. These op erators are related via a F ourier transform to (matrix) con volution op er ators on R + as p ointed o ut in Section 2 : our primary fo cus shall b e the construction of a suitable Riemann– Hilber t problem for computing the resolven t o pe rator S = −K ◦ (1 + K ) − 1 . The knowledge o f the resolven t op erator allows to wr ite v a riational formulæ for the F redholm deter mina nt of the o p e rator Id + K : L 2 ( γ + , C r ) → L 2 ( γ + , C r ) via the w ell–known v aria tional for m ula ∂ ln det (Id + K ) = T r ((Id + S ) ◦ ∂ K ) . (1.2) 4 The conto ur could b e -for example- R + prov ided that the matrices in the n umerator yield E T 1 (0) E 2 (0) = 0. 1 The situation is closely related to the IIKS theor y men tioned ab ov e (with the tensorial extension explained in [ 8 ]), w hich we brieﬂy rec all: let Σ ⊂ C b e a collection of (smo o th) contours and let f , g : Σ → Mat( q × n, C ) b e smooth (a na lytic) functions o n γ sub ject to the condition f T ( λ ) g ( λ ) ≡ 0 , λ ∈ Σ . (1.3) Consider the integral op erator N : L 2 (Σ , C n ) with k ernel given by N ( λ, µ ) := f T ( λ ) g ( µ ) λ − µ (1.4) Then the resolven t op era tor R = N ◦ (Id − N ) − 1 has a k er nel (denoted with the same s y m b o l R ) of the form 5 R ( λ, µ ) = f T ( λ )Θ T ( λ )Θ − T ( µ ) g ( µ ) λ − µ (1.5) where Θ( λ ) is the q × q matrix bo unded solution of the following Riemann–Hilb ert problem Θ( λ ) + = Θ( λ ) −  1 q − 2 i π f ( λ ) g T ( λ )  Θ( λ )= 1 q + O ( λ − 1 ) , λ → ∞ (1.6) F urthermore the so lutio n o f the RHP ( 1.6 ) exists if a nd only if the F redholm de ter minant det(Id − N ) is not zero. The op erator ( 1.1 ) is not immediately o f the form ( 1.4 ) a nd hence the IIK S theory is not directly applicable. Nev er theless the former situation is amenable -not sur prisingly- to the la tter (see also [ 23 ]). In fact one could obser ve, for exa mple, tha t K 2 is an op erator of the form ( 1.4 ) K 2 ( λ, µ )= 1 λ − µ  E T 1 ( λ ) H 1 ( µ ) − H 2 ( λ ) E 2 ( µ )  H 1 ( µ ) := Z γ + d ξ E 2 ( ξ ) E T 1 ( ξ ) E 2 ( µ ) µ + ξ , H 2 ( λ ) := Z γ + d ξ E T 1 ( λ ) E 2 ( ξ ) E T 1 ( ξ ) ξ + λ (1.7) This obse r v ation shows that the IIKS theo ry is relev a nt also to the s tudy of op er a tors of the for m ( 1.1 ): how ever it is not practical to use ( 1.7 ) as a starting point for the analysis as this r oute is imper vious a nd is not the one w e follow. W e provide a direct treatment of K as well a s K 2 in a uniﬁed fashion; the RHPs tha t are re lev ant are sp eciﬁed in Problems 3.1 , 3.2 (please refer to the statements there) for t wo matrix functions Γ , Ξ of size 2 r × 2 r . The t wo problems are intimately related to each o ther in that the jump conditions are identical while only the asymptotic b ehavior at λ = ∞ for Γ , Ξ diﬀers. The solubility of the Riemann–Hilb ert Pr oblems 3.1 , 3.2 is equiv ale nt 5 The sup erscript − T to a matrix denotes the inv erse transposed matrix. 2 to the non v anishing of the F re dholm deter minants of the op erator s Id γ + − K 2 (Thm. 3.1 ) and Id γ + + K (Thm. 3.2 ), re s pe ctively , which follows from I IK S theory ; we thus obtain the formula for the resolven ts of K a nd K 2 (Theorems 3.2 and 3.1 ) − K ◦ (Id γ + + K ) − 1 ( λ, µ ) = 2 µ  E T 1 ( λ ) , 0 p × r  Γ T ( λ )Γ − T ( µ )  0 r × p E 2 ( µ )  λ 2 − µ 2 (1.8) K 2 ◦ (Id γ + − K 2 ) − 1 ( λ, µ ) = [ E T 1 ( λ ) , 0 p × r ] Ξ T ( λ )Ξ − T ( µ ) λ − µ  0 r × p E 2 ( µ )  (1.9) The kno wledge of the resolven t op er ator allows to write v ar ia tional formulæ for the r esp ective F redholm determina nt s: howev er one ma y b ypass for m ula ( 1.2 ) a nd write the v aria tional formulæ dir e ctly in terms of the solution of the r esp e ctive RHPs (Thm. 4.1 4.2 ) using the ide a s in [ 2 ] ∂ ln det(Id γ + − K 2 )= Z γ + ∪ γ − T r  Ξ − 1 − Ξ ′ − ∂ M M − 1  d λ 2 iπ (1.10) ∂ ln det(Id γ + + K )= 1 2 Z γ + ∪ γ − T r  Γ − 1 − Γ ′ − ∂ M M − 1  d λ 2 iπ (1.11) M ( λ ) := 1 2 r − 2 i π E 1 ( λ ) E 2 ( λ ) T  χ γ + ⊗ σ + + χ γ − ⊗ σ −  , γ − := − γ + (1.12) where ′ is deriv ative w.r .t. λ , ∂ denotes a ny v aria tion of the sym bo ls E j and σ + ( σ − ) deno tes the 2 × 2 matrix with just one non-zero entry on the upper right corner (lower left corner ). In the second pa rt of the pap e r we provide so me applications to the study of matr ix conv olution op erator s; our ex ample of choice is a matrix version of the (scalar) conv olution oper ator by the Airy function [ 6 ] A i s : L 2 ( R + ) → L 2 ( R + ) f ( y ) 7→ ( A i s f )( x ) := Z R + Ai( x + y + 2 s ) f ( y )d y (1.13) The F redholm determina n t of the oper ator Id − A i s/ 2 is k nown to yield the T ra cy-Widom ga p distribution F 1 ( s ) for the GO E [ 6 ] and –o n the o ther hand- the F redho lm determinant of Id − A i 2 s yields the dis tribution F 2 ( s ) for the GUE [ 17 ] ; in fact it is w e ll known that the k ernel of the squar e of the Airy-conv olution op erator is the celebra ted Airy kernel A i 2 ( x, y ) := Z R + Ai( x + z )Ai( y + z )d z = Ai( x )Ai ′ ( y ) − Ai( y )Ai ′ ( x ) x − y =: K Ai ( x, y ) (1.14) F 2 is expr essed in terms of the Hastings-McLeo d so lution [ 9 ] to the second Painlev´ e equation [ 22 ] while F 1 can b e expressed in terms of the Miura transform o f the same transcendent. Alternatively 3 (and equiv alently) F 1 ( s ) ca n b e expressed in terms o f the unique solution of the P ainlev´ e XXXIV equation with a certain prescrib ed asymptotics 6 . F 2 ( s )= e xp  − Z ∞ s ( x − s ) u ( x ) 2 dx  , u 2 ( s ) = − ∂ 2 s ln F 2 ( s ) (1.15) F 1 ( s )= e xp  − 1 2 Z ∞ s u ( x ) dx   F 2 ( s )  1 2 (1.16) F 1 ( s )= e xp  − Z ∞ s ( x − s ) w ( x ) dx  , w ( s ) = − ∂ 2 s ln F 1 ( s ) (1.17) u ′′ ( s )= 2 u ( s ) 3 + su ( s ) , u ( s ) ∼ Ai( s ) , s − → + ∞ . (1.18) w ′′′ ( s )= 1 2 w ( s ) w ′ ( s ) + 2 w ( s ) + sw ′ ( s ) , w ( s ) ∼ − 1 2 Ai ′ ( s ) , s − → + ∞ . (1.19) w ( s ) = 1 2 u 2 ( s ) − 1 2 u ′ ( s ) (1.20) where ( 1.20 ) is the usual Miura tr a nsformation b etw een so lutions of mo diﬁed KdV and KdV e q ua- tions. Mo reov er w e refer to ( 1.19 ) a s the PXXXIV equation since, up to r escaling, this is the same as the deriv ative of equation (30) in [ 4 ] (see also [ 1 6 ]). The noncommut ative ana lo g of the who le preceding discussion arises in the study of a ma- trix version o f the Air y-conv o lution op erator (see Se c tion 5 ) which we hav e pick ed a s exempla ry application: ( A i ~ s ~ f )( x ):= Z R + Ai ( x + y ; ~ s ) ~ f ( y )d y (1.21) Ai ( x ; ~ s ):= [ c j k Ai( x + s j + s k )] j,k . (1.22) Here the matrix C = [ c j k ] j,k is a n arbitr ary r × r matrix with co mplex entries (in general) and the depe ndence o f A i ~ s on C is cons ide r ed as para metric (and it is under s to o d in the notation). The kernel o f the square of this matrix-kernel doe s deﬁne a probabilistic mo del b ecaus e it is a totally po sitive kernel on the conﬁgur ation space { 1 , . . . , r } × R as shown in Thm. 5.2 . The a nalysis of A i ~ s and A i 2 ~ s is then related to certain noncommutativ e analogs of the aforementioned Painlev´ e equations and particular so lutions thereof; in par ticular the F redho lm determinant of A i 2 ~ s is related to the noncommutativ e (matrix) Painlev ´ e II e q uation 7 D 2 U ( ~ s ) = 4 ( s U ( ~ s ) + U ( ~ s ) s ) + 8 U 3 ( ~ s ) , s := diag ( s 1 , . . . , s r ) , D := r X j =1 ∂ ∂ s j . (1.23) 6 The uniqueness of the solution w with the prescrib ed asymptotics is easily deduced from the uniqueness of the Hasting-Mc Leo d solution u of PI I. 7 Equation ( 1.23 ) reduces to ( 1.18 ) in the scalar case r = 1 with the cha nge of v ariable x = 2 s . A l so, the matrix U ( ~ s ) in the bo dy of the paper shall b e denoted by β 1 ( ~ s ). 4 which app eared recently in [ 19 ]: that pap er provided sp ecia l solutions in terms o f quasideterminants [ 7 ] in a mo re general con text of no ncommutativ e rings, but no t a Lax-pair r epresentation or a connection to Riemann–Hilb ert problems o r F redho lm determinants. The isomono dro mic approach to the ab ov e equa tion yields a Lax pa ir represe n tation contained in Section 5.1 and par ticula rly Lemma 5.1 . Of greater in terest is the fact that the particula r solutio n that is in volv ed in the computation of the F redholm determinant o f A i 2 ~ s enjoys the same smo othness pro p erties for ~ s ∈ R r as the Hastings- McLeo d solution. Mo re pre c isely w e prov e (Pr op. 5.1 ) that there is a unique solution of ( 1.23 ) with the asymptotic U kℓ ( ~ s ) = c kℓ Ai( s k + s ℓ ) + O  √ S e − 4 3 (2 S − 2 m ) 3 2  , S := 1 r X s j , m = max j | s j − S | , S → + ∞ . Additionally , this solution is po l e free for ~ s ∈ R r if the maximal singular v a lue of the matrix C = [ c kℓ ] is one or less 8 , a co ndition which is s uﬃcient if C is a n a rbitrar y c omplex matrix and bec omes als o necess a ry if C is Hermitea n (Thm. 5.1 a nd Thm. 5 .3 ). The a nalog of the third-or der ODE for F 1 is no w a system with noncommutativ e symbols (Thm. 5.4 ) that ca n b e reduced to a fourth order matrix O DE (Remark 5.6 ) and only in the scala r case is further r educed to an O DE of the third order . The F redholm determinant of A i ~ s (the ana log of F 1 ) is then computed in terms of the relev ant solution in Coro lla ries 5.2 , 5.3 . 2 Matrix con vo lution op erators on a semi– inﬁnite int erv al. Given a function C : R − → Mat( r × r ) decaying s uﬃcient ly fast at inﬁnity let’s consider the conv olution op erator C acting on L 2 ( R + , C r ) as follows:  C ϕ  ( x ) = Z ∞ 0 C ( x + y ) ϕ ( y )d y ∈ L 2 ([0 , ∞ ) , C r ) (2.1) Our aim is to study the F redho lm deter minants det(Id + C ) and det(Id − C 2 ) 9 . Here C (and hence the deter minant) may dep end on some para meters not explicitly indicated he r e (see b elow). Such t yp e of deter minants appea rs in ma ny applicatio ns; just to cite tw o of them let’s recall the Dyson formula [ 5 ] in the in verse scattering for the Sc hr¨ odinger op era tor and in the integral formula of the T racy-Widom distribution for GOE found b y F errar i and Spo hn [ 6 ] (see below) fo r r = 1. 8 The singular v alues of a matrix C are the (positive) squareroots of the eigen v al ues of C † C : they coincide with the absolute v alues of the eigen v alues of C if it i s Hermitean (or more generally normal). 9 Of course the si gn in the expression det(Id + C s ) is inessen tial since w e can alwa ys c hange C with − C . 5 Remark 2.1 In t he inverse sc att ering the ory of the Schr¨ odi nger op er ator and other applic ations the F r e dholm determinant is written as the r est riction to [ s, ∞ ) of the c onvolution by C : f 7→ Z ∞ s C ( x + y ) f ( y )d y ∈ L 2 ([ s, ∞ )) . (2.2) This is identic al to the setting ab ove up to tr anslation. In fact it is ju s t enough to r e deﬁne C ( x ) 7− → C s ( x ) := C ( x + 2 s ) and make it act on L 2 ([0 , ∞ ) . W e will consider functions C ( z ) that admit the following r epresentation (the fac tor of − i b eing purely for later conv enie nc e ) C ( z ) = − i Z γ + e izµ r ( µ )d µ (2.3) where γ + stands for a ﬁnite union of oriented contours in the upp er-half plane with po sitive distance from R and r ( µ ) is a b ounded L 1 ( γ + , Mat( r × r )) function on γ + (with resp ect to the arc- le ng th measure). This a ssumption guarantees that C ( z ) is r apidly decaying a t z = + ∞ ∈ R with a simple estimate 10 | C ( z ) | ≤ e − z dist( γ + , R ) Z γ + | r ( µ ) || d µ | . (2.4) An in teresting example is as follows Example 2. 1 L et C ( z ) = − Ai( z ) and r = 1 : t hen C ( z ) = − Ai( z + s/ 2) = − 1 2 π Z γ + e i µ 3 3 + i ( z + s/ 2) µ d µ (2.5) wher e γ + is a c ontour exten ding to inﬁ nity along t he dir e ctions arg( µ ) = π 2 ± π 3 . This example is r elevant for appli c ations s inc e, as we have written in the intr o duction, the F r e dholm determinant of the c orr esp onding c onvolution op er ator is e qual to the T r acy-Widom distribution for GOE, namely F 1 ( s ) = det(Id + C ) . W e would like to transfer the study of the F redho lm determinant of C on L 2 ( R + ) to the study of a F redholm determinant of an op erator in L 2 ( γ + ); this is a ccomplished hereafter. 10 The sym b ol | r | on a m atrix stands for an y norm on the matrices, for example the Hilb ert-Schmidt norm or the supremum of the absolute v alues of the en tries. This is so not to ov erload the notat ion when considering norms in some L p . 6 Prop ositi o n 2.1 L et C ( z ) as ab ove, with r ( µ ) = E 1 ( µ ) E T 2 ( µ ) and E j ∈ L 2 ∩ L ∞ ( γ + , Mat( r × p )) ; then the op er ator C is of tr ac e–class on L 2 ( R + , C r ) and also det(Id L 2 ( R + ) + C ) = de t (Id H 2 r + b K ) (2.6) wher e H 2 r is the Har dy sp ac e H 2 ⊗ C r (i.e. t he u n itary image of the F ourier–Plancher el tr ansform of L 2 ( R + , C r ) ) and wher e b K is the inte gr al op er ator on H 2 r ⊂ L 2 ( R , C r ) with kernel b K ( λ, µ ) := Z γ + r T ( ξ )d ξ 2 iπ ( λ − ξ )( µ + ξ ) (2.7) Pro of of Prop. 2.1 . By Paley–Wiener theor em, L 2 ( R + , C r ) is unitar ily equiv alent under F ourier– Plancherel tra nsform T to the subspace H 2 r := H 2 ⊗ C r , with H 2 the Hardy space o f the upper half plane. Hence the convolution ope r ator acts as fo llows ψ ( x ) := ( C ϕ )( x ) = Z ∞ 0 C ( x + y ) ϕ ( y )d y = − i Z ∞ 0 d y Z γ + d ξ e i ( x + y ) ξ r ( ξ ) ϕ ( y ) = = − i √ 2 π Z γ + d ξ e ixξ r ( ξ )( T ϕ )( ξ ) so that (note that the x –int egral below is con vergen t b ecaus e ξ ∈ γ + ⊂ C + ) ( T ψ )( λ ) = 1 √ 2 π Z ∞ 0 e iλx ψ ( x )d x = − i Z ∞ 0 d x e iλx Z γ + d ξ e ixξ r ( ξ )( T ϕ )( ξ ) = = Z γ + d ξ r ( ξ ) λ + ξ ( T ϕ )( ξ ) = Z γ + d ξ r ( ξ ) λ + ξ ( T ϕ )( ξ ) . (2.8) W e note that for a function in H 2 r like f ( µ ) := T ϕ ( µ ), the ev a luation at a p oint ξ ∈ C + can b e written as f ( ξ ) = Z R f ( µ ) d µ 2 iπ ( µ − ξ ) , (2.9) which is Cauch y’s theorem. W e shall thus deﬁne b K T := T − 1 C T (2.10) (the reas o n for the transp os itio n is solely for later conv enience) with kernel given by b K T f ( λ ) = Z R d µ Z γ + d ξ r ( ξ ) λ + ξ f ( µ ) 2 iπ ( µ − ξ ) ⇒ b K f ( λ ) = Z R d µ Z γ + d ξ r T ( ξ ) λ − ξ f ( µ ) 2 iπ ( µ + ξ ) (2.11) Finally , s ince the F ourier Plancherel transform from L 2 ( R + , C r ) to H 2 r is an iso metr y , the resp ective F redholm determinan ts a re eq ual (if they exist). W e note that b K extends to an in tegral op er ator on 7 the whole of L 2 ( R , C r ) with the same kernel: this extensio n automatically annihilates the o rthogona l complement of H 2 r in L 2 ( R , C r ), which is seen by clos ing the µ –integral with a ha lf cir cle in the low er half pla ne and then invoking Cauch y’s theorem. W e will understand this ex tension in what follows. Therefore to conc lude we need to show that C and b K are tra ce-class. By the unitary equiv alence given b y the F our ier-Pla ncherel transfor m it suﬃces to s how that b K is tr ace class: w e shall prese nt b K as the pro duct of tw o Hilb ert-Schmidt o p e rators , th us proving it of tr a ce clas s. T o this end recall that r ( µ ) = E 1 ( µ ) E T 2 ( µ ); thus b K f ( λ ) = Z R d µ Z γ + d ξ E 2 ( ξ ) λ − ξ E T 1 ( ξ ) f ( µ )d µ 2 iπ ( µ + ξ ) = C 2 ◦ C 1 f ( λ ) . (2.12) where the t wo ope rators are de ﬁned a s follows C 1 : H 2 r ⊂ L 2 ( R , C r ) → L 2 ( γ + , C p ) f 7→ ( C 1 f )( ξ ) := E T 1 ( ξ ) Z R f ( µ )d µ 2 iπ ( µ + ξ ) (2.13) C 2 : L 2 ( γ + , C p ) → H 2 r h 7→ ( C 2 h )( λ ) := Z γ + E 2 ( ξ ) h ( ξ )d ξ − ξ + λ (2.14) W e em bed H 2 r and L 2 ( γ + , C p ) as subspaces o f L 2 ( R ∪ γ + , C r + p ) as H 2 r ∋ f 7→  f ( λ ) χ R ( λ ) ~ 0 p  , L 2 ( γ + , C p ) ∋ h 7→  ~ 0 r h ( λ ) χ γ + ( λ )  (2.15) (they are then orthogonal but not co mplement ary) and think o f C j as extended to the whole L 2 ( R ∪ γ + , C r + p ) in the trivia l way (i.e. a cting like zero o n the or thogonal co mplements o f the H 2 r , L 2 ( γ + , C p ), respe c tively). Analogo usly we extend trivially the action of b K to this enlarg ed Hilber t spa ce. Then it is promptly seen that they b oth are Hilbe r t Schmidt in L 2 ( R ∪ γ + , C r + p ) bec ause Z γ + | d ξ | Z R | d µ | T r  E † j ( ξ ) E j ( ξ )  | ξ ± µ | 2 < + ∞ (2.16) thanks to o ur ass umption that (the entries of ) E j are a ll in L 2 ( γ + , C ). Thus b K : H 2 r → H 2 r is tr ace class, so is C and their determina nts a re the sa me. Q.E.D Recall that if A : H 1 → H 2 and B : H 2 → H 1 are (b ounded) ope r ators be tw een Hilb ert spaces and b oth AB , B A are trace class then (see for instance [ 20 ]) T r H 1 ( B ◦ A ) = T r H 2 ( A ◦ B ) . (2.1 7 ) 8 Comp osing the op erator s C j in the opp osite order we obtain an oper ator on L 2 ( γ + , C p ) as follows ( C 1 ◦ C 2 f )( µ ) = E T 1 ( µ ) 2 iπ Z R d ξ Z γ + d λ E 2 ( λ ) f ( λ ) ( ξ − λ )( ξ + µ ) (2.18) The ξ –in tegral c an b e closed with a big circle in the upper –half plane, thus picking up the re s idue at ξ = λ to give ( C 1 ◦ C 2 f )( µ ) = E T 1 ( µ ) Z γ + d λ E 2 ( λ ) f ( λ ) λ + µ =: ( K f )( µ ) . (2.19) Renaming the v ariables we obtain that K ( λ, µ ) = E T 1 ( λ ) E 2 ( µ ) λ + µ . (2.20) The op erato r K is also trac e cla ss b ecaus e the compos itio n o f tw o Hilb ert- Schmidt op erator s in L 2 ( γ + ∪ R , C r + p ); clea rly it de ﬁnes an op era tor on L 2 ( γ + , C p ) since it acts trivially on its orthogo nal complement (by co nstruction). In par ticula r Corollary 2.1 The F r e dholm determinants of C : L 2 ( R + , C r ) , b K : H 2 r → H 2 r and K : L 2 ( γ + , C p ) → L 2 ( γ + , C p ) ar e al l e qual. Given the fact that the o per ator b K (with kernel ( 2.7 )) and K (with k er nel ( 2.20 )) have the sa me F redholm determinant, we shall co n tin ue o ur discuss ion by fo cusing o n the latter. In the sequel we will simply analyze op erator s with kernels as in ( 2.20 ) and forget a bo ut their origin as F ourie r transform of conv olution op erators . Remark 2.2 It is worth mentioning that op er ators with kernel ( 2.20 ) with p = r = 1 (and E 1 = E 2 ) b elong to the same class c onsider e d in [ 23 ]. Using our formalism it is p ossible to r e-derive the c onne ction b et we en the F r e dholm determinants of these op er ators and the mKdV/KdV hier ar chies. 3 Riemann-Hilb ert problems with diﬀeren t asymptotics and their m utual relationship Given a kernel K ( λ, µ ) as in ( 2.20 ) c o rresp onding to the oper ator (denoted b y the same symbol) K : L 2 ( γ + , C p ) → L 2 ( γ + , C p ) , K ( λ, µ ) := E T 1 ( λ ) E 2 ( µ ) λ + µ (3.1) 9 we constr uct tw o related Riemann-Hilb e rt problem on the colle c tion o f contours γ := γ + ∪ γ − (here γ − := − γ + ) and with jump matrix M ( λ ) := " 1 r − 2 iπ r ( λ ) χ γ + − 2 iπ e r ( λ ) χ γ − 1 r # (3.2) r ( λ ) = E 1 ( λ ) E T 2 ( λ ) ∈ Mat( r × r ) , e r ( λ ) := r ( − λ ) . (3.3) where χ X denotes the indica tor function of the set X . Her e and b elow we denote with σ i , i = 1 , 2 , 3 the Pauli matrices σ 1 :=  0 1 1 0  , σ 2 :=  0 i − i 0  , σ 3 :=  1 0 0 − 1  , σ + := ( δ i 1 δ j 2 ) i,j =1 , 2 and σ − its transp ose. F urthermore we shall s e t b σ k = 1 r ⊗ σ k , k = ± , 1 , 2 , 3 , (3.4) where by the tensor notation ca n b e taken to mean the matrix o f size 2 r × 2 r split int o 2 × 2 blo cks of size r × r . Note that the jump matrices M ( λ ) on γ := γ + ∪ γ − satisfy M ( − λ ) = b σ 1 M ( λ ) b σ 1 . (3.5) W e ar e g oing to for mulate tw o Riema nn-Hilber t Pro blems (Problems 3.1 , 3.2 ) a nd we will show how they are related betw een themselves (Pr op. 3.2 ) and how they r elate resp ectively to the tw o F redholm deter minants det(Id γ + + K ) a nd det(Id γ + − K 2 ) and the resolvents of the resp ective op erator s. In the sequel we shall assume that E j ( λ ) are smo oth (beside the a lready imp o s ed conditions E j ∈ L 2 ( γ + ) ∩ L ∞ ( γ + ) ⊗ Mat( r × p )). Problem 3.1 Find the se ctional ly analytic function Ξ( λ ) ∈ GL (2 r, C ) on C \ ( γ + ∪ γ − ) such that (with M ( λ ) given in ( 3.2 )) Ξ + ( λ ) = Ξ − ( λ ) M ( λ ) λ ∈ γ + ∪ γ − (3.6) Ξ( λ ) = 1 2 r + Ξ 1 λ + Ξ 2 λ 2 + . . . , λ → ∞ . (3.7) 10 Problem 3.2 Find the se ctional ly analytic function Γ( λ ) ∈ GL (2 r, C ) on C \ ( γ + ∪ γ − ) such that (with M ( λ ) given in ( 3.2 )) Γ + ( λ ) = Γ − ( λ ) M ( λ ) λ ∈ γ + ∪ γ − , (3.8) Γ( λ ) = L ( λ )   1 2 r + ∞ X j =1 Γ j λ j   , λ → ∞ (3.9) Γ( λ ) L − 1 ( λ ) = O (1) , λ → 0 (3.10) Γ( − λ ) = Γ( λ ) b σ 1 (3.11) Γ 1 = a 1 ⊗ σ 3 (3.12) wher e the matrix L ( λ ) is deﬁne d as fol lows L ( λ ) := 1 r ⊗ L ( λ ) =  1 r 1 r − iλ 1 r iλ 1 r  , L ( λ ) :=  1 1 − iλ iλ  (3.13) The v a lidit y of the asymptotic expansions near inﬁnit y needs additional conditions on the jump matrices if some component of γ + extends to inﬁnit y (if this happ ens we a ssume that these co m- po nents extend to inﬁnit y along asy mpto tic directions) . A suﬃcient c ondition, which we hereby tacitly as sume, is that r ( λ ) = O ( | λ | −∞ ) as | λ | → ∞ along any such comp onent a nd extends to an analytic function on an op en secto r containing the dir ection of appro ach in such a way that the same asymptotic holds. It is clear that the t wo Problems 3.1 , 3.2 are closely related and the remainder of this section is devoted to expla ining their mutual rela tionship. It is a stra ightforw a rd res ult that, if a solution of Problem 3.1 exists, then it is unique. The uniqueness of the so lution for the Problem 3.2 comes from the following prop osition. Prop ositi o n 3.1 L et Γ( λ ) b e a se ct ional ly analytic function that solves the RHP ( 3.8 , 3.9 , 3.10 , 3.11 ). If a solution ex ists then 1. det Γ( λ ) = (2 iλ ) r ; 2. Any matrix e Γ = ( 1 2 r + c ⊗ σ − )Γ solves the same RH P with c ∈ Mat( r × r ) , c onst ant ; 11 3. Any solution has an exp ansion wher e the terms Γ j in ( 3.9 ) have the symmetry Γ( λ ) = L ( λ ) 1 2 r + ∞ X k =1 Γ k λ k ! , Γ j = ( − 1) j b σ 1 Γ j b σ 1 (3.14) and henc e Γ 2 j = a 2 j ⊗ 1 2 + b 2 j ⊗ σ 1 , Γ 2 j +1 = a 2 j +1 ⊗ σ 3 + b 2 j +1 ⊗ σ 2 , a j , b j ∈ Mat( r × r ) (3.15 ) 4. If we additional ly r e quir e t he c ondition ( 3.12 ) Γ 1 = a 1 ⊗ σ 3 (for some c onstant matrix a 1 ) then the solut ion is unique. This solut ion wil l b e r eferr e d to as the gauge- ﬁxed solution. Pro of of Prop. 3.1 . 1. It is cle a r that the determinant has no jumps b eca use the jump matrices are unimo dular. Mor eov er , from det L = (2 iλ ) r (see ( 3.13 )) and ( 3.9 ) we ha ve det Γ( λ ) = (2 iλ ) r (1 + O ( λ − 1 )) . Finally from ( 3.10 ) we hav e det Γ = O (det L ) = O ( λ r ), λ → 0, and hence it must b e det Γ ≡ (2 iλ ) r . 2. W e note that ( 1 2 r + c ⊗ σ − ) L ( λ ) = L ( λ )  1 2 r + ic 2 λ ⊗ ( σ 3 − iσ 2 )  (3.16) and hence the m ultiplication on the left b y a c o nstant matrix of such a form do es not c ha nge the form of the asymptotic expansion a nd do es not change the jump co nditions. This prov es the s e cond po int . 3. The statement is obvious once one notices that L ( − λ ) = L ( λ ) b σ 1 . 4. Suppo se e Γ is another solution satisfying the same requirements and denote by e a j , e b j the co eﬃcients in its expans io n as p er ( 3.15 ). By p oint 1, an y tw o solutions have the sa me determinant; the ratio S ( λ ) := e Γ( λ )Γ − 1 ( λ ) (3.17) m ust b e a holomorphic matrix function on C \ { 0 } . How ever, fr om the co ndition ( 3.10 ) we see that actually S ( λ ) must b e a nalytic at 0 as well. Lo oking a t the behaviour at inﬁnity o f e Γ and Γ one ﬁnds by a direct computation that S ( λ ) is b ounded a nd S ( λ ) =  1 r 0 ia 1 − i e a 1 1 r  (3.18) Suppo se now that e a 1 6 = a 1 ; then e Γ( λ ) =  1 r 0 ia 1 − i e a 1 1 r  Γ( λ ) (3.19) 12 But then one sees by direct ma trix mult iplication that e b 1 should equal e a 1 − a 1 2 which violates the normalizatio n e b 1 = 0. This pr ov es uniqueness. Q.E. D In P rop osition 3.2 and Prop os itio n 3.3 we study the relationship betw een the Riemann–Hilbe r t problems 3.1 and 3.2 : in particular w e s hall see that they are not equiv alen t , in the sense that if Problem 3.1 admits a so lution then s o do es Pr oblem 3.2 but, in g eneral, not viceversa. W e start by obser ving that the symmetry ( 3.5 ) for the jump matrices implies the s ame symmetry for Ξ Ξ( − λ ) = b σ 1 Ξ( λ ) b σ 1 (3.20) which in turns implies the following form for the co eﬃcient Ξ j in ( 3.7 ) Ξ( λ ) = 1 2 r + ∞ X k =1 Ξ k λ k , Ξ 2 j +1 = α 2 j +1 ⊗ σ 3 + β 2 j +1 ⊗ σ 2 , Ξ 2 j = α 2 j ⊗ 1 2 + β 2 j ⊗ σ 1 . (3.2 1 ) Prop ositi o n 3.2 L et Ξ b e the solution of Pr oblem 3.1 ; then the solution of Pr oblem 3.2 is Γ( λ ) =  1 r 1 r − iλ 1 r − 2 β 1 iλ 1 r − 2 β 1  Ξ( λ ) (3.22) with β 1 as in ( 3.21 ). In addition the c o eﬃcients of the exp ansions for Γ ( 3.15 ) and Ξ ( 3.21 ) satisfy a 1 = α 1 − iβ 1 , b 1 = 0 (3.23) a 2 j +1 = α 2 j +1 + iβ 1 ( β 2 j − α 2 j ) , b 2 j +1 = β 2 j +1 + β 1 ( β 2 j − α 2 j ) a 2 j = α 2 j − iβ 1 ( α 2 j − 1 − iβ 2 j − 1 ) , b 2 j = β 2 j − iβ 1 ( α 2 j − 1 − iβ 2 j − 1 ) (3.24) Pro of of Prop. 3.2 . Since Γ and Ξ hav e the same jumps we mu st hav e Γ( λ ) = R ( λ )Ξ( λ ) for some R ( λ ) at most p olyno mia l. F rom the symmetries we must have R ( − λ ) = R ( λ ) b σ 1 and det R = (2 iλ ) r . The expansion of Γ and Ξ a t inﬁnit y forces R to b e of the for m R ( λ ) =  1 r 1 r − iλ 1 r + 2 i c iλ 1 r + 2 i c  (3.25) On the other hand, as we presently show, the gauge ﬁxing ( 3.12 ) determines C ; indeed L − 1 R = 1 2 r + c λ ⊗  − 1 − 1 1 1  = 1 2 r + 1 λ ( − c ⊗ σ 3 + ic ⊗ σ 2 ) (3.26) and therefore in the expansions of Γ and Ξ a nd ma trix m ultiplica tions w e have 1 2 r + ∞ X j =0 a 2 j +1 ⊗ σ 3 + b 2 j +1 ⊗ σ 2 λ 2 j +1 + ∞ X j =1 a 2 j ⊗ 1 2 + b 2 j ⊗ σ 1 λ 2 j = (3.27) = 1 2 r + ( α 1 − c ) ⊗ σ 3 + ( β 1 + ic ) ⊗ σ 2 λ + 13 + ∞ X j =1 ( α 2 j − c ( α 2 j − 1 − iβ 2 j − 1 )) ⊗ 1 2 + ( β 2 j − c ( α 2 j − 1 − iβ 2 j − 1 )) ⊗ σ 1 λ 2 j + + ∞ X j =1 ( α 2 j +1 − cα 2 j + cβ 2 j ) ⊗ σ 3 + ( β 2 j +1 − icβ 2 j + icα 2 j ) ⊗ σ 2 λ 2 j +1 (3.28) The gauge ﬁxing ( 3.12 ) ma ndates b 1 = 0 (i.e. the co eﬃcient matrix of σ 2 in the term λ − 1 m ust b e absent) s o that we must hav e c = i β 1 and equating the co eﬃcients of the expansion a b ov e implies ( 3.24 ). It only remains to show that R ( λ )Ξ( λ ) L − 1 ( λ ) is b ounded at λ = 0 (condition ( 3.10 )). Since L − 1 = 1 r ⊗ L − 1 and L − 1 ( λ ) has only simple p ole at λ = 0, then R ( λ )Ξ( λ ) L − 1 ( λ ) = c 0 λ + O (1) as λ → 0. On the other hand the sy mmetries imply R ( λ )Ξ( λ ) L − 1 ( λ ) = R ( − λ )Ξ( − λ ) L − 1 ( − λ ) and hence c 0 = − c 0 so that c 0 = 0. Q . E.D Prop ositi o n 3.3 L et Γ( λ ) ∈ GL (2 r, C ) b e the solution of Pr oblem 3.2 and denote t he r × r blo cks of Γ by Γ ij , i , j = 1 , 2 ; then the solution of Pr oblem 3.1 for Ξ exists if and only if det Γ 11 (0) 6 = 0 . Mor e over β 1 = − 1 2 Γ − 1 11 (0)Γ 21 (0) = − i lim λ →∞ λ Ξ 12 ( λ ) Pro of o f Prop. 3.3 . Let Γ( λ ) b e the so lution of Pro blem 3 .2 . In particular Γ( λ ) is b ounded everywhere (by deﬁnition) and we want now to ﬁnd a matrix R ( λ ) of the for m ( 3.25 ) such that Ξ( λ ) := R − 1 ( λ )Γ( λ ) (3.29) solves Problem 3.1 . It is clear that the jumps will b e automatica lly satisﬁed a nd so the asymptotic behaviour at inﬁnity . The v alue of the constant matrix c must b e determined b y the requirement that Ξ is b ounde d at λ = 0. F rom the symmetry ( 3.11 ) we hav e the matrix equations Γ 11 (0) = Γ 12 (0) , Γ 21 (0) = Γ 22 (0) . (3.30) A direct computation yields R − 1 ( λ )Γ( λ ) = O (1) , λ → 0 (3.31)   1 r 2 + c λ − 1 r 2 iλ 1 r 2 − c λ 1 r 2 iλ   Γ( λ ) = O (1) (3.32) c Γ 11 (0) + i 2 Γ 21 (0) = 0 , ⇒ c = − i 2 Γ − 1 11 (0)Γ 21 (0) (3.33) c Γ 12 (0) + i 2 Γ 22 (0) = 0 , (3.34) 14 The t wo e q uations a re the same due to ( 3.3 0 ). Now, if det Γ 11 (0) 6 = 0 then the so lution fo r c is as in ( 3.33 ) and the suﬃciency is pr ov ed. As for the necessity , if det Γ 11 (0) = 0, then the equation ( 3.33 ) may still b e compatible. Howev er this would mean that there a re inﬁnitely many c that solve the matrix equation ( 3.33 ), which would viola te the uniqueness of the RHP 3.1 . Q.E .D W e conclude the section with the following tw o theorems, which we state side-by-side for the sake of easy compariso n. Theorem 3. 1 L et K ( λ, µ ) b e the inte gr al op er ator on L 2 ( γ + , C p ) with kernel K ( λ, µ ) := E 1 ( λ ) T E 2 ( µ ) λ + µ (3.35) Then t he r esolvent op er ator R ++ = K 2 ◦ (Id γ + − K 2 ) − 1 of Id γ + − K 2 on L 2 ( γ + , C p ) has kernel R ++ ( λ, µ ) given by R ++ ( λ, µ ) = [ E T 1 ( λ ) , 0 p × r ] Ξ T ( λ )Ξ − T ( µ ) λ − µ  0 r × p E 2 ( µ )  (3.36) wher e Ξ is t he solution of Pr oblem 3.1 with the jump matrix ( 3.2 ). If r ( λ ) := E 1 ( λ ) E T 2 ( λ ) is symmetric, r = r T (for example if E 1 = E 2 = E ) then t he r esolvent c an b e written mor e symmet r ic al ly as R ++ ( λ, µ ) = i [ E T 1 ( λ ) , 0 p × r ] Ξ T ( λ ) b σ 2 Ξ( µ ) λ − µ  E 2 ( µ ) 0 r × p  (3.37) The s olution t o Pr oblem 3.1 ex ists if and only if the op er ator Id γ + − K 2 is invertible. Theorem 3. 2 L et K ( λ, µ ) b e t he inte gr al same op er ator as in Thm. 3.1 . Then the r esolvent op er ator S = −K ◦ (Id γ + + K ) − 1 has kernel S ( λ, µ ) given by S ( λ, µ ) = 2 µ  E T 1 ( λ ) , 0 p × r  Γ T ( λ )Γ − T ( µ )  0 r × p E 2 ( µ )  λ 2 − µ 2 (3.38) wher e Γ is the solut ion of Pr oblem 3.2 with the jump matrix ( 3.2 ). If r ( λ ) := E 1 ( λ ) E T 2 ( λ ) is sym- metric, r = r T (for example if E 1 = E 2 = E ) then the r esolvent c an b e written mor e symmetric al ly 15 as S ( λ, µ ) =  E T 1 ( λ ) , 0 p × r  Γ T ( λ ) b σ 2 Γ( µ )  E 2 ( µ ) 0 r × p  λ 2 − µ 2 (3.39) This s olut ion t o Pr oblem 3.2 ex ists if and only if the op er ator Id γ + + K is invertible. Pro of of Thm. 3.1 . W e s tart o bs erving that the jump M ( λ ) in Problem 3.1 can b e written as M ( λ )= 1 − 2 iπ f ( λ ) g T ( λ ) (3.40) f ( λ )=  E 1 ( λ ) 0 r × p  χ γ + ( λ ) +  0 r × p e E 1 ( λ )  χ γ − ( λ ) (3.41) g ( λ )=  0 r × p E 2 ( λ )  χ γ + ( λ ) +  e E 2 ( λ ) 0 r × p  χ γ − ( λ ) , e E j ( λ ) := E j ( − λ ) . (3.42) By the I IKS theory , this RHP is a sso ciated to the kernel N acting on L 2 ( γ + ∪ γ − , C p ) with k ernel given by N ( λ, µ ) = f T ( λ ) g ( µ ) λ − µ = E T 1 ( λ ) e E 2 ( µ ) χ γ + ( λ ) χ γ − ( µ ) + e E T 1 ( λ ) E 2 ( µ ) χ γ − ( λ ) χ γ + ( µ ) λ − µ (3.43) According to the split L 2 ( γ + ∪ γ − ) = L 2 ( γ + ) ⊕ L 2 ( γ − ), using the naturally r elated matrix notation, we can write N a s N =  0 G F 0  , G : L 2 ( γ − , C p ) → L 2 ( γ + , C p ) , F : L 2 ( γ + , C p ) → L 2 ( γ − , C p ) (3.44) where the op e rators F and G a re in tegral ope r ators with kernels G ( λ, ξ ) = E T 1 ( λ ) e E 2 ( ξ ) χ γ + ( λ ) χ γ − ( ξ ) λ − ξ , F ( ξ , µ ) = e E T 1 ( ξ ) E 2 ( µ ) χ γ − ( ξ ) χ γ + ( µ ) ξ − µ (3.45) W e observe that the kernel of the comp osition reads ( G ◦ F )( λ, µ ) = E T 1 ( λ ) Z γ − e E 2 ( ξ ) e E T 1 ( ξ ) ( λ − ξ )( ξ − µ ) dξ ! E 2 ( µ ) = (3.46 ) E T 1 ( λ ) Z γ + E 2 ( ξ ) E T 1 ( ξ )d ξ ( λ + ξ )( ξ + µ ) ! E 2 ( µ ) = K 2 ( λ, µ ) (3.47) and hence our tas k of computing the r e solven t of Id γ + − K 2 is the sa me as computing the reso lven t of Id γ + − G ◦ F . T o this end we write ﬁr st the r esolven t of Id γ + ∪ γ − − N (using [ 11 ]): R ( λ, µ ) = f T ( λ )Ξ T ( λ )Ξ − T ( µ ) g ( µ ) λ − µ (3.48) 16 according to the pro jections in L 2 ( γ ± , C p ): R ( λ, µ )= R ++ ( λ, µ ) + R − + ( λ, µ ) + R + − ( λ, µ ) + R −− ( λ, µ ) = [ E T 1 ( λ ) , 0 p × r ] Ξ T ( λ )Ξ − T ( µ ) λ − µ  0 r × p E 2 ( µ )  χ γ + ( λ ) χ γ + ( µ ) + [ 0 p × r , e E T 1 ( λ )] Ξ T ( λ )Ξ − T ( µ ) λ − µ  0 r × p E 2 ( µ )  χ γ − ( λ ) χ γ + ( µ ) + +[ E T 1 ( λ ) , 0 p × r ] Ξ T ( λ )Ξ − T ( µ ) λ − µ  e E 2 ( µ ) 0 r × p  χ γ + ( λ ) χ γ − ( µ ) + [ 0 p × r , e E T 1 ( λ )] Ξ T ( λ )Ξ − T ( µ ) λ − µ  e E 2 ( µ ) 0 r × p  χ γ − ( λ ) χ γ − ( µ ) (3.49) where the four addenda app ears in the ma trix notation induced by the splitting L 2 ( γ + ∪ γ − ) = L 2 ( γ + ) ⊕ L 2 ( γ − ); (Id γ + ∪ γ − − K ) − 1 =  Id γ + + R ++ R + − R − + Id γ − + R −−  On the other hand w e hav e  (Id γ + − G ◦ F ) − 1 0 F ◦ (Id γ + − G ◦ F ) − 1 Id γ −  =  Id γ + − G ◦ F 0 −F Id γ −  − 1 = =  Id γ + −G −F Id γ −  − 1 ◦  Id γ + −G 0 Id γ −  =  Id γ + + R ++ R + − R − + Id γ − + R −−  ◦  Id γ + −G 0 Id γ −  so that the en try (1 , 1) o f the equation ab ov e gives (Id γ + − G ◦ F ) − 1 = Id γ + + R ++ and the equation ( 3.49 ) gives the precise form of the kernel R ++ ( λ, µ ). In case of symmetry r = r T this form simpliﬁes b ecause Ξ − 1 ( λ ) = b σ 2 Ξ T ( λ ) b σ 2 (whic h is prov ed along the same lines as in Thm. 3.2 ). The s ta tement ab out the existence is a dire c t applica tion of IIK S theory . Q.E .D Pro of of Thm. 3.2 . The idea of the pro of is to reduce as muc h as p ossible the theorem to the theory of integrable oper ators of Its- Izergin-K orepin-Slavnov (IIKS). W e can write the o p erator as K ( λ, µ ) := E 1 ( λ ) T E 2 ( µ ) λ + µ = ( λ − µ ) E 1 ( λ ) T E 2 ( µ ) λ 2 − µ 2 (3.50) W e no w intro duce the co ordina te z := λ 2 and w := µ 2 . Since γ + is in the uppe r half-plane, its imag e under the square map is w e ll- deﬁned and lies in C \ R + . Since the arc- length of γ + diﬀers in the z -plane and λ -plane, we must introduce the squar e-ro ots of the Jacobia ns. The in tegral op erator ( 3.50 ) reads K ( z , w ) := ( √ z − √ w ) E 1 ( √ z ) T E 2 ( √ w ) 2( z w ) 1 4 ( z − w ) = −  E T 1 ( √ z ) , − i √ z E T 1 ( √ z )   E 2 ( √ w ) − i √ w E 2 ( √ w )  2( z − w )  w z  1 4 (3.51) 17 W e hav e to construct the resolvent of Id γ + + K = Id γ + − ( −K ). W e have now an integrable kernel in the sense of Its-Izergin-K orepin-Slavnov where the matrices f , g can b e chosen as ( −K )( z , w ) = f T ( z ) g ( w ) z − w , f ( z ) = 1 4 √ z  E 1 ( √ z ) − i √ z E 1 ( √ z )  , g ( z ) = 1 2 4 √ z " E 2 ( √ z ) − iE 2 ( √ z ) √ z # (3.52) W e immediately observe that f ( z ) = 1 4 √ z L ( √ z )  E 1 ( √ z ) 0 r × p  =: L ( √ z ) f 0 ( z ) , (3.53) g ( z ) = 4 √ z ( L − 1 ) T ( √ z )  0 r × p E 2 ( √ z )  = L − T ( z ) g 0 ( z ) . (3.54) The construction of the res olven t is then as so ciated, in the standar d wa y [ 11 ], to the following Riemann–Hilb ert Pro blem Θ( z ) + = Θ( z ) −  1 2 r − 2 i π f ( z ) g T ( z )  , z ∈ γ + (3.55) Θ( z ) = 1 2 r + O ( z − 1 ) , z → ∞ (3.56) W e can r ewrite the jump ma trix as follows 1 2 r − 2 i π L ( √ z )( r ( √ z ) ⊗ σ + ) L − 1 ( √ z ) (3.57) r ( √ z ) := E 1 ( √ z ) E T 2 ( √ z ) (3.58) Consequently we introduce the ne w matrix Θ( z ) L ( √ z ), where L ( √ z ) := 1 r ⊗ L ( √ z ). In order to connect with Problem 3.2 we deﬁne b Γ( λ ) = Θ( λ 2 ) L ( λ ) (3.59) and w e see immediately that b Γ( − λ ) = b Γ( λ ) b σ 1 . F urthermore b Γ( λ ) L − 1 ( λ ) = Θ( λ 2 ) = O (1) as λ → 0. Thu s b Γ solves Problem 3.2 ex cept for the gaug e-ﬁxing ( 3.12 ), whic h we now ta ke into considera tio n: if we denote by a 1 the (1 , 2) blo ck of size r × r in Θ( z ) = 1 + 1 z  ⋆ − a 1 ⋆ ⋆  + O ( z − 2 ) (3.60) then one veriﬁes by matrix multiplication that the rela tion b etw een b Γ and the gauge–ﬁxed Γ is ( 1 2 r + a 1 ⊗ σ + ) b Γ( z ) = Γ( z ) , (3.61) 18 The resolvent op er ator, acco rding to the gene r al theor y , is S ( λ, µ ) = f T ( z )Θ T ( z )Θ − T ( w ) g ( w ) z − w √ d z d w = (3.62) = f T 0 ( z ) L T ( λ )Θ T ( λ 2 )Θ − T ( µ 2 ) L − T ( µ ) g 0 ( w ) z − w √ d z d w = (3.63) = 2 p λµ f T 0 ( λ )Γ T ( λ )Γ − T ( µ ) g 0 ( µ ) λ 2 − µ 2 p d λ d µ = (3.64) = 2 µ  E T 1 ( λ ) , 0 p × r  Γ T ( λ )Γ − T ( µ )  0 r × p E 2 ( µ )  λ 2 − µ 2 (3.65) Finally , note that L − T ( λ ) = 1 2 iλ b σ 2 L ( λ ) b σ 2 (3.66) and if r = r T then Γ − T ( µ ) = 1 2 iµ b σ 2 Γ( µ ) b σ 2 (3.67) which ca n b e chec ked by verifying that 2 iµ b σ 2 Γ − T ( µ ) b σ 2 solves the same P roblem 3.2 and hence equals Γ. In this ca se the formula fo r the re solven t takes a more symmetric form S ( λ, µ ) =  E T 1 ( λ ) , 0 p × r  Γ T ( λ ) b σ 2 Γ( µ )  E 2 ( µ ) 0 r × p  λ 2 − µ 2 (3.68) As for the statement of existence ; Γ exists if and only if Θ exis ts, which is equiv a lent to the inv ertibility of the mentioned op era tor by the IIK S gener al theor y . Q.E.D 4 T au functions and F redholm determinan ts Slightly generalizing the deﬁnition in [ 2 ] (which is itself a gener alization of the notio n of isomon- o dromic tau function intro duce d in the work of Jim bo -Miwa-Ueno [ 15 , 13 , 14 ]) we a sso ciate, to the space of deformations of the Riemann–Hilb ert problems 3.1 and 3.2 the tw o diﬀerent ials below. Deﬁnition 4.1 W e deﬁne t he two forms over the sp ac e of deformations of Pr oblem 3.2 and Pr oblem 3.1 ω Ξ ( ∂ ) := Z T r  Ξ − 1 − Ξ ′ − ∂ M M − 1  d λ 2 iπ (4.1) 19 and ω Γ ( ∂ ) := 1 2 Z T r  Γ − 1 − Γ ′ − ∂ M M − 1  d λ 2 iπ (4.2) wher e ∂ denotes any deformatio n of t he jump matric es, ’ the derivatives with r esp e ct t o the sp e ctr al p ar ameter and the int e gr ation is extende d to al l the c ontours wher e the jumps ar e supp orte d, γ + ∪ γ − . In the ca ses in w hich these tw o diﬀeren tia l forms are clo sed it is deﬁned, up to a consta nt , the corres p o nding tau function given by ∂ ln τ Ξ / Γ = ω Ξ / Γ ( ∂ ) . A particular case (of great interest) of deformations is when the jump matr ices hav e the form M ( λ ; s ) = e T ( λ ) M 0 ( λ )e − T ( λ ) , (4.3) and T ( λ ) is a diagonal matrix depending on deformatio n parameter s, while M 0 ( λ ) is assumed independent of them. A t y pical case is T ( λ ) = P N k =0 T k λ k and the diagona l matrices T k are taken as deformation parameters . The r elation b etw een the Deﬁnition 4.1 and F redholm determinants is elucidated in the following t wo theorems, stated side-by-side for compar ison. Theorem 4. 1 Given an op er ator K as in Se ction 2 and the Riemann-Hilb ert Pr oblem 3.1 (with the same r ( µ ) ) we have the e quality ∂ ln τ Ξ = Z γ + ∪ γ − T r  Ξ − 1 − Ξ ′ − ∂ M M − 1  d λ 2 iπ = ∂ ln det(Id γ + − K 2 ) (4.4) Theorem 4. 2 Given an op er ator K as in Se ction 2 and the Riemann-Hilb ert Pr oblem 3.2 (with the same r ( µ ) ) we have the e quality ∂ ln τ Γ = 1 2 Z γ + ∪ γ − T r  Γ − 1 − Γ ′ − ∂ M M − 1  d λ 2 iπ = ∂ ln det(Id γ + + K ) (4.5) Pro of of Thm. 4.1 . In [ 3 ] (see Theorem 2.1) it was pr ov ed (for the case of scala r ope r ators, but the pro of do es not diﬀer signiﬁcantly as we see below) that ∂ ln τ Ξ = ∂ ln det(Id γ + ∪ γ − − N ) (4.6) 20 where the in tegral oper ator N , acting o n L 2 ( γ + ∪ γ − ) is the o ne ex pr essed in ( 3.4 3 ). On the other hand we hav e the identit y det(Id γ + ∪ γ − − N ) = det(Id γ + − F ◦ G ) = det(Id γ + − K 2 ) (4.7) where the ﬁrst equality follows from det  Id γ + ∪ γ − −  0 G F 0  = det  Id γ + ∪ γ − −  0 G F 0  det  Id γ − G 0 Id γ +  = (4.8) det  Id γ + ∪ γ − −  0 0 F F ◦ G  = det  Id γ + − F ◦ G  . (4.9) and the second is ( 3.47 ). The ab ove computation is formal inasmuc h as one w o uld nee d to prov e that all the op erator s inv o lved ar e of trace-cla ss. T o see that we now prove that b oth F , G a re trace-clas s in L 2 ( γ + ∪ γ − , C p ). Reca lling their deﬁnition ( 3.45 ) we augmen t the Hilber t s pace as b H := L 2 ( γ + ∪ γ − , C p ) ⊕ L 2 ( R , C r ), and extend trivia lly the deﬁnition of F , G to the augmented space. This allows to repr esent them a s the comp osition o f tw o Hilber t–Schmidt op e rators (thus immediately implying the trace class pr op erty). Indeed (for example for F ) we ha ve the identit y below F ( ξ , µ ) = e E T 1 ( ξ ) E 2 ( µ ) χ γ − ( ξ ) χ γ + ( µ ) ξ − µ = Z R d ζ 2 iπ e E T 1 ( ξ ) χ γ − ( ξ ) ( ξ − ζ ) E 2 ( µ ) χ γ + ( µ ) ( ζ − µ ) (4.10) which follows fro m Cauch y ’s residue theorem by clo sing the ζ integration either in the upp er or in the lower ha lf-plane. This realizes F as the comp os ition of tw o o p er ators b etw een the subspa ces L 2 ( γ + , C p ) → L 2 ( R , C r ) → L 2 ( γ − , C p ), each of whic h is Hilb ert– Schm idt: Z γ − | d ξ | Z R | d ζ | T r  e E † 1 ( ξ ) e E 1 ( ξ )  | ξ − ζ | 2 < ∞ > Z γ + | d µ | Z R | d ζ | T r  E † 2 ( µ ) E 2 ( µ )  | µ − ζ | 2 (4.11) F or the s ake o f self-containedness we shall re-der ive ( 4.6 ) b elow. let us denote by Ξ = [ A , B ] the tw o blo ck-columns of Ξ (of sizes 2 r × r ) and by Ξ − 1 =  C D  the blo c k rows of Ξ − 1 (of s izes r × 2 r ). Then Z γ + ∪ γ − T r  Ξ − 1 − Ξ ′ − ∂ M M − 1  d λ 2 iπ = (4.12) − Z γ + T r  C − D −  [ A ′ − , B ′ − ] ∂ r ⊗ σ +  d λ − Z γ − T r  C − D −  [ A ′ − , B ′ − ] ∂ e r ⊗ σ −  d λ = (4.13) − Z γ + T r ( D A ′ ∂ r ) d λ − Z γ − T r ( CB ′ ∂ e r ) d λ (4.14) 21 where we have used that C and A are a nalytic acros s γ + and D , B a na lytic acros s γ − . On the other hand the jump relations imply A ( λ ) =  1 r 0 r × r  − Z γ − B ( µ ) e r ( µ )d µ µ − λ , B ( λ ) =  0 r × r 1 r  − Z γ + A ( µ ) r ( µ )d µ µ − λ (4.15) and these iden tities can b e diﬀerentiated on γ + (for A ) and γ − (for B ). W e thus hav e − Z γ + T r ( D A ′ ∂ r ) d λ − Z γ − T r ( CB ′ ∂ e r ) d λ = (4.16) = Z γ + d λ Z γ − d µ T r  D ( λ ) B ( µ ) e r ( µ ) ( µ − λ ) 2 ∂ r ( λ )  d λ + Z γ − d λ Z γ + d µ T r  C ( λ ) A ( µ ) r ( µ ) ( µ − λ ) 2 ∂ e r ( λ )  (4.17) On the other hand these t wo terms exactly compute 11 − T r γ + ( R + − ◦ ∂ F ) − T r γ − ( R − + ◦ ∂ G ) = − T r γ + ∪ γ −  Id γ + ∪ γ − + R  ◦ ∂ N  = = ∂ ln det(Id γ + ∪ γ − − N ) (4.18) where R is the resolvent o f the o pe rator with kernel N alrea dy us ed in ( 3.43 ) a nd R has b een decomp osed as in ( 3.49 ) and F , G deﬁned in ( 3.45 ). Q .E.D Pro of of Thm. 4.2 W e star t a nalyzing the l.h.s. of the equatio n and observing that ∂ ln τ Γ is equal to 1 2 ∂ ln τ Ξ plus an a dditional term. Indeed, from Γ = R ( λ )Ξ( λ ) with R deﬁned in ( 3.25 ), we ﬁnd Γ − 1 Γ ′ = Ξ − 1 Ξ ′ + Ξ − 1 R − 1 R ′ Ξ = Ξ − 1 Ξ ′ + 1 2 λ Ξ − 1 ( λ )( 1 − b σ 1 )Ξ( λ ) (4.19) so that 2 ∂ ln τ Γ − ∂ ln τ Ξ = Z γ + ∪ γ − T r  Γ − 1 − Γ ′ − ∂ M M − 1  d λ 2 iπ − Z γ + ∪ γ − T r  Ξ − 1 − Ξ ′ − ∂ M M − 1  d λ 2 iπ = = Z γ + ∪ γ − 1 2 λ T r  Ξ − 1 − ( λ ) b σ 1 Ξ − ( λ )  0 r × r ∂ r χ + ∂ e r χ − 0 r × r  d λ = = Z γ + 1 2 λ T r  Ξ − 1 − ( λ ) b σ 1 Ξ − ( λ ) b σ + ∂ r ( λ )  d λ + Z γ − 1 2 λ T r  Ξ − 1 − ( λ ) b σ 1 Ξ − ( λ ) b σ − ∂ e r ( λ )  d λ = = Z γ + 1 λ T r  Ξ − 1 − ( λ ) b σ 1 Ξ − ( λ ) b σ + ∂ r ( λ )  d λ = Z γ + 1 λ T r ( D ( λ ) b σ 1 A ( λ ) ∂ r ( λ )) d λ (4.20) 11 here we used the symmetries B ( λ ) = b σ 1 A ( − λ ) , D ( λ ) = C ( − λ ) b σ 1 . 22 W e want to identify this la st integral; recall the notation Ξ − 1 = h C D i and that the jumps for Ξ − T imply for the column C T C T ( λ ) =  1 r 0  + Z γ + D T ( ξ ) r T ( ξ )d ξ ξ − λ (4.21) Moreov er, by deﬁnition of inv er s es we hav e C ( λ ) A ( λ ) ≡ 1 r ≡ D ( λ ) B ( λ ) . (4.22) Consider ( R ++ ◦ ∂ K )( λ, µ ) = Z γ + R ++ ( λ, ξ ) E T 1 ( ξ ) ∂ E 2 ( µ ) + ∂ E T 1 ( µ ) E 2 ( ξ ) µ + ξ d ξ = = Z γ + E T 1 ( λ ) A T ( λ ) D T ( ξ ) E 2 ( ξ ) λ − ξ E T 1 ( ξ ) ∂ E 2 ( µ ) + ∂ E T 1 ( ξ ) E 2 ( µ ) ξ + µ d ξ = = E T 1 ( λ ) A T ( λ ) Z γ + D T ( ξ ) r T ( ξ )d ξ ( ξ + µ )( λ − ξ ) ! ∂ E 2 ( µ ) + Z γ + E 1 ( λ ) A T ( λ ) D T ( ξ ) E 2 ( ξ ) λ − ξ ∂ E T 1 ( ξ ) E 2 ( µ ) ξ + µ d ξ = = E T 1 ( λ ) A T ( λ ) µ + λ Z γ +  1 ξ + µ − 1 ξ − λ  D T ( ξ ) r T ( ξ )d ξ ! ∂ E 2 ( µ ) + + Z γ + E T 1 ( λ ) A T ( λ ) D T ( ξ ) E 2 ( ξ ) λ − ξ ∂ E T 1 ( ξ ) E 2 ( µ ) ξ + µ d ξ = = − E T 1 ( λ ) A T ( λ ) µ + λ  C T ( λ ) − C T ( − µ )  ∂ E 2 ( µ ) + Z γ + E T 1 ( λ ) A T ( λ ) D T ( ξ ) E 2 ( ξ ) λ − ξ ∂ E T 1 ( ξ ) E 2 ( µ ) ξ + µ d ξ = = − E T 1 ( λ ) A T ( λ ) µ + λ  C T ( λ ) − b σ 1 D T ( µ )  ∂ E 2 ( µ ) + Z γ + E T 1 ( λ ) A T ( λ ) D T ( ξ ) E 2 ( ξ ) λ − ξ ∂ E T 1 ( ξ ) E 2 ( µ ) ξ + µ d ξ = = E T 1 ( λ ) A T ( λ ) b σ 1 D T ( µ ) ∂ E 2 ( µ ) µ + λ − E T 1 ( λ ) ∂ E 2 ( µ ) λ + µ + Z γ + E T 1 ( λ ) A T ( λ ) D T ( ξ ) E 2 ( ξ ) λ − ξ ∂ E T 1 ( ξ ) E 2 ( µ ) ξ + µ d ξ (4.2 3) (note that we used ( 4.21 )). T aking the trace we hav e to set µ = λ and in teg rate ov er γ + : the last term in ( 4.23 ) can then be s impliﬁed as w ell Z γ + d λ Z γ + d ξ T r  E T 1 ( λ ) A T ( λ ) D T ( ξ ) E 2 ( ξ ) ∂ E T 1 ( ξ ) E 2 ( λ )  ( λ − ξ )( ξ + λ ) = (4.24) = Z γ + d λ Z γ + d ξ T r  r T ( λ ) A T ( λ ) D T ( ξ ) E 2 ( ξ ) ∂ E T 1 ( ξ )  ( λ − ξ )( ξ + λ ) = = T r " Z γ + d λ Z γ + d ξ r T ( λ ) A T ( λ ) 1 2 ξ  1 λ − ξ − 1 λ + ξ  D T ( ξ ) E 2 ( ξ ) ∂ E T 1 ( ξ ) # = (4.25) = − T r " Z γ + B T ( ξ ) − B T ( − ξ ) 2 ξ D T ( ξ ) E 2 ( ξ ) ∂ E T 1 ( ξ )d ξ # = (4.26) 23 = − Z γ + T r  ( 1 r − A T ( ξ ) b σ 1 D T ( ξ )) E 2 ( ξ ) ∂ E T 1 ( ξ )  2 ξ d ξ (4.27) T aking the trace of ( 4.23 ) we thus hav e T r ( R ++ ◦ ∂ K ) = − =T r ∂ K z }| { Z γ + d λ T r  E T 1 ( λ ) ∂ E 2 ( λ ) + ∂ E T 1 ( λ ) E 2 ( λ )  2 λ (4.28) + T r " Z γ + d ξ A T ( ξ ) b σ 1 D T ( ξ ) ∂ E 2 ( ξ ) E T 1 ( ξ ) 2 ξ + Z γ + d ξ A T ( ξ ) b σ 1 D T ( ξ ) E 2 ( ξ ) ∂ E T 1 ( ξ ) 2 ξ # (4.29) T ogether we thu s ha ve T r ( R ++ ◦ ∂ K ) = − T r ∂ K + Z γ + d λ T r  A T ( λ ) b σ 1 D T ( λ ) ∂ r T ( λ )  2 λ , (4.30) and therefore Z γ + ∪ γ − T r  Γ − 1 − Γ ′ − ∂ M M − 1  d λ 2 iπ − Z γ + ∪ γ − T r  Ξ − 1 − Ξ ′ − ∂ M M − 1  d λ 2 iπ = (4.31) = 2 T r( R ++ ◦ ∂ K ) + 2 T r ∂ K (4.32) In summary , using Theorem 4.1 , we hav e Z γ + ∪ γ − T r  Γ − 1 − Γ ′ − ∂ M M − 1  d λ 2 iπ = ∂ ln det(Id γ + − K 2 ) + 2 T r( R ++ ◦ ∂ K ) + 2 T r ∂ K (4.33) On the other ha nd we now show that the r.h.s. of ( 4.33 ) is precisely 2 ∂ ln det(Id γ + + K ) at which po int the pro of shall b e then co mplete. T o verify this last p oint we hav e (using Theorem 3.1 ) (Id γ + + K ) − 1 = (Id γ + − K 2 ) − 1 (Id γ + − K ) Thm. 3.1 = (Id γ + + R ++ )(Id γ + − K ) = Id γ + − K + R ++ − R ++ K (4.3 4) from which we can compute the v a riations of the determinant 2 ∂ ln det(Id γ + + K )= 2 T r((Id γ + + K ) − 1 ∂ K ) = = 2 T r ( ∂ K − K ∂ K + R ++ ∂ K − R ++ K ∂ K ) = = − T r((Id γ + + R ++ ) ∂ ( K 2 )) + 2 T r( ∂ K ) + 2 T r( R ++ ∂ K ) = = ∂ ln det(Id γ + − K 2 ) + 2 T r( R ++ ∂ K ) + 2 T r( ∂ K ) (4.35) Q.E.D When the depe ndence on the deformation para meter is in the for m sp eciﬁed in ( 4.3 ) then we can write the diﬀerentials in terms of formal residues as shown here. 24 Prop ositi o n 4.1 Supp ose that M ( λ ) in ( 3.2 ) c an b e written as M ( λ ) = e T ( λ ) M 0 ( λ )e − T ( λ ) wher e T ( λ ) is a p olynomial diagonal matrix without c onstant term in λ ( T (0) = 0 ) whose entries dep end on the deformations and su ch that T ( − λ ) = b σ 1 T ( λ ) b σ 1 . L et ∂ b e the deriva t ive w.r.t. one deformation p ar ameter. Then ∂ ln τ Γ = ω Γ ( ∂ ) = − 1 2 res ∞ T r  Γ − 1 ( λ )Γ ′ ( λ ) ∂ T ( λ )  d λ, (4.3 6 ) ∂ ln τ Ξ = ω Ξ ( ∂ ) = − re s ∞ T r  Ξ − 1 ( λ )Ξ ′ ( λ ) ∂ T ( λ )  d λ (4.37) wher e the r esidues ar e understo o d as formal r esidues, or the c o eﬃcient of λ − 1 in the exp ansion at inﬁnity. Pro of of Prop. 4.1 . The equiv alence of the formal residues ( 4 .36 ) with the integral represen- tation ( 4.1 ) (or ( 4.2 )) was proven in [ 2 ] in a more general context, but w e r ecall here the gist o f it. The formal residue in ( 4.36 ) (for the case o f ω Γ , the other cas e b eing completely analog ous) can b e written as an int egral o n a n expanding counterclockwise circle (with the piecewise-deﬁned Γ) and then it can b e transferre d by the use of Cauch y theorem to the in teg ral − 1 2 res λ = ∞ T r(Γ − 1 Γ ′ ∂ T )d λ = + 1 2 lim R →∞ I | λ | = R T r(Γ − 1 Γ ′ ∂ T ) d λ 2 iπ = (4.38) = 1 2 Z γ + ∪ γ − T r  − Γ − 1 + Γ ′ + + Γ − 1 − Γ ′ −  ∂ T  d λ 2 iπ + ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ 1 2 I | λ | = ǫ T r(Γ − 1 Γ ′ ∂ T ) d λ 2 iπ = (4.39) = − 1 2 Z γ + ∪ γ − T r  M − 1 Γ − 1 − Γ ′ − M + M − 1 M ′ − Γ − 1 − Γ ′ −  ∂ T  d λ 2 iπ = = − 1 2 Z γ + ∪ γ − T r  Γ − 1 − Γ ′ −  M ∂ T M − 1 − ∂ T  + M − 1 M ′ ∂ T  d λ 2 iπ (4.40) Firstly , the term cros s ed out is zero b ecause T ( λ ) = O ( λ ) ⇒ ∂ T ( λ ) = O ( λ ) (as λ → 0) and Γ − 1 Γ ′ may hav e at most a simple pole at λ = 0 (thanks to ( 3.10 )) so that the term is analy tic at λ = 0. Secondly , note that M − 1 M ′ is (piece wise) s trictly upp e r or lower triangular o n γ + ∪ γ − and ∂ T is diagonal, hence the term T r( M − 1 M ′ ∂ T ) ≡ 0 o n γ + ∪ γ − . O n the other hand fro m the formula ( 4.3 ) follows immediately that M ∂ T M − 1 − ∂ T = − ∂ M M − 1 and hence ( 4.36 ) gives exac tly ( 4.2 ). Q.E.D The follo wing cor ollary follows from direct matrix multiplications using the asymptotic for ms ( 3.9 ) and ( 3.7 ) together with the spe cial structure of the expansion matrices ( 3.15 ) and ( 3.21 ). Corollary 4.1 I f ∂ T ( λ ) = i λ e kk ⊗ σ 3 with e kk the diagonal elementary ( r × r ) matrix, then ω Γ ( ∂ ) = − 1 2 res ∞  Γ − 1 Γ ′ λe kk ⊗ σ 3  d λ = − i ( a 1 ) k,k (4.41) 25 Similarly ω Ξ ( ∂ ) = − re s ∞  Ξ − 1 Ξ ′ λe kk ⊗ σ 3  d λ = − 2 i ( α 1 ) k,k (4.42) T o conclude this section we pr ov e that, when r = 1 the rela tio n b etw een the t wo Riemann– Hilber t pr oblems, and in particular equation ( 3.23 ), can be in ter preted as a Miura transfor mation betw een the tw o tau functions. Prop ositi o n 4.2 Supp ose r = 1 (bu t p ossibly p ≥ 1 ) and M ( λ ) := e isλσ 3 M 0 ( λ )e − isλσ 3 (i.e. a sp e cial c ase of Pr op. 4.1 ). Then the t au functions (F r e dholm determinants) τ Ξ , τ Γ ar e r elate d thr ough the Miur a tr ansformation ( ∂ s ln τ Ξ − 2 ∂ s ln τ Γ ) 2 = − ∂ 2 s ln τ Ξ (4.43) Equivalently we may simply write ( ω Ξ ( ∂ s ) − 2 ω Γ ( ∂ s )) 2 = − ∂ s ω Ξ ( ∂ s ) (4.44) Pro of of Prop. 4.2 . The solution Ξ of Pro ble m 3.1 is such that ∂ s  Ξ( λ ; s )e isλσ 3  = U ( λ ; s )Ξ( λ ; s )e isλs 3 (4.45) U ( λ ; s ) := iλσ 3 + 2 β 1 ( s ) σ 1 (4.46) which can b e easily prov ed by no ticing that U has no jumps, hence it is entire, and then by lo oking at the b e havior at inﬁnit y using the expansio n o f Ξ. On the o ther hand then co mparing the ter ms in the expansion of the t wo s ides of ( 4.45 ) one ﬁnds that ∂ s α 1 = − 2 iβ 2 1 . (4.47) Now, Corollar y 4.1 (i.e. P rop. 4 .1 ) y ields ∂ s ln τ Ξ = − res λ = ∞ T r  Ξ − 1 Ξ ′ ∂ s T  d λ = − 2 iα 1 (4.48) ∂ s ln τ Γ = − 1 2 res λ = ∞ T r  Γ − 1 Γ ′ ∂ s T  d λ = − i a 1 (4.49) Rewriting ( 3.23 ) as 2 β 2 1 = (2 ia 1 − 2 iα 1 ) 2 and using the equations ( 4.4 7 ), ( 4.48 ), ( 4.49 ) w e obtain the statement of the prop os ition. Q.E.D Remark 4.1 D eﬁning u := 2 ∂ 2 s ln τ Γ and v 2 := − ∂ 2 s ln τ Ξ we obtain the u sual formulation of t he Miur a tr ansformation u = − v 2 ± ∂ s v . 26 5 Applications: F redholm d eterminan ts a nd noncomm uta- tiv e P ainlev ´ e I I, XXXIV W e now consider the F redholm determinant for the conv o lution op era tor o n L 2 ( R + , C r ) given b y ( A i ~ s f )( x ):= Z R + Ai ( x + y ; ~ s ) f ( y )d y (5.1) Ai ( x ; ~ s ):= Z γ + e θ ( µ ) C e θ ( µ ) e ixµ d µ 2 π = [ c j k Ai( x + s j + s k )] j,k (5.2) θ := iµ 3 6 1 r +      is 1 µ is 2 µ . . . is r µ      = iµ 3 6 1 + i s µ (5.3) s := diag ( s 1 , s 2 , . . . , s r ) (5.4) The matrix C is a constant r × r matrix and the cont our γ + is a con tour c o ntained in the upper half plane and extending to inﬁnit y along the directio ns ar g( z ) = π 6 , 5 π 6 . Here the matrices E 1 , E 2 can be chosen as E 1 ( λ ) = − 1 2 iπ e θ ( λ ) C, E 2 ( λ ) = e θ ( λ ) , r ( λ ) = − 1 2 iπ e θ ( λ ) C e θ ( λ ) (5.5) The ﬁr s t issue is whether the so lutions of Problems 3.2 , 3.1 exist for real v a lues of the parameters ~ s . W e s hall -in fact- show an existence theorem for Pro blem 3.1 , which immediately implies exis tence of the solution of Problem 3.2 by Prop ositio n 3.2 . Theorem 5. 1 Supp ose C = C † is a Hermite an matrix; t hen the solution to Pr oblem 3.1 with r as in ( 5.5 ) exist s for al l values of ~ s ∈ R r if and only if t he eigenvalues of C ar e al l in the interval [ − 1 , 1] . If C is an arbitr ary c omplex matrix with singular v alues in [0 , 1] then the solution stil l ex ists for al l ~ s ∈ R r . (The singular values of a m atr ix ar e the squar e r o ots of the eigenva lues of C † C ) Pro of of Thm. 5.1 The pr o of is based on the estimate of the op era torial norm of the op era tor A i ~ s (with parameter C ∈ Mat( r × r, C )); the inv er tibilit y of the op er ator Id + A i ~ s will be g uaranteed if the norm of A i ~ s is less than o ne. On the o ther hand the in vertibilit y is equiv alent to the non-v a nishing of the respec tive F redholm determinant; hence from Cor ollary 2.1 we have (in the pr e sent notation, with p = r ) det  Id R + ± A i ~ s  L 2 ( R + , C r ) = det  Id γ + ± K  L 2 ( γ + , C p ) ⇒ (5.6) det  Id R + − A i 2 ~ s  L 2 ( R + , C r ) = det  Id γ + − K 2  L 2 ( γ + , C p ) (5.7) 27 Thu s, if k|A i ~ s k| < 1 then k|A i 2 ~ s k| < 1 and th us the F redholm determinants on the line ( 5.7 ) do not v a nish. This is suﬃcient for the existence of the so lutio n o f Pro blem 3.1 as shown in Thm. 3.1 . Let us then e s timate the nor m k|A i ~ s k| ; ﬁrst of all no te that L 2 ( R + ) ≃ L 2 ([ s, ∞ )) by simple translation; with this in mind we can express the op erator A i ~ s as the op erato r A i ~ 0 but a cting on the space H ~ s = L 2 ([ s 1 , ∞ )) ⊕ . . . ⊕ L 2 ([ s r , ∞ )) (5.8) A i ~ 0 : H ~ s → H ~ s (5.9) ( f 1 , . . . f r ) 7→ r X k =1 C j k Z R Ai( x + y ) χ [ s k , ∞ ) f k ( y )d y ! j =1 ,...,r (5.10) Let P ~ s be the or thogonal pro jector P ~ s : L 2 ( R , C r ) → H ~ s , P ~ s = diag ( χ [ s 1 , ∞ ) , . . . , χ [ s r , ∞ ) ) (5.1 1) Then we have A i ~ s ≃ P ~ s A i ~ 0 P ~ s . O n the other hand it is evident that the o p er ator A i ~ 0 : L 2 ( R , C r ) → : L 2 ( R , C r ) is the tensor pro duct A i ~ 0 = C ⊗ L where we hav e denote L the sc alar conv olution op erator with the Airy function on R + L : L 2 ( R ) → L 2 ( R ) (5.12) L f ( x ) := Z R Ai( x + y ) f ( y )d y (5.13) This op erator squares to the identit y (as it is easily seen in F ourier transform, but is a lso well known [ 22 ]) and hence has unit norm (in fact it is a unita r y op erator , an e asily veriﬁed fact in F ourier transform). Ther efore k|A i ~ 0 k| = k | C ⊗ Lk | = k| C k|k|Lk| = k|C k| , (5.14) and hence k|A i ~ 0 k| = k| P ~ s A i ~ 0 P ~ s k| ≤ k | P ~ s k| |||A i ~ 0 ||| ||| P ~ s k| < k |C k | (5.15) Since ||| C ||| is the maximal singular v alue the ﬁrst part of the theorem is prov e d b eca use ||| C ||| ≤ 1 implies that the norm of our o p e r ator is strictly less than one. T o prov e neces sity in the case C = C † Hermitean, supp ose that C has an eig env alue κ ∈ R \ [ − 1 , 1 ] with eigenv ector ~ v 0 (for Hermitean (and mor e generally normal) matrices the singular v a lues are simply the abso lute v alue s of the eig env alues). W e need to show that fo r some choice of ~ s the F redho lm determinant v a nishes; w e will accomplish this b y ﬁnding a special v a lue of ~ s for which 28 the square of A i ~ s has an eigenv ector and hence it is not inv ertible, thus implying the non-solubility of Pro b. 3.1 . T o this end w e take ~ s = ( s, s, s, s . . . , s ) and ~ f ( y ) = ~ v 0 ϕ ( y ) with ϕ ( y ) ∈ L 2 ( R + ). Then ( A i 2 ~ s ~ f )( x ) = κ 2 ~ v 0 Z R + K Ai ( x + s, y + s ) ϕ ( y )d y (5.16) where K Ai is the well known Airy kernel K Ai ( x, y ) = Ai( x )Ai ′ ( y ) − Ai ′ ( x )Ai( y ) x − y = Z R + Ai( x + z )Ai( z + y )d z (5.17) It is w ell–known that K Ai on L 2 ([ s, ∞ )) is self-adjoint and of tra ce-class. Let Λ( s ) b e the max im um eigenv alue. This is a contin uous function of s and tends to 1 as s → − ∞ (it clearly tends to zero as s → + ∞ ) [ 22 ]. Let ϕ s ( y ) b e the corr esp onding eigenfunction and use now f ( y ) = ~ v 0 ϕ s ( y ). Then ( A i 2 ~ s ~ f )( x ) = Λ ( s ) κ 2 ~ f ( x ) (5.18) If κ 2 > 1 there is a v a lue o f s 0 ∈ R for which ( A i 2 ~ s f )( x ) = f ( x ), th us proving that Id − A i 2 ~ s cannot be in vertible for ~ s = ( s 0 , . . . , s 0 ). This co ncludes the pr o of. Q.E. D The reader may now wonder whether these kernels hav e a “physical” interpretation. The a nswer is in the aﬃrmative Theorem 5. 2 If C is r e al or Hermite an then the kernel A i 2 ~ s is total ly p ositive on t he set { 1 , 2 , . . . r }× R . Pro of It is a genera l result that if ( X, d µ ) is a meas ur e space then for any W ⊂ X we hav e k ! det  Z W f a ( ζ ) g b ( ζ )d µ ( ζ )  a,b =1 ..k = Z W k det[ f a ( ζ c )] det[ g b ( ζ d )] k Y j =1 d µ ( ζ j ) (5.19) In our case we take X = { 1 , . . . , r } × R with the counting measur e times Lebesgue measur e . A function on X is then equiv alently in terpr eted as a v ec to r of usual functions: ξ = ( j, x ) ⇒ f ( ξ ) = f (( j, x )) =: f j ( x ). With this understanding the kernel Ai 2 ~ s is understo o d as a sca lar function on X × X to wit A i 2 ~ s ( ξ 1 , ξ 2 ) = (5.20) = [ A i 2 ~ s ] j 1 ,j 2 ( x 1 , x 2 ) = r X k =1 c j 1 k c kj 2 Z R + d z Ai ( x 1 + z + s j 1 + s k )Ai( x 2 + z + s j 2 + s k ) = (5.21) = Z X + d µ ( ξ ) F ( ξ 1 , ζ ) F ( ζ , ξ 2 ) (5.22) 29 where w e hav e set F ( ξ 1 , ζ ) := c j 1 ,k Ai( x 1 + s j 1 + z + s k ) (5.23) ζ = ( k , z ) ∈ X + := { 1 , . . . , r } × R + ⊂ X (5.24) The statement of total p ositivity amounts to checking that for a ny K ∈ N a nd a ny ξ 1 , . . . , ξ K ∈ X we hav e det  A i 2 ~ s ( ξ a , ξ b )  a,b ≤ K > 0 (5.25) T o this end we use the previous fac t ( 5.19 ) and w e hav e det  A i 2 ~ s ( ξ a , ξ b )  1 ≤ a,b ≤ K = det " Z X + d µ ( ξ ) F ( ξ a , ζ ) F ( ζ , ξ b ) # a,b = (5.26) = 1 K ! Z X K + det [ F ( ξ a , ζ c )] det [ F ( ζ c , ξ a )] K Y c =1 d µ ( ζ c ) = 1 K ! Z X K + | det [ F ( ξ a , ζ c )] | 2 K Y c =1 d µ ( ζ c ) > 0 (5.27) where the modulus o c curs if C is complex Hermitean (in which case F ( ξ , ζ ) = F ( ζ , ξ )), while if C is any real matrix then we hav e a simple square (which is anyw ay po sitive). Q.E.D Theorem 5.2 allows us to interpret the kernel A i 2 ~ s as deﬁning a determinan tal p oint pro cess on the space o f conﬁgura tions X = { 1 , . . . , r } × R [ 21 ]. The F redholm determinant is then a m ulti-level ga p distribution for said pro ces s o n the interv al [ S, ∞ ) (after a translation x 7→ x − S ). 5.1 Noncomm utative Pa inlev ´ e I I and its p ole-free solutions W e consider ﬁrst Problem 3.1 for Ξ; the jump is wr itten ( r deﬁned in 5.5 ) Ξ + = Ξ − ( 1 2 r − 2 i π r ( λ ) ⊗ σ + ) , λ ∈ γ + (5.28) Ξ + = Ξ − ( 1 2 r − 2 i π r ( − λ ) ⊗ σ − ) , λ ∈ γ − (5.29) The matrix Ψ( λ ) := Ξ( λ )e θ ( λ ) ⊗ σ 3 , with θ ( λ ) as in ( 5.3 ), solves a RHP with constant jumps Ψ + = Ψ − ( 1 2 r + C ⊗ σ + ) , λ ∈ γ + Ψ + = Ψ − ( 1 2 r + C ⊗ σ − ) , λ ∈ γ − Ψ( λ )=  1 2 r + O ( λ − 1 )  e θ ( λ ) ⊗ σ 3 , λ → ∞ (5.30) It would b e s imple to show that Ψ( λ ) solves a poly nomial ODE in λ (of degr ee 2, see Lemma 5.1 ), which even tually w ould lead to showing that β 1 ( ~ s ) solves a nonco mmut ative v ersion of the Painlev ´ e II equation (whence the title of the section). In this persp ective, the ab ov e jumps are a p articular choice of Stokes’ m ultiplier s asso ciated to such an ODE, ex actly as in the scalar commutativ e Lax representation of PI I [ 12 ]. W e thus describ e in the next sectio n below, ex ante , the mo s t genera l set of genera lized mono dro my da ta for the O DE ( 5.46 ). 30 5.1.1 The general Stok es ’ data/Riemann–Hi lb ert problem for Ψ Denote by ∆ ⊂ C r the set o f diagonals ∆ := { ~ s ∈ C r : s j = s ℓ , j 6 = ℓ } . (5.31) Let ~ s (0) ∈ C r \ ∆ and choose a r ay γ R := R + e − ϕ R in such a way that ℑ z ( s (0) k − s (0) ℓ ) 6 = 0 for z ∈ γ R . Let γ L := − γ R . W e introduce the orde r ing k ≺ ℓ as follows k ≺ ℓ ⇔ ℑ (e iϑ R ( s (0) k − s (0) ℓ )) < 0 (5.32) F or a ﬁxed γ R this or dering is constant in a suitable op en conical neighbo rho o d of ~ s (0) not int ersecting the dia gonals ∆ (as should b e clear by a simple contin uity argument): we shall under- stand such choice o f neig h b o rho o d and keep the chosen or dering ﬁxe d. W e shall say that a matrix N is upp er(lo wer) -triangular relativ e to the ordering ( 5.32 ) if N kℓ = 0 for k ≺ ℓ ( ℓ ≺ k , resp ectively) and N kk = 0. Example 5. 1 If s 1 < s 2 < . . . s r ar e r e al and or der e d, then t he notion of u pp er(lower) triangularity r elative to any r ay a rg z = ( − π , 0) is the usual one. W e deﬁne the six additional contours γ j := R + e ikπ 3 + π 6 , k = 0 , . . . , 5 (5.33) Let C 0 , . . . C 2 three ar bitr ary r × r matrices, S u = 1 r + N u , S l = 1 r + N l with N u , N l t wo upper /low er triangular matrices relative to the ordering ( 5.32 ) determined b y the choice o f γ R , and let M = diag ( µ 1 , . . . , µ r ) ∈ S L ( r , C ) (tra c eless). The entries of M will be referre d to a s exp o nen ts of formal mono dromy . Problem 5.1 L et Ψ ( λ ) b e a se ctional ly analytic function on C \ γ 0 ∪ . . . γ 5 ∪ γ L ∪ γ R , b ounde d over c omp act sets of C and solving t he fol lowi ng Riemann–Hilb ert pr oblem Ψ + = Ψ − ( 1 2 r + C 0 ⊗ σ + ) , λ ∈ γ 0 , Ψ + = Ψ − ( 1 2 r + C 1 ⊗ σ − ) , λ ∈ γ 1 (5.34) Ψ + = Ψ − ( 1 2 r + C 2 ⊗ σ + ) , λ ∈ γ 2 , Ψ + = Ψ − ( 1 2 r + C 0 ⊗ σ − ) , λ ∈ γ 3 (5.35) Ψ + = Ψ − ( 1 2 r + C 1 ⊗ σ + ) , λ ∈ γ 4 , Ψ + = Ψ − ( 1 2 r + C 2 ⊗ σ − ) , λ ∈ γ 5 (5.36) Ψ + = Ψ − [( 1 r + S u ) ⊕ ( 1 r + S l )] , λ ∈ γ R (5.37) Ψ + = Ψ − [( 1 r + S l ) ⊕ ( 1 r + S u )] , λ ∈ γ L (5.38) Ψ + = Ψ − e − iπ M ⊗ 1 2 , λ ∈ R ± (5.39) Ψ( λ )=  1 2 r + O ( λ − 1 )  e − iǫπ M ⊗ 1 2 λ M ⊗ 1 2 e θ ( λ ) ⊗ σ 3 , λ → ∞ (5.40) 31 wher e in ( 5.40 ) ǫ = 1 in the u pp er half-plane, ǫ = 0 in t he lower half-plane and ar g( λ ) ∈ [ − π , π ) . The matric es C 0 , . . . , C r , S u , S l , M ar e chosen satisfy the no - mono dromy c ondition st ating that the pr o duct of the jumps is the identity. (We cho ose γ R 6 = R ± and al l the r ays ar e oriente d towar ds inﬁnity). Since the rays γ L , γ R may lie in b etw een diﬀerent γ j ’s depending on the v alue of ~ s the no- mo no dromy condition may take diﬀeren t for ms . F or exa mple, if ~ s ∈ R r \ ∆ we ca n choos e arg ( γ R ) = π 2 + ǫ and the no-mono dro my condition takes the form  1 + S u 0 0 1 + S l  ( 1 2 r + C 0 ⊗ σ + )( 1 2 r + C 2 ⊗ σ − )( 1 2 r + C 1 ⊗ σ + )e − iπ M × ×  1 + S l 0 0 1 + S u  ( 1 2 r + C 0 ⊗ σ − )( 1 2 r + C 2 ⊗ σ + )( 1 2 r + C 1 ⊗ σ − )e − iπ M = 1 2 r (5.41) Remark 5.1 The pr oblem asso ciate d to the F r e dholm determinant of t he op er ator as in Thm. 5.1 c orr esp onds to t he p articular choic e S u = S l = C 1 = M = 0 and C 0 = C = − C 2 . Note that the jumps satisfy the symmetr y M ( − λ ) = b σ 1 M ( λ ) b σ 1 and hence we a lso have (noticing that θ ( − λ ) ⊗ σ 3 = b σ 1 θ ( λ ) ⊗ σ 3 b σ 1 since θ ( − λ ) = − θ ( λ ) a s per ( 5.3 )) Ψ( − λ ) = b σ 1 Ψ( λ ) b σ 1 (5.42) The dimensio n of the manifold ( C 0 , C 1 , C 2 , S u , S l , M ) o f solutions of ( 5.41 ) c a n b e computed by noticing that ther e are a total of 3 r 2 + 2 r ( r − 1) 2 + r − 1 = 4 r 2 − 1 v a riables. The equation ( 5.41 ) is of the for m A b σ 1 A b σ 1 = 1 2 r and hence –due to the sy mmetry of conjuga tion by b σ 1 – there a re o nly 2 r 2 independent equations. Of these, one is redundant since the determinant of A is already unit. Hence there are 2 r 2 − 1 indep e ndent equa tions and thus the manifold of solutions ha s dimension 2 r 2 . Lemma 5.1 L et the matrix Ψ( λ ) b e the solut ion of the Pr oblem 5.1 and denote the asymptotic exp ansion at ∞ as Ψ( λ )e iǫπ M ⊗ 1 2 λ − M ⊗ 1 2 e − θ ( λ ) ⊗ σ 3 = 1 2 r + ∞ X j =1 Ξ j λ j , Ξ 2 j +1 = α 2 j +1 ⊗ σ 3 + β 2 j +1 σ 2 , Ξ 2 j = α 2 j ⊗ 1 + β 2 j σ 1 , (5.43) (r e c al l that ǫ = 1 for ℑ λ > 0 and ǫ = 0 for ℑ λ < 0 ) wher e t he exp ansion is valid se ctorial ly and indep endent of the se ct or. Then ∂ s j Ψ= U j Ψ , U j = iλ e j ⊗ s 3 + i [ α 1 , e j ] ⊗ 1 + { β 1 , e j } ⊗ σ 1 (5.44) ∂ λ Ψ= A ( λ )Ψ , A ( λ ) = λ 2 r X j =1 U j − 1 2 D ( α 1 ⊗ σ 3 + β 1 ⊗ σ 2 ) + i s ⊗ σ 3 = (5.45) A ( λ ) = i λ 2 2 b σ 3 + λβ 1 ⊗ σ 1 − 1 2 D β 1 ⊗ σ 2 + i ( β 2 1 + s ) ⊗ σ 3 (5.46) 32 wher e D := r X j =1 ∂ s j , e j := diag (0 , 0 , . . . , 1 , 0 , . . . ) (5.47) with the one in the j - th p osition. Pro of The fact that the expansion for (recall that arg( λ ) ∈ [ − π , π )) Ξ( λ ) :=    Ψ( λ )e iπ M λ − M ⊗ 1 2 e − θ ( λ ) ⊗ σ 3 ℑ λ > 0 , | λ | > 1 Ψ( λ ) λ − M ⊗ 1 2 e − θ ( λ ) ⊗ σ 3 ℑ λ < 0 , | λ | > 1 Ψ( λ ) | λ | < 1 (5.48) near λ = ∞ is o f the for m in ( 5.43 ) fo llows from the symmetry Ψ ( − λ ) = b σ 1 Ψ( λ ) b σ 1 which then implies the sa me symmetry for Ξ. The function Ξ has then no jumps on R ± \ { | λ | < 1 } and the remaining jumps a re those of Ψ conjugated b y e − iǫ M λ M ⊗ 1 2 e θ ( λ ) ⊗ σ 3 . The fact that the expansion is indep endent of the s ector is a consequence of the fact that the jumps for Ξ alo ng the eight rays are analytic in a small open s ector a round said rays and of the form Ξ + ( λ ) = Ξ − ( λ )( 1 + O ( λ −∞ )) , λ → ∞ , λ ∈ γ 0 ∪ . . . γ 5 ∪ γ L ∪ γ R (5.49) uniformly within s aid sectors. The fact that U j and A are p oly no mials is an immediate c o nsequence of the fact that the jumps of Ψ are indep endent of λ, ~ s . Using Liouville’s theorem and the fa c t that ∂ s j ΨΨ − 1 is en tire (a simple consequence of the indep endence on s j of the jumps) we deduce immediately that U j ( z ) can only b e a p olynomial of degree 1. Then ∂ j Ξ( λ ) + iλ Ξ( λ )e j ⊗ σ 3 =  U (1) j λ + U (0) j  Ξ( λ ) ∂ j ( α 1 ⊗ σ 3 + β 1 ⊗ σ 2 ) λ + . . . + iλ  1 + α 1 ⊗ σ 3 + β 1 ⊗ σ 1 λ + α 2 ⊗ 1 + β 2 ⊗ σ 1 λ 2 + . . .  e j ⊗ σ 3 = =  U (1) j λ + U (0) j   1 + α 1 ⊗ σ 3 + β 1 ⊗ σ 1 λ + α 2 ⊗ 1 + β 2 ⊗ σ 1 λ 2 + . . .  (5.50) Comparing the co eﬃcients of the p owers of λ we hav e λ : ⇒ U (1) j = iλ e j ⊗ σ 3 (5.51) λ 0 : ⇒ i [ α 1 ⊗ σ 3 + β 1 ⊗ σ 2 , e j ⊗ σ 3 ] = U (0) j (5.52) λ − 1 : ⇒ ∂ s j ( α 1 ⊗ σ 3 + β 1 ⊗ σ 3 ) = − i [ α 2 ⊗ 1 + β 2 ⊗ σ 1 , e j ⊗ σ 3 ] + U (0) j ( α 1 ⊗ σ 3 + β 1 ⊗ σ 1 ) ∂ s j ( α 1 ⊗ σ 3 + β 1 ⊗ σ 3 ) = − i [ α 2 ⊗ 1 + β 2 ⊗ σ 2 , e j ⊗ σ 3 ] + U (0) j ( α 1 ⊗ σ 3 + β 1 ⊗ σ 2 ) (5.53) If w e sum up fo r j = 1 , . . . , r we o btain the diﬀerential e q uation D ( α 1 ⊗ σ 3 + β 1 ⊗ σ 2 )= − i [ α 2 ⊗ 1 + β 2 ⊗ σ 2 , 1 ⊗ σ 3 ] + i [ α 1 ⊗ σ 3 , + β 1 ⊗ σ 2 , 1 ⊗ σ 3 ] ( α 1 ⊗ σ 3 + β 1 ⊗ σ 2 ) = = − 2 β 2 ⊗ σ 1 +2 β 1 ⊗ σ 1 ( α 1 ⊗ σ 3 + β 1 ⊗ σ 2 ) = − 2 β 2 ⊗ σ 1 + 2 i β 1 α 1 ⊗ σ 2 − 2 i β 2 1 ⊗ σ 3 (5.54) 33 In particular D α 1 = − 2 iβ 2 1 . (5.55) If w e lo ok also at the λ − 2 co eﬃcient we ﬁnd D β 1 = 2 iβ 2 − 2 β 1 α 1 , β 2 = − i 2 D β 1 − iβ 1 α 1 . (5.56) Exactly as b efore we argue that A ( z ) is a p olynomial of degree 2. Then we compute ∂ λ Ξ( λ ) + Ξ( λ )  i λ 2 2 1 r + i s  ⊗ σ 3 =  A 2 λ 2 + A 1 λ + A 0  Ξ( λ ) (5.57) . . . +  1 + α 1 ⊗ σ 3 + β 1 ⊗ σ 2 λ + α 2 ⊗ 1 + β 2 ⊗ σ 1 λ 2 + . . .   i λ 2 2 1 r + i s  ⊗ σ 3 = (5.58) =  A 2 z 2 + A 1 z + A 0   1 + α 1 ⊗ σ 3 + β 1 ⊗ σ 2 z + α 2 ⊗ 1 + β 2 ⊗ σ 1 z 2 + . . .  (5.59) Collecting the co eﬃcients λ 2 : ⇒ A 2 = i 2 λ 2 1 ⊗ σ 3 (5.60) λ 1 : ⇒ A 1 = i 2 [ α 1 ⊗ σ 3 + β 1 ⊗ σ 2 , 1 ⊗ σ 3 ] (5.61) λ 0 : ⇒ A 0 = i s ⊗ σ 3 + i 2 [ α 2 ⊗ 1 + β 2 ⊗ σ 1 , 1 ⊗ σ 3 ] − A 1 ( α 1 ⊗ σ 3 + β 1 ⊗ σ 2 ) = (5.62) A 0 = i s ⊗ σ 3 − 1 2 D ( α 1 ⊗ σ 3 + β 1 ⊗ σ 2 ) (5.63) where we hav e used formula ( 5.54 ). The second expressio n for A ( λ ) ( 5.46 ) follows from ( 5.55 ). The rest of the pro of is a simple co mputation. Q . E.D Lemma 5.2 L et Ψ b e as in L emma 5.1 and denote by β 1 = β 1 ( ~ s ) the r × r c o eﬃcient matrix in Ξ 1 = α 1 ⊗ σ 3 + β 1 ⊗ σ 2 of the exp ansion as in t he mentione d L emma. Then the matrix funct ion β 1 ( ~ s ) ∈ M at ( r × r, C ) satisﬁes t he nonc ommu tative Painlev´ e II e quation. D 2 β 1 = 4 { s , β 1 } + 8 β 3 1 , s := dia g( s 1 , . . . , s r ) , D := r X j =1 ∂ ∂ s j , { X , Y } = X Y + Y X (5.64) Pro of. W e use the zero curv a ture equatio ns D Ψ = r X j =1 U j Ψ =: U D Ψ , ∂ λ Ψ = A Ψ (5 .65) ( ∂ λ U D + U D A − D A + AU D ) ≡ 0 (5.66) 34 with the U j ’s introduced in Lemma 5.1 . W e hav e U D = iλ 1 ⊗ σ 3 + 2 β 1 ⊗ σ 1 , A = λ 2 U D − 1 2 D ( α 1 ⊗ σ 3 + β 1 ⊗ σ 2 ) + i s ⊗ σ 3 (5.67) A = i 2 λ 2 1 ⊗ σ 3 + λβ 1 ⊗ σ 1 − 1 2 D ( α 1 ⊗ σ 3 + β 1 ⊗ σ 2 ) + i s ⊗ σ 3 (5.68) W e now compute this expre s sion (for simplicity we denote with a prime the action of D , no ting that Ds = 1 r ) ∂ λ U D = i 1 ⊗ σ 3 (5.69) D A = λβ ′ 1 ⊗ σ 1 − 1 2 ( α ′′ 1 ⊗ σ 3 + β ′′ 1 ⊗ σ 2 ) + i 1 ⊗ σ 3 (5.70) [ U D , A ]=  iλ 1 ⊗ σ 3 + 2 β 1 ⊗ σ 1 , − 1 2 ( α ′ 1 ⊗ σ 3 + β ′ 1 ⊗ σ 2 ) + i s ⊗ σ 3  = (5.71) = λβ ′ 1 ⊗ σ 1 − 2 i β 1 α ′ 1 ⊗ σ 2 + i { β 1 , β ′ 1 } ⊗ σ 3 − 2 { β 1 , s } ⊗ σ 2 (5.72) Hence 0 ≡ ∂ λ U D − D A + [ U D , A ] = (5.73) = 1 2 a ′′ 1 ⊗ σ 3 + 1 2 β ′′ 1 ⊗ σ 2 − 2 i β 1 α ′ 1 ⊗ σ 2 + i { β 1 , β ′ 1 } ⊗ σ 3 − 2 { β 1 , s } ⊗ σ 2 (5.74) Using now ( 5.55 ) we hav e α ′′ 1 = − 2 i { β 1 , β ′ 1 } and hence we are le ft only with  1 2 β ′′ 1 − 4 β 3 1 − 2 { β 1 , s }  ⊗ σ 2 ≡ 0 (5.75) Q.E.D Thu s Lemma 5.1 and the matrices ( 5.6 5 ) provide a Lax ma trix repres e ntation for the gener al solution of the noncommu tative Painlev´ e equation ( 5.88 ) which is parametriz e d b y the 2 r 2 initial v a lues β 1 ( ~ s ) and D β 1 ( ~ s ) at any p oint ~ s . It should als o be clear that any solution β 1 of the noncommutative Painlev ´ e II equation ( 5.88 ) yields a compatible La x pair A ( λ ; ~ s ) ( 5.46 ) and U D ( λ ; ~ s ) ( 5.65 ); the Stokes’ phenomenon (genera lized mono dromy data ) for the O DE ∂ λ Ψ = A Ψ can b e s e e n to b e given exactly b y the data sp eciﬁed in P roblem 5 .1 (we refer to [ 24 ] fo r the general theory of Stokes’ multipliers). Thus an y solution of ( 5.88 ) is obtained via the ab ov e Lax-pair. Remark 5.2 The generic solut ion of β 1 ( ~ s ) of the nonc ommutative PII e quation wil l have non- movable singularities on the diagonals ~ s ∈ ∆ ; this is due to the pr esenc e –in gener al– of nontrivial Stokes mu ltipliers along t he r ays γ R , γ L . If those mult iplier ar e trivial as wel l as the exp onents of 35 formal mono dr omy i.e. M = 0 , S u = 0 = S l then those singularities wil l b e absent. In this c ase the no-mono dr omy c ondition ( 5.41 ) c an b e sp elt out mor e cle arly as [ C 0 , C 1 ] = 0 , [ C 1 , C 2 ] = 0 , [ C 0 , C 2 ] = 0 (5.76) C 0 + C 2 + C 1 + C 0 C 1 C 2 = 0 (5.77) which yields a manifo ld of dimension r 2 + r (the matric es c an b e generic al ly diagonalize d simulta- ne ously and a simple c oun ting yields t his numb er). The c ondition ( 5.77 ) r esembles very closely the or dinary situation r = 1 of the c ommutative L ax r epr esentation for PII [ 10 ] 12 . The impor tance of the isomo no dromic r epresentation for the noncommutativ e Painlev ´ e I I equa - tion is that it implies automa tically the Pa inlev´ e prop ert y [ 18 ] that the o nly singula rities of the solution a re p oles except -p ossibly - the singularities o n the diagonal manifold ~ s ∈ ∆ if the Stok e s’ matrices S u , S l are nonzero. Remark 5.3 Another imp ortant r emark is that the solution β 1 ( ~ s ) as a function of the b aryc entric variables S := 1 r r X j =1 s j , δ j = s j − S (5.78) has only p oles as a function of S ( n ote that D = ∂ S ) if ~ s 6∈ ∆ ; this is s o b e c ause changing S do es not change the diﬀer enc es b etwe en the s j ’s and henc e never cr osses the dia gonal manifold. Remark 5.4 T o our know le dge, the nonc ommu t ative Painlev ´ e e quation ( 5.64 ) has app e ar e d ﬁrst in the r e c ent [ 19 ] wher e the aut hors c onstruct sp e cial r ational solution u s ing the the ory of quasi- determinants [ 7 ]. Pr eviously, a version with sc alar indep endent variable (henc e r eplacing the anti- c ommutator by simply sβ 1 ) was stu die d in [ 1 ], wher e the Painlev ´ e test was applie d. It se ems that the L ax r epr esentation for t he nonc ommu t ative version of [ 19 ] app e ars in t he pr esent manuscript for the ﬁrs t time. It se ems p ossible to gener alize t he L ax- p air r epr esentation by al lowi ng a p ole at λ = 0 in the L ax matrix A ( λ ) (exactly as in the sc alar c ase). F or example the c omp atibility of t he fol lowing two L ax matric es A ( λ )= iλ 2 2 b σ 3 + λβ 1 ⊗ σ 1 + i  s + β 2 1  ⊗ σ 3 − 1 2 β ′ 1 ⊗ σ 2 + 1 λ Θ b σ 1 , U D = iλ b σ 3 + 2 β 1 ⊗ σ 1 (5.79) [ ∂ λ − A ( λ ) , D − U D ( λ )] = 0 (5.80) 12 Their matrix Ψ( λ ) has the symm etry Ψ( − λ ) = σ 2 Ψ( λ ) σ 2 , whic h means that it should b e compared with ours after conjugation b y e i π 4 σ 3 . 36 with Θ an arbitr ary sc alar (i.e. c ommu tative s ymb ol). The zer o-curvatur e e qu ations ar e e asily veriﬁe d to yield D 2 β 1 = 4 { s , β 1 } + 8 β 3 1 − 4Θ (5 .8 1) which is pr e cisely (with diﬀer en t symb ols) the Painlev ´ e II e quation studie d in [ 19 ]. F r om the isomon- o dr omic metho d, however, the ab ove e quation app e ars t o b e not the most gener al that one may obtain by al lowing a p ole in A ( λ ) . We do now dwel l further into the matt er sinc e it is p eripher al to the fo cu s of the pr esent p ap er. The compatibility equations for the op erato rs ∂ s j − U j , ∂ s k − U k and ∂ λ − A yield additional equation listed in the Coro llary below, whic h is pr oved along the same lines (but we will no t re po rt the pro of here since it is unnecessarily long , s tr aightforw ard and anyw ay this has no b earing for our goals) Corollary 5.1 The matric es β 1 , α 1 satisfy the systems ( ∂ j = ∂ s j , D = P ∂ j , j = 1 , . . . , r ) ∂ j β 1 = 1 2 { e j , D β 1 } − i { e j , β 1 a 1 } + i e j [ α 1 , β 1 ] + iα 1 e j β 1 + iβ 1 e j α 1 (5.82) 1 2 ∂ j D β 1 = i ( ∂ j β 1 ) α 1 + {{ β 1 , e j } , s } − i 2 [e j , α 1 ] D β 1 +  [ α 1 , e j ] , β 1  α 1 + + β 1 { e j β 1 } β 1 − i 2 { D β 1 α 1 , e j } + { e j , β 3 1 } (5.83) 1 − e j + i  s , [ α 1 , e j ]  + 1 2  [ β 1 , D β 1 ] , e j  = 0 (5.84) { e j , ∂ k β 1 } − { e k , ∂ j β 1 } = i [ β 1 e k , α 1 e j ] + i [e j β 1 , e k α 1 ] + i e k [ β 1 , α 1 ]e j + + i e j [ α 1 , β 1 ]e k + i [ α 1 e k , β 1 e j ] + i [e j α 1 , e k β 1 ] , (5.85) i [e k , ∂ j α 1 ] − i [e j , ∂ k α 1 ] =e k α 2 1 e j − e j α 2 1 e k + e k β 2 1 e j − e j β 2 1 e k + +[e k β 1 , e j β 1 ] + [ β 1 e k , β 1 e j ] + [e j α 1 , e k α 1 ] + [ α 1 e j , α 1 e k ] . (5.86) 5.2 P ole-free solutions of noncomm utativ e Painlev ´ e I I and F redholm de- terminan ts W e now return to the s pec iﬁc situation of the RHP asso ciated to the in teg rable kernel A i 2 ~ s ; this is the sp e c ial case of the setting as ex pla ined in Remark 5.1 . 37 Theorem 5. 3 L et Ξ = Ξ( λ ; ~ s ) b e t he solution of Pr oblem 3.1 with r as in ( 5.5 ); let β 1 ( ~ s ) := − i lim λ →∞ λ Ξ 12 ( λ ; ~ s ) (5.87) wher e Ξ ij denote the r × r blo cks of Ξ , i , j = 1 , 2 . The m atrix function β 1 ( ~ s ) ∈ Mat( r × r , C ) satisﬁes the nonc ommut ative Painlev´ e II e quation. D 2 β 1 = 4 s β 1 + 4 β 1 s + 8 β 3 1 , s := dia g( s 1 , . . . , s r ) , D := r X j =1 ∂ ∂ s j (5.88) The asymptotic b ehavior of the p articular solution asso ciate d t o the Pr oblem 3.1 is as fol lows: if S := 1 r P r j =1 s j → + ∞ and δ j := s j − S , j = 1 , . . . , r ar e kept ﬁxe d, | δ j | ≤ m , then [ β 1 ] kℓ = − c kℓ Ai( s k + s ℓ ) + O  √ S e − 4 3 (2 S − 2 m ) 3 2  (5.89) If C is Hermite an then so is the solution β 1 ( ~ s ) of the nonc ommut ative Painlev ´ e e quation ( 5.88 ) and it is p ole-fr e e for al l ~ s ∈ R r if and only if the eigenvalues of C ar e within [ − 1 , 1] . If C is arbitr ary and its singular values lie in [0 , 1] then the solution is also p ole fr e e for ~ s ∈ R r . Final ly, the F r e dholm determinant τ ( ~ s ) := det  Id − A i ~ s 2  satisﬁes det  Id − A i ~ s 2  = exp  − 4 Z ∞ S ( t − S ) T r( β 2 1 ( t + ~ δ ))d t  (5.90) wher e t + ~ δ := ( t + δ 1 , . . . , t + δ r ) . The last statement is the noncommutativ e (matrix) equiv alent of the celebr ated T r acy–Widom distribution [ 22 ]. Befor e giving the pro of of Thm. 5.3 w e pr ov e the uniqueness of the s olution. Prop ositi o n 5.1 F or any r × r matrix C ther e is a unique solution of nonc ommutative PII ( 5.88 ) with the asymptotics ( 5.89 ). Pro of o f Prop. 5.1 . The pro of do es not diﬀer signiﬁca nt ly from the scalar case as in [ 9 ]. In barycentric and relative co ordinates S, δ j as in Thm. 5.3 we hav e D = ∂ S . The reg ime we c onsider is S → + ∞ and all δ j bo unded be low. W e note that the function [ U ( ~ s )] kℓ := − c kℓ Ai( s k + s ℓ ) = − c kℓ Ai(2 S + δ k + δ ℓ ) (5.91) is a solution of the linear part of ( 5.88 ): D 2 U kℓ = − 4 c kℓ Ai ′′ ( s k + s ℓ ) = − 4 c k,ℓ ( s k + s ℓ )Ai( s k + s ℓ ) = 4 ( s k U kℓ + U kℓ s ℓ ) . (5.92 ) 38 Then any solution ( β 1 ) kℓ with the spec iﬁed asymptotic also solves the integral equation 13 [ β 1 ] kℓ = U kℓ + 4 π Z ∞ S (Ai(2 S + δ k + δ ℓ )Bi(2 t + δ k + δ ℓ ) − Ai (2 t + δ k + δ ℓ )Bi(2 S + δ k + δ ℓ )) [( β 1 ) 3 ] kℓ d t (5.93) Equation ( 5.93 ) can b e so lved b y iterations for S suﬃciently larg e , as noted in [ 9 ] for the s c alar case. The lo cal uniqueness follows from the loc al uniqueness of the solution o f the ODE ( 5.88 ) (in S ). The s olution is eas ily seen to be lo cally analytic in ~ s beca use of the analyticity of the ODE and also from the integral equa tion. W e also point out that since the genera lized mono dro my data asso ciated to this solution have S u = S l = 0 = M (see P roblem 5.1 ) then there are no critical singula r ities at all in ~ s and we may only hav e p oles a t mo s t (the F redholm determinant is analy tic in ~ s , hence can only have zero es ). Thus the solutio n is g lobally deﬁned for ~ s ∈ R by a nalytic co nt inuation. Q.E.D Remark 5.5 Be c ause of ( 5.89 ) and Pr op. 5.1 we may c al l the sp e cial solution of nonc ommutative PII arising ab ove the noncommutativ e Hastings -McLeo d solution(s). Pro of of Thm. 5.3 . The fa ct that β 1 solves the no ncommutativ e PI I equation ( 5 .88 ) follows fro m Lemma 5.2 since this is a sp ecia l case o f that with S u = S l = C 1 = M = 0 a nd C = C 0 = − C 2 . Asymptotics. Supp ose that S = 1 r P s j is lar g e and p ositive and δ j := s j − S are b ounded by -let’s say- m . W e r ewrite the RHP in the s caled v ar iable z := λ √ S . The jump on the contours γ ± of the for m 1 − 2 iπ r ⊗ σ + , 1 − 2 π ˜ r ⊗ σ − can b e factored into (comm uting) matrices (here b elow e kℓ is the element ary matrix) 1 − 2 iπ r ⊗ σ + = r Y k,ℓ =1  1 + c kℓ e iS 3 2  1 3 z 3 +(2+ δ k + δ ℓ S ) z  e k,ℓ ⊗ σ +  (5.94) 1 − 2 iπ r ⊗ σ − = r Y k,ℓ =1  1 + c kℓ e − iS 3 2  1 3 z 3 +(2+ δ k + δ ℓ S ) z  e k,ℓ ⊗ σ −  (5.95) Each factor has a saddle p oint at z = ± i q 2 + δ k + δ ℓ S and the co n tours γ ± suppo rting the single jump can b e split a c c ording to the factorizatio n ( 5.94 ) s o that each o f the factor is s upp o r ted o n a diﬀerent contour γ ( k,ℓ ) ± of steep est descent for the corr esp onding pha ses. P ro ceeding this w ay the reader realiz e s that each factor M ( k,ℓ ) ± ( √ S z ) = 1 + c kℓ e ± iS 3 2  1 3 z 3 +  2+ δ k + δ ℓ S  z  e k,ℓ ⊗ σ ± , z ∈ γ ( k,ℓ ) ± (5.96) is close to the iden tit y jump in any L p ( γ ( k,ℓ ) ± ), 1 ≤ p ≤ ∞ with the supremum norm given by k 1 − M ( k,ℓ ) k ∞ = | c kℓ | e − 2 3 (2 S + δ k + δ ℓ ) 3 2 → 0 . (5.97) 13 W e recall that Bi is the solution of f ′′ = xf suc h that W r(Ai , Bi) = π − 1 . 39 This shows that the RH problem may be s olved by iterations; the ﬁrst iteration for Ξ yields Ξ( √ S z )= 1 − Z γ + r ( √ S w ) ⊗ σ + d w ( w − z ) − Z γ − ˜ r ( √ S w ) ⊗ σ − d w ( w − z ) + O e − 4 3 (2 S − 2 m ) 3 2 1 + | z | ! = ( 5.98) = 1 − i λ [ c kℓ Ai( s k + s ℓ )] ⊗ σ + + i λ [ c kℓ Ai( s k + s ℓ )] ⊗ σ − + O √ S e − 4 3 (2 S − 2 m ) 3 2 √ S + | λ | ! (5.99) where the notation [ A kℓ ] sta nds for a matr ix with entries A kℓ . This yields [ β 1 ] kℓ = − i lim λ →∞ λ Ξ 12 ( λ ) = − c kℓ Ai( s k + s ℓ ) + O  √ S e − 4 3 (2 S − 2 m ) 3 2  (5.100) P ole s. It follows from Thm. 5.1 that under the sta ted conditions β 1 exists and ﬁnite fo r all ~ s = R r and hence cannot hav e po les. Symmetry . If C = C † then (for ~ s ∈ R r ) r T ( λ ) = − r ( − λ ) (se e ( 5.5 )) and the jump matrices M ( λ ) then satisfy M † ( λ ) = b σ 3 M − 1 ( − λ ) b σ 3 , M ( λ ) = 1 2 r + e θ ( λ ) C e θ ( λ ) ⊗  σ + χ γ + ( λ ) + σ − χ γ − ( λ )  . (5.101) The contours of jump s atisfy (also) γ + = γ − . Then Ξ −† ( λ ) := Ξ − T ( λ ) = b σ 3 Ξ( λ ) b σ 3 . This implies ( σ 3 σ 2 σ 3 = − σ 2 ) Ξ( λ ) = 1 + α 1 ⊗ σ 3 + β 1 ⊗ σ 2 λ + . . . ⇒ Ξ − 1 ( λ ) = 1 − α 1 ⊗ σ 3 + β 1 ⊗ σ 2 λ + . . . (5.1 02) Ξ −† ( λ ) = 1 − α † 1 ⊗ σ 3 + β † 1 ⊗ σ 2 λ + . . . ⇒ b σ 3 Ξ −† ( λ ) b σ 3 = 1 + − α † 1 ⊗ σ 3 + β † 1 ⊗ σ 2 λ + . . . (5 .103) which shows immediately β 1 = β † 1 (as w ell a s α 1 = − α † 1 ). F orm ula for the determinan t. F rom Corolla ry ( 4.1 ) we deduce that D ln det(Id − A i 2 ~ s ) = − 2 i T r α 1 and together with eq. ( 5.55 ) we hav e D 2 ln det(Id − A i 2 ~ s ) = − 4 T r( β 2 1 ) (5.104) from whic h the formula follows immediately b y integration as in the usua l T r acy–Widom distribu- tion. Q.E .D 5.3 Noncomm utative PXXX IV Similarly we can prov e 40 Theorem 5. 4 L et a 1 , a 2 b e the c o eﬃcient matric es in the exp ansion of the solution Γ of Pr oblem 3.2 as in formula ( 3.15 ) and deﬁne Φ( λ ) := Γ( λ )e θ ( λ ) ⊗ σ 3 . The n ∂ s j Φ= V j Φ , ∂ λ Φ = B Φ (5.105) V j := λ 2 e j ⊗ σ − + i [ a 1 , e j ] ⊗ 1 − 2 { b 2 , e j } ⊗ σ − − e j ⊗ σ + . (5.106) B ( λ )=  0 − λ 2 λ 3 2 + λ  s − i 2 a ′ 1  0  + 1 λ " i [ a 1 , s ] − i 4 a ′′ 1 − s − i 2 a ′ 1 2 ia 1 + 2[ a 2 , s ] + 2[ s , a 1 ] a 1 − 1 2 ( a ′ 1 ) 2 1 + i [ a 1 , s ] + i 4 a ′′ 1 # (5.107) Denoting by D = P r j =1 ∂ s j , so that Ds = 1 , we have D Φ = V D Φ , V D = λ 2 1 ⊗ σ − − 2 i D a 1 ⊗ σ − − 1 ⊗ σ + =  0 − 1 λ 2 − 2 i D a 1 0  (5.108) The m atric es a 1 , a 2 satisfy the e quations (prime denotes action of D ) a ′′′ 1 = 8 i [ a 1 , s ] a 1 + 8 a 1 + 8 i [ s , a 2 ] + 6 i ( a ′ 1 ) 2 + 4 { a ′ 1 , s } (5.109) a ′ 2 = a ′ 1 a 1 (5.110) Remark 5.6 D iﬀer entiating ( 5.109 ) and using ( 5.110 ) one obtains an ODE for the matrix a 1 a iv 1 = 6 i { a ′′ 1 , a ′ 1 } + 8 ia ′ 1 [ s , a 1 ] + 8 i [ a 1 , σ a ′ 1 ] + 8 i s [ a ′ 1 , a 1 ] + 4 { s , a ′′ 1 } + 16 a ′ 1 (5.111) Remark 5.7 If r = 1 , then we ar e in the c ommutative setting and s = s is just a sc alar. Then the term involvi ng a 2 in ( 5.109 ) dr ops out and we obtain ( ′ = ∂ s ) a ′′′ 1 = 8 a 1 + 6 i ( a ′ 1 ) 2 + 8 s a ′ 1 (5.112) If we take the e qu ation ( 5.112 ) and we diﬀer entiate onc e we obtain, for a ′ 1 , the e quation ( 1.19 ) (up to r esc aling). F or this r e ason we wil l c al l the system ( 5.109 , 5.110 ) the noncom mutativ e P ainle v ´ e XXXIV equation . Corollary 5.2 D enoting by F ( nc ) 1 ( ~ s ) the F r e dholm determinant of the op er ator Id + A i ( • ; ~ s ) on L 2 ( R + , C r ) we have ∂ s j F ( nc ) 1 ( ~ s ) = i ( a 1 ) j j (5.113) and a 1 is a solution of the nonc ommutative PXXXIV e quation ( 5.109 , 5.110 ). In p articular D F ( nc ) 1 ( ~ s ) = i T r a 1 . (5 .114) 41 Pro of of Cor. 5.2 . The F redholm determinant equa ls the deter minant of Id + K as expla ined already in the pro of of Thm. 5.1 . Then the form ula s ab ov e follow simply fro m Thm. 4.2 and Corollar y 4.1 . Q.E.D Before concluding the pap er with the pro o f of Thm. 5.4 we p o int out tha t –in a sense– all the relev ant information is alre ady contained in Thm. 5.3 be c a use of the matr ix Miur a relation a 1 ( ~ s ) Prop. 3.2 = α 1 ( ~ s ) − iβ 1 ( ~ s ) eq. ( 5.55 ) = − 2 i Z ∞ S β 2 1 ( t + δ 1 , . . . , t + δ r )d t − iβ 1 ( ~ s ) (5.115) where ~ s = ( S + δ 1 , . . . S + δ r ) and β 1 is the nonco mm utative Hastings– McLeo d family of solutions (depending on C ) in Theo rem 5.3 . This immediately yields the Corollary 5.3 The F r e dholm determinant of the m atr ix Airy c onvolution kernel A i ~ s satisﬁes det ( I d + A i ~ s ) = ex p  Z ∞ S T r  β 1 ( t + δ 1 , . . . , t + δ r ) + 2( t − S ) β 2 1 ( t + δ 1 , . . . , t + δ r )  d t  (5.116) wher e β 1 ( ~ s ) is t he Hastings-McL e o d family of solutions to nonc ommu tative Painlev ´ e II as in Thm. 5.3 . Pro of of Thm. 5.4 . Recall that L = 1 ⊗ L and Lσ 3 L − 1 = i λ σ + − iλσ − , Lσ 2 L − 1 = 1 λ σ + + λσ − , Lσ 1 L − 1 = σ 3 (5.117) ∂ j Γ( λ ) + iλ Γ( λ )e j ⊗ σ 3 =  V (2) j λ 2 + V (1) j λ + V (0) j  Ξ( λ ) L ( λ ) ∂ j ( a 1 ⊗ σ 3 ) λ + . . . + iλ L ( λ )  1 + a 1 ⊗ σ 3 + b 1 ⊗ σ 1 λ + α 2 ⊗ 1 + b 2 ⊗ σ 1 λ 2 + . . .  e j ⊗ σ 3 = =  V (2) j λ 2 + V (1) j λ + V (0) j  L  1 + a 1 ⊗ σ 3 λ + a 2 ⊗ 1 + b 2 ⊗ σ 1 λ 2 + . . .  (5.118) − i∂ j a 1 ⊗ σ − + i λ 2 ∂ j a 1 ⊗ σ + + a 2 ⊗ 1 + b 2 ⊗ σ 3 λ 2 + . . . + (5.119) + iλ  1 − ia 1 ⊗ σ − + i λ 2 a 1 ⊗ σ + + a 2 ⊗ 1 + b 2 ⊗ σ 3 λ 2 + . . .  e j ⊗  i λ σ + − iλσ −  =  V (2) j λ 2 + V (1) j λ + V (0) j   1 − ia 1 ⊗ σ − + i λ 2 a 1 ⊗ σ + + a 2 ⊗ 1 + b 2 ⊗ σ 3 λ 2 + . . .  (5.120) = λ 2 ( 1 − ia 1 ⊗ σ − )e j ⊗ σ − + ( ia 1 ⊗ σ + + a 2 ⊗ 1 + b 2 ⊗ σ 3 ) e j ⊗ σ − − ( 1 − ia 1 ⊗ σ − )e j ⊗ σ + = = λ 2 V (2) j ( 1 − ia 1 ⊗ σ − ) + V (2) j ( ia 1 ⊗ σ + + a 2 ⊗ 1 + b 2 ⊗ σ 3 ) + V (0) j ( 1 − ia 1 ⊗ σ − ) (5.121) 42 W e th us ha ve V (2) j ( 1 − ia 1 ⊗ σ − ) = ( 1 − i a 1 ⊗ σ − )e j ⊗ σ − ⇒ V (2) j = e j ⊗ σ − (5.122) V (0) j = i [ a 1 , e j ] ⊗ 1 +  [ a 2 , e j ] − { b 2 , e j } − [ a 1 , e j ] a 1 − i∂ j a 1  ⊗ σ − − e j ⊗ σ + (5.123) Lo oking at the λ − 1 co eﬃcient one ﬁnds the following iden tity − i∂ j a 1 = [e j , a 2 ] + [ a 1 , e j ] a 1 − { e j , b 2 } (5.1 24) which allows us to rewrite V (0) j = i [ a 1 , e j ] ⊗ 1 − 2 { b 2 , e j } ⊗ σ − − e j ⊗ σ + (5.125) Summing up ( 5.124 ) for j = 1 , . . . r w e a lso hav e D a 1 = − 2 ib 2 . (5.126) W e will need also mor e informatio n fro m ( 5.118 ) b y lo ok ing at the co eﬃcients of the nega tive p ow e r s of λ in particular we will need this for V D := P r j =1 . T o do so it is conv enient to m ultiply ( 5.118 ) on the left by L − 1 . Below we list the results of lengthy but co mpletely straightforward insp ections . W e list the en try of the co eﬃcient of λ j in the form [ λ j ] k,ℓ . [ λ − 1 ] 1 , 2 b 2 = i 2 a ′ 1 [ λ − 2 ] 1 , 2 b 3 = − 1 2 a ′ 1 a 1 − i 4 a ′′ 1 [ λ − 2 ] 1 , 1 a ′ 2 = 1 2 a ′ 1 a 1 − ib 2 a 1 = a ′ 1 a 1 [ λ − 3 ] 1 , 2 b 4 = i 4 a ′ 1 b 2 + 1 2 b 2 2 + i 4 a ′ 1 a 2 + 1 2 b ′ 3 + 1 2 b 2 a 2 = − 1 2 ( a ′ 1 ) 2 + i 2 a ′ 1 a 2 − 1 4 a ′′ 1 a 1 − i 8 a ′′′ 1 (5.127) A similar and c ompletely straig htforward computatio n (inv olving longer algebra) yields B ( λ ) =  0 − λ 2 λ 3 2 + λ ( s − b 2 ) 0  + (5.128) + 1 λ  b 3 − ib 2 a 1 + i [ a 1 , s ] − s − b 2 ia 1 − b 4 − { s , b 2 } − ib 3 a 1 + [ a 2 , s ] + [ s , a 1 ] a 1 + b 2 ( b 2 − a 2 1 + a 2 ) i [ a 1 , s ] + ib 2 a 1 − b 3 + 1  (5.129) The expressio n can b e simpliﬁed using ( 5.127 ) to give B ( λ ) =  0 − λ 2 λ 3 2 + λ  s − i 2 a ′ 1  0  + (5.130) + 1 λ  i [ a 1 , s ] − i 4 a ′′ 1 − s − i 2 a ′ 1 ia 1 + [ s , a 2 ] − i 2 { s , a ′ 1 } + [ x, a 1 ] a 1 + 1 4 ( a ′ 1 ) 2 + i 8 a ′′′ 1 1 + i [ a 1 , s ] + i 4 a ′′ 1  (5.131) 43 One then ha s to write the z ero-cur v a ture equations ∂ λ V D − D B + [ V D , B ] = 0 (5.132) which yield a ′′′ 1 = 8 i [ a 1 , s ] a 1 + 8 a 1 + 8 i [ s , a 2 ] + 6 i ( a ′ 1 ) 2 + 4 { a ′ 1 , s } (5.133) a ′ 2 = a ′ 1 a 1 (5.134) where the ﬁrst eq uation comes from the entry (1 , 1) of the coeﬃcient in λ − 1 of ( 5.132 ), while the second equation co mes from ( 5.127 ); all other entries of ( 5.1 32 ) are then automatically zero. 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