* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Full Text - J
Capelli's identity wikipedia , lookup
Cartesian tensor wikipedia , lookup
Quadratic form wikipedia , lookup
Linear algebra wikipedia , lookup
Determinant wikipedia , lookup
System of polynomial equations wikipedia , lookup
Jordan normal form wikipedia , lookup
Eigenvalues and eigenvectors wikipedia , lookup
Non-negative matrix factorization wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Matrix (mathematics) wikipedia , lookup
Gaussian elimination wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Four-vector wikipedia , lookup
System of linear equations wikipedia , lookup
Singular-value decomposition wikipedia , lookup
Matrix calculus wikipedia , lookup
Perron–Frobenius theorem wikipedia , lookup
Funkcialaj Ekvacioj, 59 (2016) 383–433 On Stokes Matrices in terms of Connection Coe‰cients By Davide Guzzetti* (SISSA, Italy) Abstract. We study the classical problem of the computation of a complete system of Stokes matrices in terms of connection coe‰cients. Stokes matrices refer to a linear system of ODEs with Poincaré rank one and semi-simple leading matrix, while the connection coe‰cients connect solutions of the associated hypergeometric system of ODEs. The problem here is solved with no assumptions on the residue matrix at zero of the system of Poincaré rank one, so extending method and results of [4]. Key Words and Phrases. Stokes matrices, Connection matrices, Monodromy linear di¤erential systems, Laplace transform. 2010 Mathematics Subject Classification Numbers. 34M40, 34M35. 1. Introduction We consider an n n linear system of ODEs of Poincaré rank one at z¼y dY A1 ¼ AðzÞY ; where AðzÞ ¼ A0 þ ð1Þ : dz z Here A0 ; A1 A GLðn; CÞ, and A0 has distinct eigenvalues. gauge, we assume that A0 is already diagonal, namely: ð2Þ A0 ¼ diagðl1 ; . . . ; ln Þ; Up to a constant li 0 lj for i 0 j; li A C; 1 a i a n: The following Fuchsian system ð3Þ ðA0 lÞ dC ¼ ðA1 þ I ÞC; dl I :¼ n n identity matrix; is associated to (1) by Laplace transformation, as it is well known [6], [14]. ~ðzÞ of (1) can be expressed in terms of convergent LaplaceVector solutions Y ~ ðlÞ of (3): type integrals of vector solutions C ð ~ ðlÞdl; ~ Y ðzÞ ¼ e lz C ð4Þ g ~ ðlÞj ¼ 0. where g is a suitable path such that e lz ðl A0 ÞC g * ORCID ID: 0000-0002-6103-6563. 384 Davide Guzzetti By means of Laplace transform, in [4] a complete system of Stokes multipliers for (1) is computed in terms of connection coe‰cients, connecting selected vector solutions of (3) (called associated functions) at di¤erent Fuchsian singularities. The definition of associated functions, connection coe‰cients, and the results of [4] depend upon the assumption that the diagonal entries of A1 are not integers (this is called assumption (i) in [4]). Special systems of the form (1) appear in the analytic theory of semisimple Frobenius manifolds [8], [9], where A0 has distinct eigenvalues as in (2). The matrix A1 has special form V þ cI , where V is skew symmetric and c A C. The analytic description of Frobenius manifolds in terms of isomonodromic deformations of (1), with the above special A1 , requires to consider all possible values of c. Therefore assumption (i) of [4] may fail, depending on the value of c. In [10] all possible values of c are analysed, obtaining the relation between Stokes matrices and connection coe‰cients, by means of Laplace transformation. See also [21]. Other applications to integrable systems involve systems (1) and (3): an example is the isomonodromic approach to the sixth Painlevé equation in terms of a 3 3 system of type (1) (see [18]). The computation of the relation between Stokes matrices of (1) and monodromy data of (3) may have important applications in this case, such as the understanding of the monodromy data associated to special solutions of the sixth Painlevé equation, including the solution associated to the quantum cohomology of CP 2 . Monodromy data can be computed in terms of Stokes matrices, once the relation with the latter and connection coe‰cients is known (see Corollary 6 in this paper). Since matrix A1 may violate assumption (i), the results of [4] are not enough to study this example. And indeed such study has not been done yet. The above are motivations for our paper. Accordingly, our purpose is to extend the results of [4] when no assumptions on A1 are made, by extending the same technique of [4], namely Laplace transformation. When no assumptions on A1 are made, new types of logarithmic behaviours occur among vector solutions of (3), which come from resonances. They are not studied in [4]. As a first step, we construct selected vector solutions of (3), which include the new logarithmic behaviours (Section 3). This class of solutions extends that of [4]. As a second step, we define the connection coe‰cients for the selected solutions. Finally, we obtain fundamental solutions of (1) in terms of suitable Laplace integrals of the selected solutions (or their modifications). As a result, in Theorem 1 of Section 7, we obtain the explicit relation between a complete system of Stokes multipliers of (1) and connection coe‰cients, which holds for any A1 . Conversely, in Corollary 6 we express traces of products of monodromy matrices of system (3) in terms of Stokes multipliers of (1). These traces On Stokes Matrices in terms of Connection Coe‰cients 385 are invariant monodromy data of (3), and Corollary 6 allows their computability in terms of Stokes matrices, independently of the knowledge of connection coe‰cients. In order to prove Theorem 1, higher order primitives of the selected solutions need to be studied. This requires a non trivial extension of the technique of [4], and a consistent technical e¤ort. As a side result of Theorem 1, the monodromy and connection relations of higher order primitives are obtained (see Section 8). We stress that our purpose is to use and extend the classical technique of Laplace transformation of [4]. A more refined technique, namely the theory of summation and resurgence [12], was developed years later. In [16], this theory is applied to obtain the explicit relation between Stokes-Ramis matrices and connection constants for a general system of rank one with the assumptions of a single level equal to one (this includes the case of diagonalizable A0 , with possibly coinciding eigenvalues). The same is done in [19] with the assumptions of an arbitrary single level. To our knowledge, the case when no assumptions at all are made on A0 , possibly involving a ramified singularity at infinity (and the system of rank one may not be in Birkho¤ normal form), has yet to be studied. Finally, we remark two facts. First, that the results of the paper can be generalized to those systems of Poincaré rank r b 1 (namely AðzÞ ¼ z r1 A~ðzÞ, A~ðzÞ analytic in a neighborhood of infinity), that are formally meromorphically equivalent to (1) in the sense of [3]. This can be done as in section 4 of [4] by a generalization of the definition of associated functions to include functions in ~k ðlÞ’s or C ~ ðkÞ ðlÞ’s as in our Section 6.2, points 1), 2), 3) and the form of the C k 4). Secondly, that any system of Poincaré rank r can be reduced to a system of Poicaré rank one by enlarging the size of the system to r n (see [15], and also [20], [17], [5]). In [15] the explicit relation between Stokes matrices of the initial and the reduced systems is given. In this sense, results obtained for rank one can be extended to any rank r. (I thank M. Loday-Richaud for drawing my attention to the papers [15], [16], [19]). 1.1. Organization of the paper Section 2: We state the problem and give the main results. The example in the end of the section shows how the general result applies to the case of Frobenius manifolds. ~k ðlÞ, 1 a k a n, to Section 3: We construct selected vector solutions C system (3)–(5), and define the connection coe‰cient, with no assumptions on A1 . Section 4: We construct two matrix solutions CðlÞ and C ðlÞ of system (3)–(5), with no assumptions on A1 . They generalize the matrices Y ðtÞ and 386 Davide Guzzetti Y ðtÞ of [4]. We establish when they are fundamental, and compute their monodromy in terms of connection coe‰cients. Section 5: We discuss the dependence of C and C on the choice of the branch cuts. Section 6: We recall the definition of complete set of Stokes multipliers for (1). We write a fundamental matrix of system (1), having canonical asymp~k ðlÞ’s, 1 a k a n (or their totics in a wide sector, as Laplace integrals of the C modifications including singular contributions—see formula (44)). We also ~k ðlÞ’s in terms of the coe‰cients of the former asymptotics. express the C Section 7: We state the main theorem (Theorem 1), which gives Stokes matrices and Stokes factors of (1) in terms of connection coe‰cients of (3), and express the first monodromy invariants of system (3) in terms of Stokes matrices (Corollary 6). Section 8: We prove Theorem 1, and find relations and monodromy for q-primitives of vector solutions of (3)–(5). It is in this section that the most technical e¤ort is required in order to generalize the results and method of [4] when no assumptions on A1 are made. 2. Setting and results Let us denote the diagonal entries of A1 as follows diagðA1 Þ ¼ ðl10 ; . . . ; ln0 Þ: In [4] it is assumed that l10 ; . . . ; ln0 are not integers (assumption (i), page 693). In this paper, we allow any values of ðl10 ; . . . ; ln0 Þ A C n . Therefore, in order to generalize the method and the results of [4], we need to characterize the solutions of (3) for any ðl10 ; . . . ; ln0 Þ A C n . System (3) can be rewritten as ð5Þ n dC X Bk ¼ C; dl l lk k¼1 Bk :¼ Ek ðA1 þ I Þ; 1 a k a n; where Ek is a n n matrix with entries ðEk Þkk ¼ 1 and ðEk Þij ¼ 0 otherwise. A vector, or a matrix solution of (5) is multivalued in Cnfl1 ; . . . ; ln g, with regular singularities in l1 ; . . . ; ln . Let U be the universal covering of Cnfl1 ; . . . ; ln g. Following [4], we fix parallel branch cuts Lk , oriented from lk to y. Choose a real number h such that ð6Þ h 0 argðlj lk Þ mod 2p; for all 1 a j; k a n: The cuts are defined as follows Lk :¼ fl A U j argðl lk Þ ¼ hg; 1 a k a n: On Stokes Matrices in terms of Connection Coe‰cients 387 Condition (6) means that a cut Lk does not contain other poles lj , j 0 k. See figure 1. Such values of h are called admissible in [4]. We fix the branch of any lnðl lk Þ by considering the following sheet of Ph U, obtained with the above cuts: Ph :¼ fl A U j h 2p < argðl lk Þ < h; 1 a k a ng: We prove in Section 3 that system (3) admits a matrix solution of the form: ~n ðlÞ; ~1 ðlÞ j j C CðlÞ ¼ ½C l A Ph ; ~k ðlÞ, k ¼ 1; . . . ; n, are the selected vector solutions we are whose columns C ~k ðlÞ’s are interested in. This matrix generalizes matrix Y ðtÞ of [4]. The C uniquely characterized by their behaviours for l in a neighbourhood of lk , as follows: ~k ðlÞ C 8 X ðkÞ 0 ~ > bl ðl lk Þ l Þðl lk Þ lk 1 ; ek þ if lk0 B Z; ðGðlk0 þ 1Þ~ > > > > lb1 > > > ! > Nk < X ðkÞ ð1Þ l ~ ~ ¼ bl ðl lk Þ ðl lk Þ Nk 1 ; if lk0 ¼ Nk A Z ; ek þ > ðNk 1Þ! > lb1 > > > X ðkÞ > ðkÞ > ~ ~ > dl ðl lk Þ l ; if lk0 ¼ Nk A N: > : d0 þ lb1 Here N ¼ f0; 1; 2; . . .g and Z ¼ f1; 2; 3; . . .g are the non negative and negative integers respectively, and ~ ek is the k-th unit column vector of C n . The Taylor series in ðl lk Þ converge in a neighbourhood of lk . The ðkÞ coe‰cients ~ bl A C n are uniquely determined by the choice of the normalizaðkÞ 0 tions Gðl þ 1Þ~ ek and ðð1Þ Nk =ðNk 1Þ!Þ~ ek . The coe‰cients d~ A C n are k l uniquely determined by the existence of a singular vector solution at lk with behaviour ek þ Oðl lk Þ ~k ðlÞ lnðl lk Þ þ Nk !~ C ; ðl lk Þ Nk þ1 Nk ¼ lk0 A N: We show in Section 3 that for any j there always exist n 1 vector solutions which are locally analytic at l ¼ lj , and there exists at most one solution which is singular at l ¼ lj (see expression (16), with k replaced with j). Therefore, a ~k ðlÞ has in general a singular contribution close to a lj , with j 0 k, solution C which behaves as a multiple of the singular solution at lj . The corresponding multiplicative constant allows to define the connection coe‰cients, as follows. ð jÞ Let PNj ðlÞ denote a vector function with polynomial entries in ðl lj Þ of degree Nj A N, and let regðl lj Þ be a vector function analytic (regular) in a 388 Davide Guzzetti neighbourhood of lj . We will show (Definition 1, Section 3) that there exist unique connection coe‰cients cjk A C such that, in a neighbourhood of any ~k ðlÞ behaves as follows: lj 0 lk , C 8 ~j ðlÞcjk þ regðl lj Þ; > lj0 B Z; C > > > > > ~j ðlÞ lnðl lj Þcjk þ regðl lj Þ; <C lj0 A Z ; ~ ð7Þ Ck ðlÞ ¼ ! ð jÞ > > PNj ðlÞ > > C ~j ðlÞ lnðl lj Þ þ > cjk þ regðl lj Þ; lj0 ¼ Nj A N: > Nj þ1 : ðl lj Þ The connection coe‰cients multiply, in the above formula, the singular contribution at lj . Note that they depend on the choice of the branch cuts, namely on h (see Section 5). In particular, ckk ¼ 1 if lk0 B Z; ckk ¼ 0 if lk0 A Z: The connection coe‰cients determine the monodromy of the matrix CðlÞ, as follows: Proposition I (Proposition 2). Let the branch cuts L1 ; . . . ; Ln be fixed. Let A1 be any matrix. The monodromy matrix of CðlÞ for a small loop in anticlockwise direction around lk , not encircling all the other points lj 0 lk , ðkÞ j ¼ 1; . . . ; n is the matrix Mk ¼ ðmij Þi; j¼1... n with entries: ðkÞ mjj ¼ 1 ðkÞ mkj ¼ ak ckj ; 1 a j a n; j 0 k; 1 a j a n; j 0 k; ðkÞ 0 mkk ¼ e 2pilk ; ðkÞ mij ¼ 0 otherwise: Fig. 1. The poles lj , 1 a j a n of system (3), and branch cuts Lj . On Stokes Matrices in terms of Connection Coe‰cients 389 where ( 0 ak :¼ ðe 2pilk 1Þ; ak :¼ 2pi; if lk0 B Z; if lk0 A Z: Equivalently, the e¤ect of the loop around lk is ~k ðlÞ; ~k ðlÞ 7! e 2pilk0 C C ~j ðlÞ 7! C ~j ðlÞ þ ak ckj C ~k ðlÞ; C The matrix solution CðlÞ is not necessarily fundamental. gives a su‰cient condition: j 0 k: The following Proposition II (Propositions 3 and 4). If A1 has no integer eigenvalues, then CðlÞ is a fundamental matrix and M1 ; . . . ; Mn generate the monodromy group of system (3). Moreover, the matrix C :¼ ðcjk Þ is invertible if and only if A1 has no integer eigenvalues. Remark 1. There are cases when A1 has integer eigenvalues and C is fundamental. We prove that in these cases, necessarily, some lk0 A Z. To explain the relation between connection coe‰cients and Stokes multipliers of (1), recall that solutions of the latter are characterized by the Stokes phenomenon. Let t ¼ 3p=2 h. According to [2], there are three unique fundamental matrices of ð1Þ, say YI ðzÞ, YII ðzÞ and YIII ðzÞ, with canonical asymptotic behaviour ðI þ Oð1=zÞÞ expfA0 z þ A1 ln zg in the three sectors fz j t p a arg z a tg, fz j t a arg z a t þ pg and fz j t þ p a arg z a t þ 2pg respectively. They are related by two Stokes matrices Sþ and S such that YII ðzÞ ¼ YI ðzÞSþ ; arg z ¼ t; YIII ðzÞ ¼ YII ðzÞS ; arg z ¼ t þ p: Introduce in f1; 2; . . . ; ng the partial ordering 0 given by j 0 k , <ðzðlj lk ÞÞ < 0 for arg z ¼ t; i 0 j; i; j A f1; . . . ; ng: We prove the following main result: Theorem I (Theorem 1). The Stokes matrices of system (1), without assumptions on A1 , and the connection coe‰cients cjk , 1 a j; k a n, of system (3)–(5) are related by the formulae 8 8 0 for j 0 k; > > < 0; < e 2pilk ak cjk ; for j 0 k; 1 for j ¼ k; ½Sþ jk ¼ 1; ½S jk ¼ 1; for j ¼ k; > > : e 2piðlk0 lj0 Þ a c ; for j k: : 0; for j k; k jk Corollary I (Corollary 6). Let A1 be any matrix. hold for the monodromy matrices of CðlÞ: The following equalities 390 Davide Guzzetti 0 ( TrðMj Mk Þ ¼ TrðMk Þ ¼ n 1 þ e 2pilk ; 0 0 0 2pilj0 2pilk0 2pilk0 n 2 þ e 2pilj þ e 2pilk e 2pilj ½Sþ jk ½S1 kj ; n2þe þe e ½S1 jk ½Sþ kj ; if j 0 k; if j k: Example. The general results above apply to semisimple Frobenius manifolds, where A0 has distinct eigenvalues. In this case, the relation between Stokes matrices and connection coe‰cients was computed in [8] and [9] when A1 does satisfy assumption (i), and in [10] when it does not. The matrix A1 has a special form, namely it is expressed in terms of a skew symmetric matrix V as follows: 1 þ n I; n A C; V T ¼ V : A1 ¼ V 2 We show how our general results above apply to this case. Since lk0 ¼ n 1=2, 1 a k a n, it follows that ð1 þ e 2pin Þ if n B Z þ 1=2; a1 ¼ a2 ¼ ¼ an ¼ a; where a :¼ 2pi if n A Z þ 1=2: From Theorem I above (and the fact that the ckk ¼ 0 when lk0 A Z), we deduce that e 2pin Sþ þ S1 ¼ aC; where C :¼ ðcij Þ: Since V is an n n skew symmetric matrix, it can be easily verified that SþT ¼ S1 : Thus e 2pin Sþ þ SþT ¼ aC: ð8Þ The above, and Proposition II, allow us to conclude that if e 2pin Sþ þ SþT is invertible, then A1 has no integer eigenvalues and so CðlÞ is invertible. This is part of the first assertion of Theorem 4.3 of [10], namely if detðe 2pin Sþ þ SþT Þ 0 0; ~1 ; . . . ; C ~n . From (8) and then system (3) has n linearly independent solutions C Proposition I, it follows that for an anticlockwise loop around li , the monodromy of the above solutions is ~i 7! e 2pin C ~i ; C ~j 7! C ~j e ipn ðe ipn Sþ þ e ipn S T Þ C ~ C þ ij i ; The above is formula (4.11) in Theorem 4.3 of [10]. j 0 i: On Stokes Matrices in terms of Connection Coe‰cients 391 Note. After this paper was completed, the works [13] and [23] appeared, showing that for the Frobenius manifold given by the Quantum Cohomology of Grassmannians, there may be cases (depending on the dimension) when A0 is still diagonalizable, but with some coinciding eigenvalues. 3. Local solutions of system (5) (i.e. (3)) The matrix Bk in system (5) has zero entries, except for the k-th row. Indeed, letting A1 ¼ ðAij Þi; j¼1;...; n , a straightforward computation yields 1 0 0 0 0 .. C .. B .. B . . C . C B 0 C B A k-th row: Bk ¼ B k1 Ak; k1 lk 1 Ak; kþ1 Akn C C B .. C .. B .. @ . . A . 0 0 0 A fundamental matrix solution of (5) is multivalued in Cnfl1 ; . . . ; ln g and single-valued in Ph , for any admissible direction h. If l is in a neighbourhood of a lk not containing other poles, there exists a fundamental matrix solution ~ ðkÞ ðlÞ j j C ~ ðkÞ ðlÞ; C ðkÞ ðlÞ ¼ ½C n 1 which can be computed in a standard way, depending on the value of lk0 (see [22]). In [4], only the case lk0 B Z is considered (point 1) below). Here we need to analyse also the case lk0 A Z (points 2), 3) and 4) below). 1) [Generic Case, as in [4]] If lk0 B Z, then Bk is diagonalizable, with diagonal form T ðkÞ ¼ ½G ðkÞ 1 Bk G ðkÞ ¼ diagð0; . . . ; 0; lk0 1; 0; . . . ; 0Þ; where the non zero entry lk0 1 is at k-th position. The k-th column of the diagonalizing matrix G ðkÞ can be chosen to be a multiple of the k-th vector ~ ek of the canonical basis of C n . As in [4] we choose normalization Gðlk0 þ 1Þ~ ek . A fundamental matrix is then C ðkÞ ðlÞ ¼ G ðkÞ ðI þ Oðl lk ÞÞðl lk Þ T ðkÞ ~ðkÞ ðlÞ j j c ~ðkÞ ðlÞðl lk Þ T ðkÞ ¼ ½c n 1 0 ~ðkÞ ðlÞ j j c ~ðkÞ ðlÞ j c ~ðkÞ ðlÞðl lk Þ lk 1 j c ~ðkÞ ðlÞ j j c ~ðkÞ ðlÞ; ¼ ½c n 1 k1 k kþ1 where Oðl lk Þ is a matrix valued Taylor series, converging in the neigh~ðkÞ ðlÞ j j c ~ðkÞ ðlÞ :¼ bourhood of lk and vanishing as l ! lk . Here ½c n 1 ðkÞ G ðI þ Oðl lk ÞÞ is analytic in a neighbourhood of lk . The columns of 392 Davide Guzzetti C ðkÞ form n independent vector solutions, n 1 being analytic. singular and we denote it ð9Þ The k-th is 0 ~ðkÞ ðlÞðl lk Þ lk 1 ; ~k ðlÞ :¼ C ~ ðkÞ ðlÞ ¼ c C k k where ~ðkÞ ðlÞ ¼ Gðl 0 þ 1Þ~ c ek þ k k X ðkÞ ~ bl ðl lk Þ l : lb1 ðkÞ The vector coe‰cients ~ bl can be computed rationally from the matrix ~k is called associated coe‰cients Bl ’s of system (5). See [22]. The above C function in [4]. 2) [Jordan Case] If lk0 ¼ 1, then Bk has Jordan form 1 0 0 C B .. C B . C B C B 0 1 C B 1 J ðkÞ ¼ ½G ðkÞ Bk G ðkÞ ¼ B C: C B 0 0 C B .. C B A @ . 0 The entry 1 is at row ðk 1Þ and column k. The ðk 1Þ-th column of G ðkÞ can be normalized to be ~ ek . There exists a fundamental matrix solution with local representation C ðkÞ ðlÞ ¼ G ðkÞ ðI þ Oðl lk ÞÞðl lk Þ J ðkÞ ðkÞ ~ ðlÞ j j c ~ðkÞ ðlÞðl lk Þ J ¼ ½c n 1 ðkÞ ~ðkÞ ðlÞ j j c ~ðkÞ ðlÞ j c ~ðkÞ ðlÞ lnðl lk Þ ¼ ½c 1 k1 k1 ~ðkÞ ðlÞ j j c ~ðkÞ ðlÞ; ~ðkÞ ðlÞ j c þc n k kþ1 ðkÞ ~ are analytic in a neighbourhood of lk . The columns where the columns c j are n independent vector solutions, n 1 being analytic and the k-th singular. ~k to the non-singular factor of lnðl lk Þ, as follows We assign the symbol C ð10Þ ~ðkÞ ðlÞ ¼ ~ ~k ðlÞ :¼ c ek þ C k1 X ðkÞ ~ bl ðl lk Þ l : lb1 Note that this is a solution of (5). ð11Þ Then, the k-th column of C ðkÞ is ~k ðlÞ lnðl lk Þ þ regðl lk Þ; ~ ðkÞ ðlÞ ¼ C C k where regðl lk Þ means an analytic (vector) function in a neighbourhood of lk . On Stokes Matrices in terms of Connection Coe‰cients 393 3) [First Resonant Case] If lk0 ¼ Nk b 0 is integer, then Bk is diagonalizable as in case 1), but now a fundamental solution has the form ðkÞ ðkÞ C ðkÞ ðlÞ ¼ G ðkÞ ðI þ Oðl lk ÞÞðl lk Þ T ðl lk Þ R ; ðkÞ where R ðkÞ is a matrix with zero entries expect for Rjk , j ¼ 1; . . . ; n, and j 0 k, because ðEigenðBk ÞÞj EigenðBk ÞÞk ¼ Nk þ 1 > 0. Thus, only the k-th column ðkÞ ðkÞ of R ðkÞ may be non zero. Let rj :¼ Rjk , so that the k-th column is ðkÞ ðkÞ ðkÞ ~ r ðkÞ :¼ ðr1 ; . . . ; rk1 ; 0; rkþ1 ; . . . ; rnðkÞ Þ T ; ðkÞ where T means transposition. The entries rj are computed as rational functions of the entries of the matrices Bl , l ¼ 1; . . . ; n (see [22]). From the above, it follows that ~ðkÞ ðlÞ j j c ~ðkÞ ðlÞðl lk Þ T ðkÞ ðI þ R ðkÞ lnðl lk ÞÞ C ðkÞ ðlÞ ¼ ½c n 1 ðkÞ ðkÞ ðkÞ ðkÞ ~ ðlÞ j C ðlÞ j c ~ ðlÞ j j c ~ðkÞ ðlÞ; ~ ðlÞ j j c ¼ ½c n 1 k1 k kþ1 where ( ðkÞ C k ðlÞ ¼ X ) ðkÞ ~ðkÞ rj c j ðlÞ lnðl lk Þ þ j0k ~ðkÞ ðlÞ c k ðl lk Þ Nk þ1 ; ~ðkÞ ðlÞ ¼ Nk !~ c ek þ Oðl lk Þ; k the factor Nk ! coming from a chosen normalization of G ðkÞ . The columns are n ~ðkÞ , j 0 k) and the independent vector solutions, n 1 being analytic (i.e. the c j ðkÞ ~k to the non-singular factor of k-th singular (i.e. C k ). We assign the symbol C lnðl lk Þ as follows ~k ðlÞ :¼ C X ðkÞ ~ðkÞ rj c j ðlÞ ¼ j0k X ðkÞ d~l ðl lk Þ l : lb0 Note that this is a solution of (5), being a linear combination of regular ~ðkÞ . Special cases can occur when ~ ~k ðlÞ 1 0. We solutions c r ðkÞ ¼ 0, so that C j ðkÞ conclude that the k-th column of C is ðkÞ ð12Þ ~ ðkÞ ðlÞ ¼ C ~k ðlÞ lnðl lk Þ þ C k PNk ðlÞ ðl lk Þ Nk þ1 þ regðl lk Þ; where ðkÞ PNk ðlÞ ¼ Nk !~ ek þ Nk X l¼0 ðkÞ ~ bl ðl lk Þ l ; 394 Davide Guzzetti ~ðkÞ . The vector represents the first Nk þ 1 terms in the expansion of c k ðkÞ coe‰cients ~ bl are computed rationally from the coe‰cients of (5). The solution (12) is not uniquely determined, because we can add a linear com~ðkÞ , but the singular part is uniquely deterbination of regular solutions c j ~k ðlÞ is uniquely mined by the normalization of P ðkÞ ðlÞ. Consequently, also C determined. 4) [Second Resonant Case] If lk0 ¼ Nk a 2 is integer, then Bk is diagonalizable as in case 1), but now a fundamental solution has the form ðkÞ ðkÞ C ðkÞ ðlÞ ¼ G ðkÞ ðI þ Oðl lk ÞÞðl lk Þ T ðl lk Þ R ; ðkÞ where R ðkÞ is a matrix with zero entries expect for Rkj , j ¼ 1; . . . ; n, and j 0 k, because ðEigenðBk ÞÞk EigenðBk ÞÞj ¼ Nk 1 > 0. Thus, only the k-th row of ðkÞ ðkÞ R ðkÞ may be non zero. Let rj :¼ Rkj , so that the k-th row is ðkÞ ðkÞ ðkÞ r ðkÞ :¼ ½r1 ; . . . ; rk1 ; 0; rkþ1 ; . . . ; rnðkÞ ; ðkÞ where the entries rj are computed as rational functions of the entries of the matrices Bl , l ¼ 1; . . . ; n (see [22]). Thus, ~ðkÞ ðlÞ j j c ~ðkÞ ðlÞðl lk Þ T ðkÞ ðI þ R ðkÞ lnðl lk ÞÞ; C ðkÞ ðlÞ ¼ ½c n 1 ~ðkÞ ðlÞ are analytic and Taylor expanded in a neighbourhood of where the c j lk . The columns of the above matrix are ~ðkÞ ðlÞðl lk Þ Nk 1 lnðl lk Þ þ c ~ðkÞ ðlÞ; ~ ðkÞ ðlÞ ¼ r ðkÞ c C j j j k ~ ðkÞ ðlÞ C k ¼ ~ðkÞ ðlÞðl c k j ¼ 1; . . . ; n; j 0 k; lk Þ Nk 1 : There are at most n 1 independent singular solutions at lk , and at least one ~ ðkÞ . In special cases, it may happen that r ðkÞ ¼ 0, so that analytic solution C k there are n independent solutions analytic at lk . We show below (Lemma 1) that in fact we can always find n 1 independent solutions analytic at lk , ~k to the k th column: whatever r ðkÞ is. We assign the symbol C ~ðkÞ ðlÞðl lk Þ Nk 1 ; ~k ðlÞ :¼ C ~ ðkÞ ðlÞ ¼ c C k k with normalization ð13Þ ~ðkÞ ðlÞ ¼ c k X ðkÞ ð1Þ Nk ~ bl ðl lk Þ l ; ek þ ðNk 1Þ! lb1 where the convergent Taylor series has coe‰cients determined rationally by the matrices Bl ’s of (5). The logarithmic solutions are rewritten as ~ðkÞ ðlÞ; ~k ðlÞ lnðl lk Þ þ c ~ ðkÞ ðlÞ ¼ r ðkÞ C C j j j j 0 k; 1 a j a n: On Stokes Matrices in terms of Connection Coe‰cients ðkÞ It follows that if at least one rj 395 0 0, we can pick up the singular solutions ~k ðlÞ lnðl lk Þ þ regðl lk Þ: C ð14Þ ~ðkÞ ’s, 1 a j a n, The regular part is an arbitrary linear combination of the c j j 0 k. The singular part is determined uniquely by the normalization (13). Lemma 1. Let lk0 be an integer Nk a 2. If r ðkÞ ¼ 0, system (5) has n independent solutions analytic at lk . If r ðkÞ 0 0, system (5) has n independent solutions, of which n 1 are analytic and one is log-singular at lk . Proof. Let 0 a s a n 1 be the number of non zero values ri1 ; . . . ; ris . If r ðkÞ ¼ 0, then s ¼ 0 and by the preceding construction there exist n independent solutions ~ðkÞ ; . . . ; c ~ðkÞ ; C ~ðkÞ ; . . . ; c ~ðkÞ : ~k ; c c n 1 k1 kþ1 If s > 0, there are s singular (at lk ) solutions ~ ðkÞ ; C ~ ðkÞ ; . . . ; C ~ ðkÞ ; C i1 i2 is and the remaining analytic (at lk ) solutions ~ðkÞ ; c ~ðkÞ ; . . . ; c ~ðkÞ : c j1 j2 jl Note that fi1 ; i2 ; . . . ; is g [ f j1 ; j2 ; . . . ; jl g is a partition of f1; 2; . . . ; k 1; k þ 1; . . . ; ng. We construct another set of s 1 independent analytic (at lk ) solutions: ðkÞ ji1 :¼ ðkÞ ji2 :¼ 1 ~ ðkÞ 1 ~ ðkÞ C i1 ðkÞ C is ; ris ðkÞ ri1 1 ~ ðkÞ 1 ~ ðkÞ C i2 ðkÞ C is ; ris ðkÞ ri2 .. . ðkÞ jis1 :¼ 1 ~ ðkÞ 1 ~ ðkÞ C is1 ðkÞ C is : ris ðkÞ ris1 It follows that there always exist n 1 linearly independent vector solutions which are analytic at lk , namely ðkÞ ðkÞ ji1 ; . . . ; jis1 ; ~ðkÞ ; c ~ðkÞ ; . . . ; c ~ðkÞ ; c j1 j2 jl ~k : C ~k ðlÞ lnðl lk Þ þ regðl lk Þ. Moreover, there also exists the singular solution C This proves the lemma. r 396 Davide Guzzetti Conclusion. The four cases above are summarized below (letting 0! ¼ 1): 8 0 > ðGðlk0 þ 1Þ~ ek þ Oðl lk ÞÞðl lk Þ lk 1 ; Case 1Þ: lk0 B Z; > > > ! > > > > ð1Þ Nk < ~ ek þ Oðl lk Þ ðl lk Þ Nk 1 ; Case 2Þ; 4Þ: ~k ðlÞ ¼ ðNk 1Þ! ð15Þ C lk0 ¼ Nk A Z ; > > >X > > ðkÞ ðkÞ > > rj cj ðlÞ ¼ regðl lk Þ; Case 3Þ: lk0 A N: > : j0k Moreover, there exists a singular solution given by 8 ~ > > > Ck ðlÞ; > > ~k ðlÞ lnðl lk Þ þ regðl lk Þ; > C > > > > > ðkÞ > > PNk ðlÞ >~ < ðlÞ lnðl l Þ þ C k k ~ ðsingÞ ðlÞ :¼ ð16Þ C ðl lk Þ Nk þ1 k > > > > þ regðl lk Þ; > > > ( > > ~k ðlÞ lnðl lk Þ þ regðl lk Þ; > C > > > > : C ~ ðsingÞ 1 0; if r ðkÞ ¼ 0; k lk0 B Z; i:e: ð9Þ; lk0 ¼ 1; i:e: ð11Þ; lk0 A N; i:e: ð12Þ; lk0 A N 2; i:e: ð14Þ: ~ ðsingÞ ðlÞ is uniquely determined. In logarithmic case The singular part of C k ~ ðsingÞ ðlÞ is defined modulo the addition of a linear of (11), (12) and (14), C k combination of regular solutions. 3.1. Connection coe‰cients—definition Definition 1. The connection coe‰cients cjk , 1 a j; k a n, are uniquely defined by ( ~k ðlÞ ¼ C ~ ðsingÞ ðlÞcjk þ regðl lj Þ; C j cjk :¼ 0; 1 a k a n; ~ ðsingÞ ðlÞ 1 0 for l 0 A N 2: when C j j Observe that: a) ckk ¼ 1 for lk0 B Z, ckk ¼ 0 for lk0 A Z: ~k 1 0. This occurs when b) In case lk0 A N, it may happen that C ðkÞ ~ r ¼ 0. In this case cjk ¼ 0 for any j ¼ 1; . . . ; n, namely the k-th column of the matrix C ¼ ðcjk Þ is zero. c) In case lj0 A N 2, it may happen that there is no logarithmic ~ ðsingÞ 1 0. This occurs if r ð jÞ ¼ 0. In such a case, we singularity, namely C j need to define cjk :¼ 0, for any k, so that the matrix C ¼ ðcjk Þ has zero j-th row. d) Letting cjk :¼ 0, for any k, when r ð jÞ ¼ 0, a more explicit way to write the definition of connection coe‰cients is (7). On Stokes Matrices in terms of Connection Coe‰cients 4. 397 Matrix solutions C and C* of system (3)–(5), monodromy and invertibility In the previous section, we have constructed a matrix solution ~1 ðlÞ j j C ~n ðlÞ: CðlÞ :¼ ½C ð17Þ This generalizes matrix Y ðtÞ of [4]. which conditions it is fundamental. In Section 4.2 we will establish under Remark 2. System (3), (5) may have vector solutions that are analytic at all l1 ; . . . ; ln . Such solutions must be polynomials in l, because y is a Fuchsian singularity. The following holds: Lemma 2. System (3), (5) has no polynomial vector solution if and only if A1 has no negative integer eigenvalues. Equivalently (see Remark 2), System (3), (5) has a singular solution at any lk , 1 a k a n, if and only if A1 has no negative integer eigenvalues. Proof. This Lemma is proved in remark 1.1 of [4]. r In [4] it is proved, under the assumption (i) of non integer lk0 ’s, that (3) admits a matrix solution C ðlÞ, whose k th column, k ¼ 1; . . . ; n, is analytic at all poles lj 0 lk . This matrix is called Y ðtÞ in [4]. The existence of C can be proved without any assumption on l10 ; . . . ; ln0 . It has new type of logarithmic behaviours when compared to Y ðtÞ. Proposition 1. Let the matrix A1 be any (no assumptions). Then ~ ðlÞ j j C ~ ðlÞ such that i) There exists a matrix solution C ¼ ½C 1 n ð18Þ ~ ðlÞ ¼ regðl lj Þ C k Ej 0 k: ii) C ðlÞ is a fundamental matrix solution if and only if none of the ~ ðsingÞ ðlÞ 0 0 for any k, eigenvalues of A1 is a negative integer. In this case, C k ~ ðlÞ has the following behaviour for l close to lk and C k ð19Þ ~ ðsingÞ ðlÞ þ regðl lk Þ ~ ðlÞ ¼ C C k k 8 ~k ðlÞ þ regðl lk Þ; > if lk0 B Z; C > > > > ~k ðlÞ lnðl lk Þ þ regðl lk Þ; <C if lk0 A Z ; ¼ ðkÞ > > PNk ðlÞ > > ~k ðlÞ lnðl lk Þ þ > C þ regðl lk Þ; if lk0 A N: : ðl lk Þ Nk þ1 398 Davide Guzzetti C ðlÞ is uniquely defined by (18) and (19), and CðlÞ ¼ C ðlÞC; ð20Þ C :¼ ðcjk Þ: Proof. We have proved in Section 3 that at any lj there exist at least n 1 regular solutions, whatever are l10 ; . . . ; ln0 . This is enough to apply the same procedure of proof of proposition 1 in [4]. The singular behaviour of ~ ðlÞ is directly obtained from C ~ ðsingÞ ðlÞ. C r k k Remark 3. From the above proposition singular at lk . This implies that if none of the integer and lk0 A N 2, then r ðkÞ 0 0, namely 4.1. ~ is always it follows that C k eigenvalues of A1 is a negative ~ ðsingÞ 0 0. C k Monodromy of C and C* associated to a loop around lk Consider a small loop in Ph around a pole lk in counter-clockwise direction, not encircling the other poles; for example l lk 7! ðl lk Þe 2pi , ~1 j j C ~n is easily computed from (15), jl lk j small. Monodromy of C ¼ ½C which immediately implies ( 2pilk0 ~ ; lk0 B Z; ~k ðlÞ 7! Ck ðlÞe C ~k ðlÞ; lk0 A Z: C and from (7), which implies (note that j and k are exchanged here): ( 2pilk0 ~k ðlÞ; l 0 B Z ~ 1Þckj C k ~j ðlÞ 7! Cj ðlÞ þ ðe C ~j ðlÞ þ 2pickj C ~k ðlÞ; C lk0 A Z These formulae make sense also when ckj ¼ 0 for any k in the special case ~j ¼ 0, possibly occurring when l 0 A N, and when ckj ¼ 0 for any j in the C k ~ ðsingÞ 1 0, possibly occurring when l 0 A N 2. special case C k k ~ j j C ~ , which exists Next, we compute the monodromy of C ¼ ½C 1 n when A1 has no negative integer eigenvalues. We consider again a small loop around lk as above. From (18) we have ~ ðlÞ 7! C ~ ðlÞ; C j j Ej ¼ 1; . . . ; n; j 0 k: Then, from the above and (20), 8 X 0 ~ ðlÞ þ ðe 2pilk0 1Þ ~ ðlÞ; > e 2pilk C cjk C > k j > < j0k ~ Ck ðlÞ 7! X > ~ ðlÞ; ~ ðlÞ þ 2pi > cjk C >C k j : j0k Summarizing: lk0 B Z; lk0 A Z: On Stokes Matrices in terms of Connection Coe‰cients 399 Proposition 2. The monodromy matrices representing the monodromy of C and C for a small counter-clockwise loop around lk in Ph are as follows. a) The matrix C is always defined. The monodromy is 1 0 0 0 0 0 B .. .. C .. .. B . . C . . C B C Bc 1 a k a n; C 7! CMk ; Mk ¼ I þ ak B k1 ck2 ckk ckn C; B . .. C .. .. C B . @ . . A . . 0 0 0 0 where I is the n n identity matrix, only the k-th row in the second matrix is non zero, and ( 0 ak :¼ ðe 2pilk 1Þ; if lk0 B Z; if lk0 A Z: ak :¼ 2pi; b) If A1 has no negative integer eigenvalues, then C dromy is 0 0 0 c1k B B0 0 c2k B B .. .. . . .. B. . . . C 7! C Mk ; Mk ¼ I þ ak B B B 0 0 ckk B. . .. B. . @. . . 0 0 cnk exists. The mono- 1 0 C 0C C .. C .C C; C 0C .C .. C . .. A 0 where only the k-th column in the second matrix is non zero. ðkÞ Remark 4. The matrix Mk is the matrix ðmij Þ in Proposition I of Section 2. For a clockwise loop, we analogously find that ½Mk1 kj ¼ bk ckj ; ½Mk1 jj ¼ 1; j 0 k; j 0 k; ½Mk1 kj ¼ 0 otherwise; 0 ½Mk1 kk ¼ e 2pilk ; and ½ðMk Þ 1 jk ¼ bk cjk ; j 0 k; ½ðMk Þ 1 ji ¼ 0 ½ðMk Þ 1 jj ¼ 1; j 0 k; ½ðMk Þ 1 kk ¼ ½Mk1 kk ; otherwise; 0 where b k ¼ e 2pilk ak . Corollary 1. tion 2 are The first invariants of the monodromy matrices in Proposi- 400 Davide Guzzetti 0 TrðMk Þ ¼ n 1 þ e 2pilk ; 0 0 0 0 TrðMj Mk Þ ¼ n 2 þ e 2pilj þ e 2pilk þ aj ak cjk ckj 0 0 ¼ n 2 þ e 2pilj þ e 2pilk þ e 2piðlj þlk Þ b j bk cjk ckj : If A1 has no negative integer eigenvalues, then TrðMk Þ ¼ TrðMk Þ; TrðMj Mk Þ ¼ TrðMj Mk Þ: From Proposition 1 we know that C is fundamental if and only if A1 has no negative integer eigenvalues. Thus: Corollary 2. Suppose that A1 has no negative integer eigenvalues; then M1 ; . . . ; Mn generate the monodromy group of equation (3–5). 4.2. On the invertibility of C and C(l) We establish necessary and su‰cient conditions for the matrices CðlÞ and C ¼ ðcjk Þ to be invertible. Let l A Ph . Remark 5. If ~ r ðkÞ ¼ 0 (case lk0 A N) then C has zero k-th column and also CðlÞ has zero k-th column, so it is not a fundamental matrix. If r ðkÞ ¼ 0 (case lk0 A N 2), then C has zero k-th row. In both cases, C is not invertible. Lemma 3. i) If A1 has no negative integer eigenvalues and CðlÞ is fundamental, then C is invertible. ii) Conversely, if C is invertible, then: – A1 has no negative integer eigenvalues, – CðlÞ is fundamental, – the matrix defined by C ðlÞ :¼ CðlÞC 1 , is the unique fundamental solution C of Proposition 1, – in case lk0 A N then ~ r ðkÞ 0 0, in case lk0 A N 2, then r ðkÞ 0 0. Proof. i) If A1 has no negative integer eigenvalues, then the fundamental matrix C ðlÞ exists form Proposition 1. If CðlÞ is invertible, then C ¼ C ðlÞ 1 CðlÞ is invertible. ii) C invertible implies that ~ r ðkÞ 0 0 and r ðkÞ 0 0, when defined (Remark 5), thus in any row and any column of C there is a cij 0 0 for some i 0 j. Write CðlÞ at lk : ðsingÞ ~1 j j C ~n ¼ ½C ~ C ¼ ½C k ðsingÞ ~ ck1 j j C k ckn þ regðl lk Þ ~ ðsingÞ j 0 j j 0 þ regðl lk ÞgC: ¼ f½0 j j 0 j C k On Stokes Matrices in terms of Connection Coe‰cients 401 The last step is possible because existence of C 1 allows to write regðl lk Þ ¼ regðl lk ÞC 1 C 1 regðl lk ÞC: Thus ~ ðsingÞ j 0 j j 0 þ regðl lk Þ: CC 1 ¼ ½0 j j 0 j C k This is equivalent to (18) and (19), which implies that there exist the unique fundamental matrix C 1 CC 1 . From Proposition 1 we conclude that A1 has no negative integer eigenvalues. Obviously, it follows also that C ¼ C C is invertible. r Proposition 3. C is invertible , A1 has no integer eigenvalues. Proof. The ‘‘)’’ is proved in the previous lemma, point ii). The proof of ‘‘(’’ is the following generalization of that of proposition 2 in [4]. Since A1 has no negative integer eigenvalues, the monodromy group is generated by M1 ; . . . ; Mn . Enumerate the poles in such a way that the ray Lkþ1 is to the left of the ray Lk , which implies that the monodromy at infinity for an anticlockwise loop encircling all the poles is My ¼ Mn . . . M1 . The behaviour of system (5) at y implies that A1 has no integer eigenvalues if and only if My has no eigenvalue ¼ 1. This is equivalent to the fact that C is invertible. Indeed, existence of an eigenvalue equal to 1 means that there exists a non zero row ^ ¼ ½w1 ; . . . ; wn , such that w ^ My ^ . Using the explicit expression of vector w ¼w the Mk in terms of the ak cjk ’s, we compute ^þ ^ Mn . . . M1 ¼ w w n X bj e^j ; j¼1 where the e^j ’s are the basis rows, and ^ CÞn ; bn ¼ an ð w " # n X ^ CÞi þ cji bj ; bi ¼ ai ð w i ¼ 1; 2; . . . ; n 1: j¼iþ1 Since all the ai , for 1 a i a n, are not zero (this is the crucial point), we ^ My ^ if and only if w ^C ¼ 0. conclude that w ¼w r Proposition 4. i) If A1 has no integer eigenvalues, then CðlÞ is a fundamental matrix solution. ii) With the additional assumption that lk0 B Z, Ek ¼ 1; . . . ; n, also the converse holds: if CðlÞ is a fundamental matrix solution, then A1 has no integer eigenvalues. 402 Davide Guzzetti Proof. i) If A1 has no integer eigenvalues, C is invertible (Proposition 3). Therefore, the statement follows from Lemma 3, point ii). ~1 j j C ~n be fundamental. Observe that under the ii) Let C ¼ ½C 0 hypothesis that lk B Z for any k, the columns are singular. Namely: 0 ~k ðlÞ 1 C ~ ðsingÞ ðlÞ ¼ ðGðl 0 þ 1Þ~ C ek þ Oðl lk ÞÞðl lk Þ lk 1 ; k k 1 a k a n: The monodromy of CðlÞ at infinity is My ¼ Mn Mn1 . . . M1 . Suppose that there is an integer eigenvalue of A1 . It follows that there exists a non zero column vector ~ v ¼ ðv1 ; . . . ; vn Þ T (T means transpose) such that My~ v ¼~ v. As in the proof of Proposition 3, making use of the explicit form of the Mk ’s in terms of the cjk ’s, we see that My~ v ¼~ v is equivalent to C~ v ¼~ 0. Take the vector Pn ~ ~ cðlÞ ¼ l¼1 vl C l ðlÞ. At every lk it behaves like ! n n X X ~ ~ ~k þ regðl lk Þ: vl Ck ckl þ regðl lk Þ ¼ ckl vl C cðlÞ ¼ l¼1 But Pn l¼1 ckl vl l¼1 ¼ 0, thus ~ðlÞ ¼ regðl lk Þ; c close to any lk ; k ¼ 1; . . . ; n: ~ðlÞ is a polynomial solution. This contradicts the fact that This implies that c ~ ~ ~k ðlÞ being singular at lk , 1 a k a n. r C1 ðlÞ; . . . ; Cn ðlÞ is a basis, each C Corollary 3. If A1 has no integer eigenvalues, then M1 ; . . . ; Mn of Proposition 2 generate the monodromy group of system (3). Corollary 4. Suppose A1 has some integer eigenvalues and CðlÞ is a fundamental matrix solution (consequently, M1 ; . . . ; Mn generate the monodromy group). In such cases, at least some lk0 is necessarily integer. 5. Dependence of matrix solutions and connection coe‰cients on h Following [4], we call critical values the inadmissible values for h, namely argðlj lk Þ mod 2p: We numerate them as in [4], as follows. In the angular interval ðp=2; 3p=2 there is an even number m ¼ 2m, m integer, of critical values, ordered as 3p p b h0 > h1 > > hm1 > : 2 2 All the possible critical values are then hnþhm :¼ hn 2hp; n ¼ 0; . . . ; m 1; h A Z: On Stokes Matrices in terms of Connection Coe‰cients 403 In each interval ðy 2p; y lie m successive values belonging to the set of critical values fhn j n A Zg. There is an ordering of the poles with respect to an admissible h, given by the dominance relation 0 below: Definition 2 (as in [4]). Let h be admissible. We say that j 0 k, whenever in the plane Ph the cut Lj lies to the right of the cut Lk . Equivalently, choose the determinations hjk :¼ determination of argðlj lk Þ s:t: h 2p < hjk < h; j 0 k; 1 a j; k a n: Then ð21Þ j0k , p þ h < hjk < h: The reason for the nomenclature ‘‘dominance’’ will be explained in section 6.1. Remark. l1 ; . . . ; ln are in lexicographical order with respect to the admissible h when the labelling order j < k coincides with the dominance order j 0 k. The matrices C ðkÞ ðlÞ, CðlÞ and C ðlÞ defined in the plane Ph , with h admissible, and the connection matrix C, depend on h. Therefore we write C ðkÞ ðlÞ ¼ C ðkÞ ðl; hÞ; CðlÞ ¼ Cðl; hÞ; C ðlÞ ¼ C ðl; hÞ; C ¼ CðhÞ: For two values h < h~, we consider the plane with both the cuts of Ph and Ph~. We denote by Ph \ Ph~ the simply connected set of reference points w.r.t. Ph and Ph~ (nomenclature as in [4]), namely the points in the doubly cut plane such that argðl lk Þ B ½h; h~, Ek ¼ 1; . . . ; n. A pole lj is called accessible if it is on the boundary of Ph \ Ph~. See figure 2. The following generalizes propositions 3 of [4] without any assumptions on the values of l10 ; . . . ; ln0 . Proposition 5. Then i) Let lk be accessible w.r.t. some admissible h and h~. ~ ðkÞ ðl; hÞ ¼ C ~ ðkÞ ðl; h~Þ C k k and ~k ðl; hÞ ¼ C ~k ðl; h~Þ; C El A Ph \ Ph~: ii) Let h and h~ lie between two consecutive critical values: namely hnþ1 < h < h~ < hn . Then CðhÞ ¼ Cð~ hÞ: iii) Let again hnþ1 < h < h~ < hn . Then C ðl; hÞ ¼ C ðl; h~Þ; El A Ph \ Ph~; 404 Davide Guzzetti Fig. 2. Picture of Ph \ Ph~. whenever C is uniquely defined (namely, when A1 has no negative integer eigenvalues). Proof. i) is immediate, because lk is accessible and l A Ph \ Ph~. ii) is proved noticing that l1 ; . . . ; ln are all accessible from Ph \ Ph~, therefore i) holds for any k. iii) follows from i) and ii), noticing that C is uniquely defined by ~k ’s and C (Proposition 1). the C r The above implies that the dependence on h is discrete, changing only when a critical value is crossed. Therefore, if hnþ1 < h < hn , we follow [4] and write C n ðlÞ :¼ Cðl; hÞ; C n ðlÞ :¼ C ðl; hÞ; ðnÞ Cn ¼ ðcjk Þ :¼ CðhÞ: Change of C n when a critical value is crossed is given by the following generalization, without assumptions on l10 ; . . . ; ln0 , of proposition 4 of [4] Proposition 6. Suppose that A1 has no negative integer eigenvalues (but no assumptions on l10 ; . . . ; ln0 ), so that the C n ðlÞ’s exist, for any n A Z. Let hnþ1 < h < hn < h~ < hn1 . Then ð22Þ ðlÞ ¼ C n ðlÞWn ; C n1 El A Ph \ Ph~; On Stokes Matrices in terms of Connection Coe‰cients 405 ðnÞ where the invertible matrix Wn ¼ ðWjk Þ is ð23Þ ð24Þ ðnÞ ðnÞ Wjk ¼ ak cjk ; ðnÞ Wjj ¼ 1; for j k such that argðlj lk Þ ¼ hn ; ðnÞ j ¼ 1; . . . ; n; Wjk ¼ 0 where 0 is the dominance relation w.r.t. h. C n ðlÞ ¼ C n1 ðlÞWn1 ; otherwise; In the same way, El A Ph \ Ph~; where Wn1 has zero entries except for ½Wn1 jj ¼ 1; ðn1Þ ½Wn1 jk ¼ bk cjk ; j ¼ 1; . . . ; n; for j k s:t: argðlj lk Þ ¼ hn : Proof. The proof is as for proposition 4 in [4]. Just observe that now we are using monodromy matrices (Proposition 2) and C which are defined in more general terms. The detailed proof is in the arXiv version of this paper [11]. r Recall that in an angular interval ðy 2p; y, there are m ¼ 2m critical values. Let hnþ1 < h < hn and introduce, as in [4], the matrices Cnþ and Cn such that ð25Þ ðlÞ ¼ C n ðlÞCnþ ; C nþm ð26Þ ðlÞ ¼ C nþm ðlÞCn ; C nþm l A Ph \ Php ; l A Php \ Ph2p : Immediately it follows that ð27Þ Cnþ ¼ ðWnþm Wnþ1 Þ 1 ; Cn ¼ Wnþm . . . Wnþmþ1 : Remark 6. Ph \ Php is the half plane to the left hand side of all lines whose positive parts are the cuts of direction h, while Php \ Ph2p is the half plane to the right hand side of all lines whose positive parts are the cuts of direction h 2p. The following is a restatement of remark 3.3 of [4] with no assumptions on A1 : Lemma 4. Let L 0 :¼ diagðl10 ; . . . ; ln0 Þ, lk0 A C, 1 a k a n. Then 0 C nþm ðlÞ ¼ C n ðlÞe 2piL ; for any l in the universal covering of Cnfl1 ; . . . ; ln g. 0 0 Cn ¼ e 2piL Cnþm e 2piL ; Moreover 0 0 ðnamely: CðhÞ ¼ e 2piL Cðh 2pÞe 2piL Þ: 406 Davide Guzzetti Proof. We write lh for l A Ph . Then lh lk ¼ ðlh2p lk Þe 2pi . It fol0 lows from the definition of C that Cðl; h 2pÞ ¼ Cðl; hÞe 2piL , whatever the values l10 ; . . . ; ln0 are. From the above and the connection relations we find two ~k ðl; hÞ: expressions for C ~k ðl; hÞ ¼ C ~ ðsingÞ ðl; hÞcjk ðhÞ þ regðl lj Þ; C j ð28Þ and ð29Þ ~k ðl; h 2pÞ ~k ðl; hÞ ¼ e 2pilk0 C C 0 ðsingÞ ~ ¼ e 2pilk C j ðl; h 2pÞcjk ðh 2pÞ þ regðl lj Þ: ~ ðsingÞ ðl; h 2pÞ ¼ C ~j ðl; h 2pÞ ¼ When lj0 B Z we substitute in the above C j 2pilj ~ 2pilj ~ ðsingÞ e ðl; hÞ. Otherwise, we observe that Cj ðl; hÞ 1 e Cj ~ ðsingÞ ðl; h 2pÞ C j " ~j ðl; h 2pÞ lnðlh2p lj Þ þ ¼C ¼e 2pilj0 0 ð jÞ PNj ðlh2p Þ ðl lj Þ Nj þ1 ( ~j ðl; hÞ ½lnðlh lj Þ 2pi þ C ðsingÞ ~ ¼ e 2pilj C j # þ regðlh2p lj Þ ð jÞ PNj ðlh Þ ) ðl lj Þ Nj þ1 þ regðlh lj Þ ðl; hÞ þ regðl lj Þ: Therefore, confronting (28) and (29), we obtain ~ ðsingÞ ðl; hÞcjk ðhÞ þ regðl lj Þ ¼ e 2piðlj0 lk0 Þ C ~ ðsingÞ ðl; hÞcjk ðh 2pÞ þ regðl lj Þ: C j j ~ ðsingÞ ðl; hÞ is singular at lj , the statement of the lemma for Cn Finally, since C j follows. r We generalize proposition 5 of [4], with no assumptions on diagðA1 Þ ¼ ðl10 ; . . . ; lk0 Þ. ðnÞ Proposition 7. Let hnþ1 < h < hn , and let cjk tion (25) and (26). The connection matrices Cnþ , 8 0 ðnÞ ðnÞ > < bk cjk ¼ e 2pilk ak cjk ; þ ½Cn jk ¼ 1; ð30Þ > : 0; ¼ cjk ðhÞ. Cn are for j 0 k; for j ¼ k; for j k; Consider the rela- On Stokes Matrices in terms of Connection Coe‰cients 8 > < 0; ½Cn jk ¼ 1; > : e 2pilj0 b c ðnÞ ¼ e 2piðlk0 lj0 Þ a c ðnÞ ; k jk k jk ð31Þ 407 for j 0 k; for j ¼ k; for j k; 0 0 where ak ¼ e 2pilk 1 if lk0 B Z, ak ¼ 2pi if lk0 A Z, bk ¼ e 2pilk ak . Proof. One proceeds as in the proof of proposition 5 of [4], with the more general monodromy matrices of Proposition 2 and Remark 4. The detailed proof is in the arXiv version of this paper [11]. r The matrices Cnþ and Cn can be defined by formulae (30) and (31), independently of the fact that A1 has no negative integer eigenvalues, namely independently of (25) and (26). The following corollary is a direct computation. Corollary 5. (31). Then Let the matrices Cnþ and Cn be defined by formulae (30) and 0 ( TrðMj Mk Þ ¼ TrðMk Þ ¼ n 1 þ e 2pilk ; 0 0 0 2pilj0 2pilk0 2pilk0 n 2 þ e 2pilj þ e 2pilk e 2pilj ½Cnþ jk ½Cn kj ; n2þe þe e ½Cn jk ½Cnþ kj ; if j 0 k; if j k: If moreover A1 has no integer eigenvalues, then TrðMk Þ ¼ TrðMk Þ; 6. TrðMj Mk Þ ¼ TrðMj Mk Þ: Fundamental solutions of (1) as laplace integrals 6.1. Fundamental solutions of (1) and stokes matrices The basic definitions of Stokes rays and matrices are well known [22], [2], so we just recall them Definition 3. Stokes rays are the oriented rays from 0 to y contained g defined by the condition in the universal covering of Cnf0g (denoted Cnf0g Cnf0g) <ðzðlj lk ÞÞ ¼ 0, =ðzðlj lk ÞÞ < 0 for j 0 k. Let h A R be admissible. We choose the Stokes rays 3p hjk rjk :¼ z A C j z ¼ r exp i ;r > 0 ; j 0 k; 1 a j; k a n; 2 where hjk ¼ determination of argðlj lk Þ s:t: hjk A ðh 2p; h: 408 Davide Guzzetti If follows that <ðzðlj lk ÞÞ < 0 for z in the half plane to the right of rjk . According to the definition, all the Stokes rays are characterized by arg z ¼ 3p=2 hjk mod 2p. For any ð j; kÞ such that hjk < h < hjk þ p <ðzðlj lk ÞÞ < 0 if arg z ¼ 3p h mod 2p: 2 This means that when an admissible h for system (3)–(5) is fixed, then ð32Þ <ðzðlj lk ÞÞ < 0 for arg z ¼ 3p h mod 2p , j 0 k; 2 where 0 is the partial ordering previously defined, which therefore coincides with the dominance relation in the sense of the theory of ODE with singularity of the second kind. All the Stokes rays can be represented as arg z ¼ tn , with tn :¼ 3p=2 hn , n A Z. It is easily seen that all Stokes rays are generated by 0 a t0 < t1 < < tm1 < p; tnþhm ¼ tn þ ph; h A Z: g Introduce the following notations for sectors of Cnf0g: Cnf0g g j a < argðzÞ < bg; Sða; bÞ :¼ fz A Cnf0g Sn :¼ Sðtn p; tnþ1 Þ; a < b A R; n A Z: For any n A Z, equation (1) has a fundamental matrix solution ð33Þ 0 Yn ðzÞ ¼ Y^n ðzÞe A0 zþL ln z ; z A Sn ; where L 0 ¼ diagðA1 Þ, and Y^n ðzÞ is an invertible matrix, analytic in a neighbourhood of y, with asymptotic expansion ð34Þ F1 F2 Y^n ðzÞ @ I þ þ 2 þ ¼ I þ z z y X Fk k¼1 zk ; for z ! y in Sn : The sector Sn is the maximal sector where the asymptotic behavior holds, and Yn ðzÞ is unique, namely it is uniquely determined by its asymptotic behavior. The n n matrices Fk are determined as rational functions of A0 and A1 , by formal substitution into (1) (see [22], [2]). Definition 4 (Stokes Matrices). Given two fundamental matrices Yn and Y as above, whose maximal sectors Sn and Sn 0 intersect in such a way that no Stokes rays are contained in Sn \ Sn 0 , then the connection matrix S such that Yn 0 ðzÞ ¼ Yn ðzÞS, z A Sn \ Sn 0 , is called a Stokes matrix. n0 On Stokes Matrices in terms of Connection Coe‰cients 409 It is easy to see that n 0 ¼ n þ m, therefore Stokes matrices are the matrices Sn , n A Z, such that Ynþm ðzÞ ¼ Yn ðzÞSn ; z A Sn \ Snþm ¼ Sðtn ; tnþ1 Þ: Since <ðzðlj lk ÞÞ < 0 for j 0 k when z A Sðtn ; tnþ1 Þ, where the dominance relation is referred to any h A ðhnþ1 ; hn Þ, then from the asymptotic behaviours of Ynþm ðzÞ and Yn ðzÞ, it follows that djk @ e ðlj lk Þz ðSn Þjk and thus ðSn Þjj ¼ 1; Definition 5 (Stokes Factors). matrices Vn such that Yn1 ðzÞ ¼ Yn ðzÞVn ; ðSn Þjk ¼ 0 for j k: The Stokes factors are the connection z A Sn1 \ Sn ¼ Sðtn p; tn Þ: Asymptotic behaviours and dominance relations in Sðtn p; tn Þ yield: ðVn Þjj ¼ 1, and ðVn Þjk ¼ 0 for all j 0 k, except possibly for ð j; kÞ s.t. argðlj lk Þ ¼ 3p=2 tn and j k with respect to h A ðhnþ1 ; hn Þ. From the definitions above, it follows that ð35Þ Sn ¼ ðVnþm . . . Vnþ1 Þ 1 : The monodromy of Yn ðzÞ is completely described by Sn , Snþm and L 0 , because the following holds ð36Þ 0 Yn ðze 2pi Þ ¼ Yn ðzÞe 2piL ðSn Snþm Þ 1 ; z A Sn : Definition 6. Sn and Snþm are a complete set of Stokes multipliers, because any other Stokes matrix can be expressed in terms of entries of Sn , Snþm and L 0 0 0 (see [2], [3]), by iterations of Snþ2m ¼ e 2piL Sn e 2piL . 6.2. Solutions of (1) as laplace integrals We consider a path gk ðhÞ which comes from infinity along the left side of the cut Lk of direction h, encircles lk with a small loop excluding all the other poles, and goes back to infinity along the right side of Lk (where Lk is oriented from lk to y). See figure 3. 1) Case of lk0 B Z We define ð ð ~k ðl; hÞdl 1 1 ~ ðkÞ ðl; hÞdl: ~k ðz; hÞ :¼ 1 Y ð37Þ e zl C e zl C k 2pi gk ðhÞ 2pi gk ðhÞ ~k ðl; hÞ, the exponential ensures that the Since l ¼ y is a regular singularity of C integral converges in the sector ð38Þ g j <ðze ih Þ < 0g ) SðhÞ :¼ fz A Cnf0g p 3p h < arg z < h: 2 2 410 Davide Guzzetti Fig. 3. The path gk ðhÞ. The asymptotic behaviour of (37) can be computed by expanding the integrand ~k ðlÞ in series at lk and then formally exchanging integration and series (see C [7]). Namely, for any N > 0 integer, " # ð X ðkÞ 0 1 l 0 zl ~ ~ Yk ðz; hÞ ¼ bl ðl lk Þ ðl lk Þ lk 1 dl ¼ ðÞ: e Gðlk þ 1Þ~ ek þ 2pi gk ðhÞ lb1 Ð P PN P a zl Now write dl l¼1 þ lb1 ¼ l>N . The known formula gk ðhÞ ðl lk Þ e ¼ z a1 e lk z =GðaÞ yields ! ðkÞ N ~ X 0 bl 1 ð39Þ þ RðzÞ e lk z z lk ; ðÞ ¼ ~ ek þ 0 l Gðlk þ 1 lÞ z l¼1 P where RðzÞ is the integral of l>N . It is standard computation to show that RðzÞ ¼ Oðz N Þ. Thus, formula (39) allows us to write the asymptotic expansion ~k ðz; hÞe lk z z lk0 @~ ek þ Y y X l¼1 ðkÞ ~ bl 1 ; 0 Gðlk þ 1 lÞ z l z ! y; z A SðhÞ: Lemma 5. Assume lk0 B Z. Let h A ðhnþ1 ; hn Þ, and tn :¼ 3p=2 hn . Then, ~ Yk ðz; hÞ defined by (37) is the k-th column of the unique fundamental solution of (1) identified by the asymptotic behavior (33), (34) in the sector Sn ¼ Sðtn p; tnþ1 Þ: Proof. If hnþ1 < h < h~ < hn , then Yk ðz; hÞ ¼ Yk ðz; h~Þ. This defines the analytic continuation of (37) to [ Sðtn p; tnþ1 Þ ¼ SðhÞ; hnþ1 <h<hn On Stokes Matrices in terms of Connection Coe‰cients 411 with the required asymptotic behaviour. It remains to prove that Yk ðz; hÞ is a vector solution of (1). This follows from integration by parts, as shown in the ~k ðlÞj ¼ 0. Introduction, since gk ðhÞ is such that e lz ðl A0 ÞC r gk If we write coe‰cients in (34) as ðkÞ f1 j j ~ fnðkÞ ; Fk ¼ ½ ~ then, for hnþ1 < h < hn , solution (9) reads X ~k ðl; hÞ 1 C ~ ðkÞ ðlÞ ¼ C k 0 ðkÞ Gðlk0 þ 1 lÞ ~ fl ðl lk Þ llk 1 ; ðkÞ ~ f0 ¼ ~ ek : lb0 2) Case of lk0 ¼ 1 ~k ðz; hÞ :¼ Y ð40Þ We define ð ~k ðl; hÞdl ¼ e zl C ð ~k ðl; hÞdl; e zl C Lk Lk along the cut Lk from lk to infinity. This is convergent in SðhÞ as before. Its ~k in the convergent asymptotic behaviour is obtained as before by expanding C series (10), and then exchanging integration and series: ð X ðkÞ ð zl ~ Yk ðz; hÞ @~ bl ek e dl ðl lk Þ l e zl dl Lk Lk lb1 " # y X e lk z lþ1 ~ðkÞ 1 ~ ¼ ð1Þ l!bl ; ek þ zl z l¼1 where we have used the fact that ð Lk ðl lk Þ l e lz dl ¼ e lk z z lþ1 ð0 x l e x dx ¼ þye if e lk z l!ð1Þ l ; z lþ1 p 3p <f< : 2 2 The same proof of Lemma 5 yields the following Lemma 6. Assume lk0 ¼ 1. Let h A ðhnþ1 ; hn Þ, and tn :¼ 3p=2 hn . Then, Yk ðz; hÞ defined by (40) is the k-th column of the unique fundamental solution of (1) identified by the asymptotic behavior (33), (34) in the sector Sn ¼ Sðtn p; tnþ1 Þ: By virtue of the lemma, we rewrite (10), for hnþ1 < h < hn , as follows ~k ðl; hÞ ¼ ~ C ek þ X ð1Þ lþ1 lb1 l! ðkÞ ~ fl ðl lk Þ l : 412 Davide Guzzetti In case lk0 ¼ 1, the solution (40) has also the represen- Lemma 7. tation ~k ðz; hÞ ¼ Y ð41Þ ð ~k ðl; hÞdl 1 1 e C 2pi Lk zl ð gk ðhÞ ~ ðkÞ ðlÞe zl dl; C k where gk ðhÞ is the same as (37). ðkÞ Ð gk ~ Proof. Recall that C k regðl lk Þe zl dl ¼ 0, we have ð gk ðhÞ ~k ðlÞ lnðl lk Þ þ regðl lk Þ. ¼C ð ~ ðkÞ ðlÞe zl dl ¼ C k Since ~k ðlÞ lnðl lk Þe zl dl: C gk ðhÞ Indicate with LkL and LkR the left and right sides of Lk (oriented from lk to y), and with ðl lk ÞR=L the branch of ðl lk Þ to the right/left of Lk . Then ð ð ð ð ð ¼ þ ¼ ¼ ðÞ: gk ðhÞ LkL LkR LkR LkL Moreover ðl lk ÞL ¼ e 2pi ðl lk ÞR , where argððl lk ÞR Þ ¼ h. Therefore ðÞ ¼ ð LkR ~k ðlÞ lnðl lk Þ e zl dl C R ( þ 2pi ð ~k ðlÞe zl dl C ð LkR Lk ð ) ~k ðlÞ lnðl lk Þ e zl dl C R ~k ðlÞe zl dl: C 1 2pi r Lk 3) Case of lk0 A N ~k ðz; hÞ :¼ 1 Y 2pi ð42Þ ~ ðkÞ ðlÞ is (12). where C k 1 2pi Define the convergent in SðhÞ integral ð gk ðhÞ ð ðkÞ gk ðhÞ ~ ðl; hÞdl; e zl C k For z ! y, the logarithmic part yields ~k ðlÞ lnðl lk Þdl ¼ e zl C ð ~k ðlÞdl @ e zl C Lk y X ðkÞ 1 ð1Þ lþ1 l!dl e zlk : lþ1 z l¼0 On the other hand, by Cauchy theorem, the terms with poles yield On Stokes Matrices in terms of Connection Coe‰cients 1 2pi ð e zl gk ðhÞ 413 P ðkÞ ðlÞ ðl lk Þ Nk þ1 1 d Nk ðP ðkÞ ðlÞe zl Þjl¼lk Nk ! dl Nk " # ðkÞ Nk ~ðkÞ ~ X bNk q q bNk Nk lk z lk z z ¼ ~ ¼e ek þ þ N z e ; q! z k q¼0 ¼ ðkÞ ~ b0 ¼ Nk !~ ek : ~k ðz; hÞ has the correct asymptotics. We conclude that Y Lemma 5 yields the following The same proof of Lemma 8. Assume lk0 A N. Let h A ðhnþ1 ; hn Þ, and tn :¼ 3p=2 hn . Then, ~ Yk ðz; hÞ defined by (42) is the k-th column of the unique fundamental solution of (1) identified by the asymptotic behavior (33), (34) in the sector Sn ¼ Sðtn p; tnþ1 Þ: Accordingly, we rewrite for hnþ1 < h < hn : " # ðkÞ Nk y ð1Þ lþ1~ X X fNk þlþ1 l ðkÞ ðkÞ lN 1 k ~ ðl; hÞ ¼ C u lnðuÞ þ regðuÞ; ðNk lÞ! ~ fl u þ k l! l¼0 l¼0 where u :¼ l lk . 4) Case of lk0 ¼ Nk A N 2 We define ð ~k ðlÞdl: ~k ðz; hÞ :¼ e zl C Y ð43Þ Lk The asymptotic behaviour of the above is readily computed: ð ð X ðkÞ zl ~ ~ bl ðl lk Þ lNk 1 dl e Ck ðlÞdl ¼ e zl Lk Lk lb0 @ e lk z ð1Þ Nk ðkÞ X ð1Þ l ðl Nk 1Þ!~ b l z lNk lb0 " ¼ ~ ek þ ð1Þ Nk # ðkÞ 1 ~ ð1Þ ðl Nk 1Þ!bl z Nk e l k z : l z lb1 X l ðkÞ Here we have used the normalization ~ b0 ¼ ð1Þ Nk~ ek =ðNk 1Þ!. We con~ clude that Yk ðz; hÞ has the correct asymptotics. The same proof of Lemma 5 yields the following Lemma 9. Assume lk0 ¼ Nk A N 2. Let h A ðhnþ1 ; hn Þ, and tn :¼ ~k ðz; hÞ defined by (43) is the k-th column of the unique 3p=2 hn . Then, Y 414 Davide Guzzetti fundamental solution of (1) identified by the asymptotic behaviour (33), (34) in the sector Sn ¼ Sðtn p; tnþ1 Þ: Accordingly: ~k ðlÞ ¼ C X ð1Þ lNk ~ðkÞ fl ðl lk Þ lNk 1 : ðl N 1Þ! k lb0 Also in this case we have ð ð 1 ~k ðlÞ lnðl lk Þdl ¼ ~k ðlÞdl: e zl C e zl C 2pi gk ðhÞ Lk ~k ðlÞ lnðl lk Þ þ regðl lk Þ exists, we Therefore, when the singular solution C have ð ~k ðlÞ lnðl lk Þ þ regðl lk ÞÞdl: ~k ðz; hÞ ¼ Y e zl ðC gk ðhÞ Proposition 8. The following are the fundamental matrix solutions of (1) uniquely identified by the asymptotic behaviour (33), (34) in Sn , n A Z: ~1 ðz; hÞ j j Y ~n ðz; hÞ; Yn ðzÞ ¼ ½Y ð44Þ ~k ðz; hÞ ¼ 1 Y 2pi ð ðsingÞ gk ðhÞ ~ e zl C k ðsingÞ ~ where C is defined in (16). k replaced by (43). ðl; hÞdl; 1 a k a n; hn < h < hnþ1 ; ~ ðsingÞ ¼ 0, for l 0 A N 2, then (44) is In case C k k Proof. The above is a consequence of the preceding discussion. Linear independence of the columns of Yn ðzÞ follows from the independence of the first term of the asymptotic behaviour of each column. Uniqueness follows from the maximality of the sector. r ~ ðsingÞ in the Lemma 10. If A1 has no negative integer eigenvalues, then, C k ~ . Namely: integral (44) can be replaced by C k ð ~ ðl; hÞdl; ~k ðz; hÞ ¼ 1 Y e zl C 1 a k a n; hn < h < hnþ1 : k 2pi gk ðhÞ Proof. Recall that if A1 has no negative integer eigenvalues, then ~ ðsingÞ 0 0 also for l 0 A N 2. Since C ~ ðsingÞ C ~ ¼ regðl lk Þ, we have C k k k k ð ~ ðsingÞ ðlÞ C ~ ðlÞÞe zl dl ¼ 0: ðC r k k gk ðhÞ On Stokes Matrices in terms of Connection Coe‰cients 7. 415 Stokes factors and matrices in terms of C —main theorem (Th. 1) In this section we state the main result of the paper, which is Theorem 1 and Corollary 6. Consider as in [4] a new path of integration gðhÞ homotopic to the product gkn ðhÞ . . . gk1 ðhÞ, k1 0 k2 0 0 kn , namely a path coming from y in direction h to the left of all the poles l1 ; . . . ; ln , encircling all the poles, and going back to y in direction h to the right of all the poles. The following Proposition is the generalization of Theorem 2 0 of [4] when no assumptions are made on l10 ; . . . ; ln0 . Proposition 9. If A1 has no negative integer eigenvalues (but no assumptions on l10 ; . . . ; ln0 ) the fundamental matrix of Proposition 8 is ð 1 Yn ðzÞ ¼ e zl C ðl; hÞdl; hnþ1 < h < hn ; 2pi gðhÞ and Yn1 ðzÞ ¼ Yn ðzÞWn ; z A Sn1 \ Sn ¼ Sðtn p; tn Þ; where the Wn ’s are given in Proposition 6. Vn ¼ Wn ; The Stokes factors and matrices are Sn ¼ Cnþ ; 1 Snþm ¼ Cn : Proof. The proof is technically as in Theorem 2 0 of [4], so for brevity we do not repeat it. The crucial point is that now the matrices C, C , Wn , and CG n have been defined—as in the construction of previous sections—for any l10 ; . . . ; ln0 . So the proof holds for any l10 ; . . . ; ln0 . r With much more technical e¤ort—which requires a non trivial generalization of the technique of [4]—we are now going to prove that the statement of the above Proposition holds without any assumptions on A1 , namely: Theorem 1. Let A1 be any n n matrix. The (complete set of ) Stokes multipliers and matrices of system (1) are given in terms of the connection ðnÞ coe‰cients cjk of system (3) according to the formulae Vn ¼ Wn ; Sn ¼ Cnþ ; 1 Snþm ¼ Cn ; En A Z; where Wn is defined by formulae (23), (24), and Cnþ and Cn are defined by formulae (30) and (31). Remark. Here formulae (23), (24), (30) and (31) are taken as the definitions of Wn , Cnþ and Cn , independently of the existence of C ðlÞ. Corollary 6. Let A1 be any n n matrix. The following equalities hold for the monodromy matrices of CðlÞ of system (3)–(5), defined in (17): 416 Davide Guzzetti Fig. 4. The paths gðhÞ and gð~ hÞ. 0 ( TrðMj Mk Þ ¼ TrðMk Þ ¼ n 1 þ e 2pilk ; 0 0 0 2pilj0 2pilk0 2pilk0 1 kj n 2 þ e 2pilj þ e 2pilk e 2pilj ½Sn jk ½Snþm n2þe þe e 1 ½Snþm jk ½Sn kj The corollary above is a restatement of Corollary 5. in a few steps. 8. if j 0 k; if j k: We prove Theorem 1 Proof of Theorem 1 We define g Y ðzÞ :¼ z g Y ðzÞ; which yields a gauge transformation of the linear systems (1): ð45Þ d A1 g ðg Y Þ ¼ A0 þ gY : dz z The fundamental solutions g Yn ðzÞ ¼ z g Yn ðzÞ, have the same Stokes multiplier and Stokes matrices than Yn ðzÞ, and their columns are obtained as Laplace transforms of solutions of ð46Þ ðA0 lÞ d ðg CÞ ¼ ðA1 g þ I Þg C: dl On Stokes Matrices in terms of Connection Coe‰cients 417 If A1 has diagonal entries l10 ; . . . ; ln0 , some of which may be integers, then we can always find a su‰ciently small g0 > 0 such that, for any 0 < g < g0 , A1 g has diagonal entries l10 g; . . . ; ln0 g which are not integers, and moreover has no integer eigenvalues, so that C exists. In the following, we assume that g has this property. ðnÞ For system (3), the matrix Cn ¼ ðcjk Þ is defined by (7). The matrices Cnþ and Cn can always be defined by the formulae of Proposition 7, independently of the existence of C and formulae (25) and (26). On the other hand, for system (46) the matrices Cnþ and Cn (which depend on g, so we write Cnþ ½g and Cn ½g), are well defined by formulae (25) and (26). According to Proposition 7, their entries are again given in terms of g-dependent connection coe‰cients ðnÞ ðnÞ cjk ¼ cjk ½g’s. The latter are defined by the first equality of (7) applied to the ~k , namely: solutions g C ð47Þ ~ g Ck ðlÞ ðnÞ ~j ðlÞc ½g þ regðl lj Þ: ¼ gC jk The following Proposition is the key step to prove Theorem 1 Proposition 10. Let g0 > 0 be small enough such that the diagonal part of A1 gI has no integer entries and A1 has no integer eigenvalues for any ðnÞ 0 < g < g0 . Let hnþ1 < h < hn be fixed. Let cjk be the corresponding connecðnÞ tion coe‰cients of system (3), defined by (7), and cjk ½g be the connection coe‰cients of (46), defined by (47). Finally, let ( 0 0 e 2pilk 1; lk0 B Z; ak ¼ ak ½g ¼ e 2piðlk gÞ 1: 0 2pi; lk A Z; Then, the following equalities hold ðnÞ ðnÞ ak cjk ¼ e 2pig ak ½gcjk ½g; ðnÞ ðnÞ ak cjk ¼ ak ½gcjk ½g; if k j; if k 0 j; where the partial ordering 0 refers to h. Corollary 7. Let g be as in Proposition 10. Let Cnþ ½g and Cn ½g be the connection matrices defined in (25) and (26) for system (46). Let Cnþ and Cn be ðnÞ the matrices for system (3) defined by (30) and (31), where the cjk are defined by (7). Then Cnþ ¼ Cnþ ½g; Cn ¼ Cn ½g; En A Z: Also, let Wn be defined by (23) and (24) for system (3), and Wn ½g be the matrix defined by (22) for system (46). Then Wn ¼ Wn ½g; En A Z: 418 Davide Guzzetti Proof of Corollary 7. It is enough to compare the formulae of Proposition 10 with those of Propositions 7 and 6. r Before proving Proposition 10, we give the proof of Theorem 1. Proof of Theorem 1. The Sn ’s are unchanged by the gauge g Y ðzÞ ¼ z Y ðzÞ. Moreover, Proposition 9 applies to the system (46), therefore g Sn ¼ Cnþ ½g; 1 Snþm ¼ Cn ½g; Vn ¼ Wn ½g: Thus, Corollary 7 implies Theorem 1. 8.1. r Proof of Proposition 10, by steps Proposition 10 is the analogous of Lemma 2 0 in [4], but it requires much more technical e¤orts. The proof is based on the properties of higher order primitives of vector solutions of (3)–(5). Since l10 ; . . . ; ln0 are any complex numbers, including integers, the description and the proof of these properties require a highly non trivial extension of the classical techniques used in [4]. We proceed in a few steps, namely Proposition 11, Lemmas 11, 12, and Proposition 12. First, we introduce higher order primitives of vector solutions of system (5). For lk0 A CnN, we recall that there are solutions ~k ðlÞ ¼ C y X ðkÞ 0 Gðlk0 þ 1 lÞ ~ fl ðl lk Þ llk 1 ; lk0 B Z; l¼0 ~k ðlÞ ¼ C y X l¼0 ð1Þ lNk ~ðkÞ f ðl lk Þ lNk 1 ; ðl Nk 1Þ! l lk0 ¼ Nk A Z : ~k Þ½q ðlÞ, q A N, given ~k . This is the function ðC We define the q primitive of C by analytic continuation of the series obtained by q-fold term wise integration of ~k ðlÞ. Namely: the corresponding term in C ð48Þ ~k Þ½q ðlÞ :¼ ð1Þ q ðC y X 0 ðkÞ Gðlk0 þ 1 lÞ ~ flq ðl lk Þ llk 1 ; lk0 B Z; l¼q ~k Þ½q ðlÞ :¼ ð1Þ q ð49Þ ðC y X ð1Þ lNk ~ðkÞ flq ðl lk Þ lNk 1 ; ðl N 1Þ! k l¼q lk0 ¼ Nk a 1: ~k has They above converge in a neighbourhood of lk , contained in Ph , where C convergent series. Indeed, if l0 0 lk is in the neighbourhood, then ðl ð s1 ð sq1 ~k Þ½q ðlÞ Qq1 ðl l0 Þ; ~k ðsq Þ ¼ ðC ð50Þ ds1 ds2 . . . dsq C l0 l0 l0 On Stokes Matrices in terms of Connection Coe‰cients 419 where ~k Þ½q ðl0 Þ þ ðC ~k Þ½1q ðl0 Þðl l0 Þ Qq1 ðl l0 Þ ¼ ðC þ ~k Þ½2q ðl0 Þ ~k Þ½1 ðl0 Þ ðC ðC ðl l0 Þ 2 þ þ ðl l0 Þ q1 ; 2! ðq 1Þ! is a polynomial in ðl l0 Þ of degree q 1. The path of integration is any in ~k to converge. Once Ph , such that jl l0 j is small enough for the series of C ½q ~k Þ ðlÞ is defined by the convergent series, then it is analytically continued ðC to Ph . For lk0 ¼ Nk A N integer, we recall that we have the solution ðkÞ ~ ðkÞ ðlÞ ¼ C ~k ðlÞ lnðl lk Þ þ C k ðkÞ PNk ðlÞ ¼ Nk X PNk ðlÞ ðl lk Þ Nk þ1 þ regðl lk Þ; ðkÞ ðNk lÞ! ~ fl ðl lk Þ l ; l¼0 ~k ðlÞ ¼ C y X l¼0 ð1Þ lþ1 ~ðkÞ fNk þlþ1 ðl lk Þ l : l! The series is convergent in a neighbourhood of lk contained in Ph . Let l0 belong to the neighbourhood. Let q b 0 integer, and compute q times the ~ ðkÞ ðlÞ. Due to convergence of the series, we can take integration integral of C k term by term. We obtain: i) For q a Nk : ðl ð s1 ð sq1 ~ ðkÞ ðsq Þ ds1 ds2 . . . dsq C k l0 l0 ¼ l0 y X ð1Þ lþ1q l! l¼0 ðkÞ ðkÞ ~ fNk þ1þlq ðl lk Þ l lnðl lk Þ þ ðkÞ PNk q ðlÞ ¼ ð1Þ q PNk q ðlÞ ðl lk Þ Nk þ1q þ regðl lk Þ; N k q X ðkÞ ðNk l qÞ! ~ fl ðl lk Þ l : l¼0 ii) ðl l0 For q ¼ Nk þ 1: ð s1 ð sN k ^ k ðlÞ lnðl lk Þ þ regðl lk Þ; ~ ðkÞ ðsN þ1 Þ ¼ C ds1 ds2 . . . dsNk þ1 C k k l0 l0 where we defined ð51Þ ^ k ðlÞ :¼ C y X ð1Þ lþNk l¼0 l! ðkÞ ~ fl ðl lk Þ l : 420 Davide Guzzetti ^ k ðlÞ is defined by the series, that converges in the neighbourhood The function C ~ ðkÞ converges. Then it is analytically of lk in Ph , where the series of C k ðkÞ continued in Ph . Note that if all rj ¼ 0, Ej, namely when there is no P Nk ðkÞ ^ k ðlÞ is truncated to ~ , then the sum in C logarithmic term in C l¼0 , giving k a polynomial of degree Nk . iii) For q ¼ Nk þ 1 þ q~, with q~ b 0: ðl ð s1 ð sq1 ^ k Þ½~q ðlÞ lnðl lk Þ þ regðl lk Þ; ~ ðkÞ ðsq Þ ¼ ðC ds1 ds2 . . . dsq C k l0 l0 l0 ^ k Þ½~q ðlÞ ¼ ð1Þ q~ ðC y X ð1Þ lþNk ðkÞ ~ fl~q ðl lk Þ l : l! l¼~ q ½q ^ k Þ ðlÞ is the q primitive of C ^ k ðlÞ, and the same computation The function ðC of (50) yields ðl ð s1 ð sq1 ^ k Þ½q ðlÞ Qq1 ðl l0 Þ; ^ k ðsq Þ ¼ ðC ð52Þ ds1 ds2 . . . dsq C l0 l0 l0 ~k ! ^ k. 7 C where Qq1 is as in (50) with substitution C Remark 7. Let ^ k Þ½r ðlÞ :¼ ðC y X dr ^ ð1Þ lþNk r ~ðkÞ ð C ðlÞÞ ¼ flþr ðl lk Þ l ; dl r l! l¼0 ^ k Þ½Nk þ1 ðlÞ ¼ C ~k ðlÞ. In particular, ðC 0 a r a Nk þ 1: Then r d ^ k ðlÞ lnðl lk Þ þ regðl lk ÞÞ ðC dl r ðkÞ ^ k Þ½r ðlÞ lnðl lk Þ þ ¼ ðC ðkÞ with Pr1 ðlÞ ¼ ð1Þ Nk þ1r P r1 l¼0 ðr Pr1 ðlÞ þ regðl lk Þ; ðl lk Þ r ðkÞ 1 lÞ! ~ fl ðl lk Þ l . We summarize (50) and (52) and the computations involving logarithmic solutions in the following Proposition 11. Let l0 0 lj for any j ¼ 1; 2; . . . ; n. For a given k A f1; . . . ; ng define ( ~k ðlÞ; if l 0 A CnN; C k ð53Þ fk ðlÞ :¼ ^ C k ðlÞ; if lk0 A N; ð s1 ðl ð sq1 ½q Fk ðl; l0 Þ :¼ ds1 ds2 . . . dsq fk ðsq Þ; l0 l0 l0 On Stokes Matrices in terms of Connection Coe‰cients ~k are defined in (15), and C ^ is defined in (51). where C ½q 421 Then ðkÞ Fk ðl; l0 Þ ¼ ðfk Þ½q ðlÞ Qq1 ðl l0 Þ; ð54Þ ðkÞ where Qq1 is a polynomial of degree q 1 in ðl l0 Þ. definition that ðl ½q ½q1 ð55Þ ds Fk ðs; l0 Þ ¼ Fk ðl; l0 Þ: It follows from the l0 ~ ðsingÞ of system (3): For lk0 A Z, consider the singular solutions C k lk0 A Z ; ~k ðlÞ lnðl lk Þ þ regðl lk Þ; ek C ~k ðlÞ lnðl lk Þ þ C P ðkÞ ðlÞ ðl lk Þ Nk þ1 lk0 ¼ Nk A N: þ regðl lk Þ; ~ ðsingÞ 1 0, otherwise ek ¼ 1. In the above, ek ¼ 0 if C k ð s1 ðl ð56Þ ds1 l0 ð sq1 ds2 . . . l0 Then, for lk0 A Z : ~k ðsq Þ lnðsq lk Þ þ regðsq lk ÞÞ dsq ðek C l0 ~k Þ½q ðlÞ lnðl lk Þ þ regðl lk Þ; ¼ ek ð C q b 0; and for lk0 ¼ Nk A N: ð57Þ ðl ð sq1 ð s1 ds1 ds2 . . . l0 l0 ¼ ! ðkÞ ~k ðsq Þ lnðsq lk Þ þ dsq C l0 8 > > > ^ k Þ½Nk þ1q ðlÞ lnðl lk Þ þ > < ðC PNk ðsq Þ þ regðsq lk Þ ðsq lk Þ Nk þ1 ðkÞ PNk q ðlÞ ðl lk Þ Nk þ1q > þ regðl lk Þ; > > > : ^ ½qþNk þ1 ðC k Þ ðlÞ lnðl lk Þ þ regðl lk Þ; 0 a q a Nk ; q b Nk þ 1: The above expressions hold by analytic continuation for l A Ph . ðnÞ Corollary 8. Let lk0 ¼ Nk A N. Let cjk denote cjk ðhÞ ¼ cjk . The vector ^ k ðlÞ in (51), l A Ph , has the following behaviours at lj 0 lk . function C For lj0 B Z: ^ k ðlÞ ¼ C ~ ½Nk 1 ðlÞcjk þ regðl lj Þ: C j ð58Þ For lj0 ¼ Nj A N: 422 ð59Þ Davide Guzzetti ^ k ðlÞ ¼ C 8 ! > PNj Nk 1 ðlÞ > ½N N j k > ^ > cjk ðlÞ logðl lj Þ þ C j > < ðl lj Þ Nj Nk > þ regðl lj Þ; > > > > :C ^ ½Nk þNj ðlÞ logðl lj Þcjk þ regðl lj Þ; j Nj b Nk þ 1; Nk b Nj : For lj0 ¼ Nj A Z : ~ ½Nk 1 lnðl lj Þcjk þ regðl lj Þ: ^ k ðlÞ ¼ C C j ð60Þ Note that (60) always makes sense, because cjk ¼ 0 for any k ¼ 1; . . . ; n ~ ðsingÞ 1 0. when C j Proof of Corollary 8. observe that ð61Þ ðl l0 ð xN dxNk þ1 k l0 We use the formulae of Proposition 11. We ð ^ ½Nk þ1 ðx1 Þ . . . dx1 C k ^ k ðlÞ ¼C Nk X ð1Þ lþNk l! l¼0 ðkÞ ~ ^ k ðlÞ þ regðlÞ; fl ðl lk Þ l QNk ðl l0 Þ ¼ C where QNk is a polynomial in ðl l0 Þ of degree Nk and regðlÞ is analytic of ^ ½Nk þ1 ¼ C ~k . Using (7), we have l A C. Now recall that C k 8 ~ ðx Þc þ regðx1 lj Þ; > C lj0 B Z; > > j 1 jk > ! > > > ð jÞ > P ðx Þ 1 < C ~j ðx1 Þ logðx1 lj Þ þ cjk lj0 ¼ Nj A N; Nj þ1 ^ ½Nk þ1 ðx1 Þ ¼ C ðx l Þ k j 1 > > > > > þ regðx1 lj Þ; > > > : ~j ðx1 Þ logðx1 lj Þcjk þ regðx1 lj Þ; C lj0 A Z : When lj0 B Z, from (7) and (61), we have ^ k ðlÞ ¼ regðlÞ þ C l0 ¼ regðlÞ þ ð xN ðl dxNk þ1 k ð ~j ðx1 Þcjk þ regðx1 lj Þ . . . dx1 ½C l0 ½N 1 Fj k ðlÞcjk þ regðl lj Þ ~ ½Nk 1 ðlÞ QN ðl l0 ÞÞcjk þ regðl lj Þ ¼ regðlÞ þ ðC k j ½Nk 1 ~ ¼C j When lj0 ¼ Nj A N, ðlÞcjk þ regðl lj Þ: On Stokes Matrices in terms of Connection Coe‰cients ^ k ðlÞ ¼ regðlÞ þ C ð xN ðl l0 dxNk þ1 k ð . . . dx1 l0 " ¼ ð jÞ ~ j ðx1 Þ lnðx1 lj Þ þ C 423 PNj ðx1 Þ ðx1 lj Þ Nj þ1 ! # cjk þ regðx1 lj Þ 8 ! > P ðlÞ N N 1 > ½N N j k j k > ^ > ðlÞ logðl lj Þ þ cjk C j > < ðl lj Þ Nj Nk > þ regðl lj Þ; > > > > ½N :C ^ k þNj ðlÞ logðl lj Þcjk þ regðl lj Þ; j Nj b Nk þ 1; Nk b Nj ; where the last step follows form Proposition 11. When lj0 ¼ Nj A Z , again from Proposition 11 we have: ^ k ðlÞ ¼ regðlÞ þ C l0 ¼ ~ ½Nk 1 C j ð xN ðl dxNk þ1 k ð ~j ðx1 Þ lnðx1 lj Þcjk þ regðx1 lj ÞÞ . . . dx1 ðC l0 lnðl lj Þcjk þ regðl lj Þ: r ~k . For Next, we introduce the ‘‘g deformed’’ series corresponding to g C g0 > 0 su‰ciently small and 0 < g < g0 , A1 g has non integer diagonal entries and no integer eigenvalues, therefore: ~k Þ½q ðlÞ ¼ ð1Þ q ðg C y X 0 ðkÞ Gðlk0 g þ 1 lÞ ~ flq ðl lk Þ llk þg1 ; Eq b 0; l¼q ~k Þ½0 ðlÞ ¼ g C ~k ðlÞ. and in particular ðg C same for any g A C. ðkÞ Recall that the coe‰cients ~ flq are the Lemma 11. Let 0 < g < g0 be such that ðA1 gÞ has no integer diagonal entries and no integer eigenvalues. Let q1 ; q2 A N. Then ðl ~k Þ½q2 ðsÞ dsðl sÞ q1 þg1 ðC lk ¼ ðl Gðq1 þ gÞ sin pðlk0 gÞ ~ ½q1 q2 ðg C k Þ ðlÞ; sin plk0 lk0 B Z; ~k Þ½q2 ðsÞ dsðl sÞ q1 þg1 ðC lk ¼ Gðq1 þ gÞ sin pg ~ ½q1 q2 ðg C k Þ ðlÞ; p lk0 ¼ Nk A Z : 424 Davide Guzzetti The branch of ðl sÞ g in the integrals, for l A Ph , is given by h 2p < argðl sÞjs¼lk < h, and the continuous change along the path of integration. The integrals are well defined for 0 < g < g0 , q1 b 0 and q2 su‰ciently big. Proof. If lk0 B Z the statement is proved in [4], Lemma 2 0 . Analogous computations bring the result for lk0 ¼ Nk A Z . It is enough to integrate expressions (48) and (49) term by term (where jl lk j is small enough to make the series converge). In each term, the following integral appears ðl 0 ðl sÞ q1 þg1 ðs lk Þ llk 1 ds ¼ ðÞ: lk Since one can integrate along a line from lk to l, we parametrize the line with parameter x A ½0; 1 as follows: s ¼ lk þ xðl lk Þ. This yealds the integral representation of the Beta function Bða; bÞ ¼ GðaÞGðbÞ=Gða þ bÞ. Indeed ð1 0 0 ðÞ ¼ ðl lk Þ q1 þgþllk 1 ð1 xÞ q1 þg1 x llk 1 dx 0 0 ¼ ðl lk Þ q1 þgþllk 1 Gðq1 þ gÞGðl lk0 Þ : Gðq1 þ g þ l lk0 Þ The formula holds for any value of lk0 . Note that l b q2 , thus if q1 and q2 are big enough, the integrals converge. Note also that for lk0 ¼ Nk a 1, the integrals converge for q2 b 0. Moreover, since we have assumed g > 0, the integrals converge for any q1 b 0. Finally, some manipulations using GðxÞGð1 xÞ ¼ p=sinðpxÞ, yield the result. For example, in case lk0 ¼ Nk a 1, we have ðl ~k Þ½q2 ðsÞ dsðl sÞ q1 þg1 ðC lk ¼ ð1Þ q2 X lbq2 ð1Þ lNk ~ðkÞ Gðq1 þ gÞGðl Nk Þ flq2 ðl lk Þ q1 þgþlNk 1 ¼ ðÞ: Gðq1 þ g þ l Nk Þ ðl Nk 1Þ! We use Gðl Nk Þ ¼ ðl Nk 1Þ!, 1 GðNk þ 1 g l q1 Þ sinðq1 þ l Nk þ gÞ ; ¼ Gðq1 þ g þ l Nk Þ p and change l 7! l q1 . ðÞ ¼ ð1Þ q1 þq2 We get Gðq1 þ gÞ sin pg X ðkÞ GðNk g1 lÞ ~ flq1 q2 ðl lk Þ lðNk gÞ1 p lbq þq 1 ¼ 2 Gðq1 þ gÞ sin pg ~ ½q1 q2 ðg Ck Þ ðlÞ: p r On Stokes Matrices in terms of Connection Coe‰cients 425 Lemma 12. Let 0 < g < g0 be such that ðA1 gÞ has no integer diagonal entries and no integer eigenvalues. Then ðl ^ k Þ½q ðsÞ ¼ GðgÞ sin pg ðg C ~k Þ½Nk 1q ðlÞ; dsðl sÞ g1 ðC p lk lk0 ¼ Nk A N: The integral is well defined for 0 < g < g0 and q b 0 integer. ðl sÞ g is defined in the same way as in Lemma 11. Proof. The branch of Integration term by term yields ^ k Þ½q ðsÞ ¼ ð1Þ Nk þ1þq ðC X ð1Þ lþ1 lbq l! ðkÞ ~ flq ðl lk Þ l ; ~k Þ½Nk 1q ðlÞ ðg C ¼ ð1Þ Nk þ1þq X ðkÞ GðNk g þ 1 lÞ ~ flNk 1q ðl lk Þ lNk þg1 : lbNk þ1þq As in the previous lemma, we compute ðl dsðl sÞ g1 ðs lk Þ l ¼ ðl lk Þ gþl lk GðgÞGðl þ 1Þ ; Gðg þ l þ 1Þ and use 1=Gðg þ l þ 1Þ ¼ ð1Þ lþ1 sinðpgÞGðg lÞ=p. This implies that ðl ^ k Þ½q ðsÞ ¼ ð1Þ Nk þ1þq dsðl sÞ g1 ðC lk GðgÞ sin pg X ðkÞ Gðl gÞ ~ flq ðl lk Þ lþg : p lbq After redefining l 0 ¼ l þ Nk þ 1 we obtain the final result. r ^ ½q in the following ~ ½q and C We establish the monodromy of C k k Proposition 12. Let l A Ph . Let q b 0 be an integer. Let aj ¼ 2pi when 0 A Z, and aj ¼ e 2pilj 1 when lj0 B Z. The following transformations hold for a loop gj around a pole lj . a) If lk0 B Z or lk0 A Z : 8 < aj cjk C ~ ½q ðlÞ; lj0 B Z or lj0 A Z ; j ½q ½q ~ ~ Ck ðlÞ 7! Ck ðlÞ þ : aj cjk C ^ ½qþNj þ1 ðlÞ; l 0 ¼ Nj A N: j j lj0 b) If lk0 A N: ^ ½q ðlÞ C k 7! ^ ½q ðlÞ C k þ 8 < aj cjk C ~ ½q1Nk ðlÞ; j : aj cjk C ^ ½qþNj Nk ðlÞ; j lj0 B Z or lj0 A Z ; lj0 ¼ Nj A N: 426 Davide Guzzetti ½q Proof. We consider the function Fk ðl; l0 Þ defined in (53) for l; l0 A Ph . For simplicity of notation, we omit l0 , namely we write ½q Fk ðlÞ ¼ ð xq ðl dxq ð x2 dxq1 . . . l0 l0 dx1 fk ðx1 Þ; l0 ½q Fk ðlÞ ¼ fk ðlÞ; if q ¼ 0: In the cut-plane Ph , we consider l close to lj 0 lk in such a way that the series representations of ðfj Þ½q ðlÞ converge. We also consider a loop gj around lj in counter-clockwise direction, represented by ðl lj Þ 7! ðl lj Þe 2pi . The new path of integration from l0 to l is represented in figure 5. We have the following transformation after the loop ½q ½q Fk ðlÞ 7! Fk ðlÞ þ ð xq þ dxq gj ð x2 dxq1 . . . l0 dx1 fk ðx1 Þ: l0 By formula (54) it follows that the analytic continuation of ðfk Þ½q ðlÞ along the loop gj is ðfk Þ ½q ðlÞ 7! ðfk Þ ½q ðlÞ þ ð xq þ dxq gj ð x2 dxq1 . . . l0 dx1 fk ðx1 Þ; Eq b 0: l0 Next, we express fk ðx1 Þ in terms of the solutions fj at lj . two cases in the proposition. We distinguish the Fig. 5. The paths of integration after the analytic continuation, starting from l0 , going around a loop gj around lj and ending in l. On Stokes Matrices in terms of Connection Coe‰cients 427 ~k ðx1 Þ, therefore we use Case a). lk0 B Z or lk0 A Z We have fk ðx1 Þ ¼ C (7), namely 8 > ~ lj0 B Z; > > Cj ðx1 Þcjk þ regðx1 lj Þ; > > > ~j ðx1 Þ lnðx1 lj Þcjk þ regðx1 lj Þ; <C lj0 A Z ; fk ðx1 Þ ¼ ! > > P ð jÞ ðx1 Þ > > ~ > Cj ðx1 Þ lnðx1 lj Þ þ cjk þ regðx1 lj Þ; lj0 ¼ Nj A N: > : ðx1 lj Þ Nj þ1 When lj0 B Z, we have a.1) þ ð xq dxq gj ð x2 dxq1 . . . l0 ¼ þ ½ðq1Þ gj ~j ðx1 Þcjk þ regðx1 lj ÞÞ dx1 ðC l0 dxq Fj ðxq Þcjk þ þ dxq regðxq lj Þ 1 gj þ ½ðq1Þ gj dxq Fj because the loop integral of regular terms at lj vanishes. (54), we have þ ðll Þe 2pi ½ðq1Þ ½q dxq Fj ðxq Þcjk ¼ cjk Fj ðxq Þjllj j ðxq Þcjk þ 0; Now, by (55) and gj ~ ½q ðxq Þj ðllj Þe 1 cjk C j llj 2pi ~ ½q ðlÞðe 2pilj0 1Þcjk ; ¼C j The last step follows from the series representation (48). Proposition in case a.1). a.2) When lj0 A Z , we use (56) and compute þ ð xq dxq gj ð x2 dxq1 . . . l0 q b 0: This proves the ~j ðx1 Þ lnðx1 lj Þcjk þ regðx1 lj ÞÞ dx1 ðC l0 ~ Þ½q ðxq Þ lnðxq lj Þj ðllj Þe ¼ cjk ðC j llj 2pi ~ Þ½q ðlÞ; ¼ 2picjk ðC j q b 0: This implies the Proposition in case a.2). a.3) When lj0 ¼ Nj A N, we use (57) and compute " # ! þ ð xq ð x2 P ð jÞ ðx1 Þ ~ dxq dxq1 . . . dx1 Cj ðx1 Þ lnðx1 lj Þ þ cjk þ regðx1 lj Þ ðx1 lj Þ Nj þ1 gj l0 l0 ^ ½qþNj þ1 ðxq Þ lnðxq lj Þj ðllj Þe ¼ cjk C j llj 2pi This proves the Proposition in case a.3). ^ ½qþNj þ1 ðlÞ; ¼ 2picjk C j q b Nj þ 1: 428 Davide Guzzetti when lk0 ¼ Nk A N, we need to compute Case b) þ ð62Þ ð xq dxq gj dxq1 . . . l0 ^ k ðx1 Þ: dx1 C l0 In the case of lj0 B Z, we use (58), and find b.1) ð62Þ ¼ ð x2 þ ð x2 ð xq dxq gj dxq1 . . . l0 l0 ½Nk 1q ~ ¼ ðcjk C j ½Nk 1 ~ dx1 ðC j ðx1 Þcjk þ regðx1 lj ÞÞ e 2pi ðllj Þ ðxq Þ þ regðxq lj ÞÞjðllj Þ 0 ½Nk 1q ~ ¼ cjk ðe 2pilj 1ÞC j ðlÞ: In the case of lj0 A N, we use (59) and Proposition 11, and find " ! ð xq ð x2 þ PNj Nk 1 ðx1 Þ ½Nj Nk ^ dxq1 . . . dx1 Cj ðx1 Þ lnðx1 lj Þ þ ð62Þ ¼ dxq cjk ðx1 lj Þ Nj Nk gj l0 l0 # b.2) þ regðx1 lj Þ ; where PNj Nk 1 ¼ 0 for Nk b Nj . For 0 a q a Nj Nk 1, the integral is 1 2 0 3 e 2pi ðllj Þ ð jÞ P ðx Þ ~ ½Nj Nk q ðxq Þ lnðxq lj Þ þ Nj Nk 1q q A þ regðxq lj Þ5 4cjk @C : j Nj Nk q ðxq lj Þ ðllj Þ For q b Nj Nk b 0, the integral is 2pi ~ ½Nj Nk q ðxq Þ lnðxq lj Þ þ regðxq lj Þj e ðllj Þ : ½cjk C j ðllj Þ In both cases, the above expressions yield ~ ½Nj Nk q ðlÞ: ð62Þ ¼ 2picjk C j b.3) ð62Þ ¼ In case lj0 A Z , we use (60) and Proposition 11, and find þ ð xq dxq gj ð x2 dxq1 . . . l0 l0 ~ ½Nk 1 ðx1 Þ lnðx1 lj Þcjk þ regðx1 lj ÞÞ dx1 ðC j 2pi ~ ½Nk 1q ðxq Þ lnðxq lj Þcjk þ regðxq lj Þj e ðllj Þ ¼ 2picjk C ~ ½Nk 1q ðlÞ: ¼ ½C j j ðllj Þ The above computations prove the Proposition in case b). r Proof of Proposition 10. Given a function f ðlÞ, l A Ph , we denote with fþ ðlÞ the value on the left side of Lj , where argðl lj Þ ¼ h 2p. We denote On Stokes Matrices in terms of Connection Coe‰cients 429 with f ðlÞ the value on the right side, where argðl lj Þ ¼ h. By Lemmas 11 and 12 we have: (ð ) ðl lj 0 ½qðl Þ ½q ½q g1 g1 0 k ~k Þ ðg C ðlÞ ¼ F ðlk Þ ðl sÞ fk ðsÞds þ ðl sÞ ðfk ÞGðsÞds ; G lj lk where q; if lk0 B Z or lk0 A Z ; q þ Nk þ 1; if lk0 ¼ Nk A N; ( ~k ; if l 0 B Z or l 0 A Z ; C k k fk ¼ 0 ^ C k ; if lk A N; 8 > sin plk0 > > ; if lk0 B Z; < 0 GðgÞ sin pðl gÞ k F ðlk0 Þ ¼ > p > > ; if lk0 A Z: : GðgÞ sin pg qðlk0 Þ ¼ In the integral, argðl sÞ, s A Ph , has the value obtained by the continuous change along the path of integration from lk up to s belonging to Lj . Change from f to fþ is obtained along a small loop encircling only lj . Therefore, Proposition 12 yields ð63Þ 0 0 0 ½qðlk Þ ½qðlk Þ ~k Þ½qðlk Þ ðlÞ ðg C ~k Þþ ~j Þþ ðg C ðlÞ ¼ aj ½gcjk ½gðg C ðlÞ: By Lemmas 11 and 12 we write ð64Þ ½qðl 0 Þ 0 ~k Þ½qðlk Þ ðlÞ ðg C ~k Þþ k ðlÞ ðg C ðl ½q ½q ¼ F ðlk0 Þ ðl sÞ g1 ½ðfk Þ ðsÞ ðfk Þþ ðsÞds: lj We need to distinguish two cases. 1) lk0 B Z, or lk0 A Z In this case (63), (64) and Proposition 12 applied to the integrand yield the following equalities. 1.a) for lj0 B Z or lj0 A Z : ð65Þ ~j Þ½q ðlÞ ¼ F ðl 0 Þ aj ½gcjk ½gðg C k þ ðl lj ~j Þ½q ðsÞ: dsðl sÞ g1 aj cjk ðC þ 1.b) for lj0 A N: ð66Þ ~j Þ½q ðlÞ ¼ F ðl 0 Þ aj ½gcjk ½gðg C k þ ðl lj ½qþNj þ1 ^ j Þþ dsðl sÞ g1 aj cjk ðC ðsÞ: 430 Davide Guzzetti We apply again Lemmas 11 and 12 to express the r.h.s. of the above equalities. To this end, we need ðl sÞþ in the integrand. Observe that ( ðl sÞg1 ¼ ½e 2pi ðl sÞþ g1 ; when k j; g1 in the integrand ¼ ðl sÞ ðl sÞþg1 ; when k 0 j: Indeed, when l belongs to the left side of Lj and s A Ph , then h 2p < argðl sÞ < h. When s reaches Lj from the left, then argðl sÞ ! h if Lk is to the left of Lj , namely k j: in this case we obtain ðl sÞ . On the other hand, argðl sÞ ! h 2p if Lk is to the right of Lj , namely k 0 j: in this case we obtain ðl sÞþ . See figure 6. Applying Lemmas 11 and 12 we find 8 F ðlk0 Þ 2pig > > ~ ½q > > F ðl 0 Þ e aj cjk ðg Cj Þþ ðlÞ; k j; < j r:h:s: of ð65Þ and ð66Þ ¼ > F ðlk0 Þ > > ~j Þ½q ðlÞ; > k 0 j: : F ðl 0 Þ aj cjk ðg C þ j Namely: ð67Þ aj ½gcjk ½g ¼ 8 F ðlk0 Þ 2pig > > > > F ðl 0 Þ e aj cjk ; k j; < j > F ðlk0 Þ > > > : F ðl 0 Þ aj cjk ; k 0 j: j Finally, we compute the ratio F ðlk0 Þ=F ðlj0 Þ. For lj0 B Z: 0 0 F ðlk0 Þ sin plk sin pðlj gÞ ð1 e 2pilk Þð1 e 2piðlj gÞ Þ ak aj ½g : ¼ ¼ ¼ 0 F ðlj0 Þ sin plj0 sin pðlk0 gÞ ð1 e 2pilj Þð1 e 2piðlk0 gÞ Þ aj ak ½g 0 Fig. 6. The figure shows argðl sÞ as s ! lj . 0 On Stokes Matrices in terms of Connection Coe‰cients 431 For lj0 A Z: 0 aj ½g ak F ðlk0 Þ sin pg sin plk0 e 2pig 1 e 2pilk 1 ; ¼ ¼ ¼ 0 0 0 2piðl gÞ k 2pi e aj ak ½g F ðlj Þ p sin pðlk gÞ 1 where we have used the fact that aj ¼ 2pi and aj ½g ¼ e 2pig 1. The above computations imply the statement of Proposition 10 when lk0 B Z and lk0 A Z . 2) lk0 A N In this case (63), (64) and Proposition 12 applied to the integrand yield the following equalities. 2.a) For lj0 B Z or lj0 A Z : ðl ~j Þ½qNk 1 ðsÞ: ~j Þ½qNk 1 ðlÞ ¼ F ðl 0 Þ dsðl sÞ g1 aj cjk ðC ð68Þ aj ½gcjk ½gðg C k þ þ lj 2.b) ð69Þ For lj0 A N: ~j Þ½qNk 1 ðlÞ ¼ F ðl 0 Þ aj ½gcjk ½gðg C k þ ðl lj ½qNk 1þNj þ1 ^ j Þþ dsðl sÞ g1 aj cjk ðC ðsÞ: We apply again Lemmas 11 and 12 to express the r.h.s. of the above equalities, keeping into account the branch of ðl sÞ g1 as before. We find 8 F ðlk0 Þ 2pig > > ~j Þ½qNk 1 ðlÞ; k j; > e aj cjk ðg C > þ < F ðlj0 Þ r:h:s: of ð68Þ and ð69Þ ¼ 0 > > > F ðlk Þ aj cjk ðg C ~j Þ½qNk 1 ðlÞ; > k 0 j: : F ðl 0 Þ þ j Namely, we obtain again ð70Þ aj ½gcjk ½g ¼ 8 F ðlk0 Þ 2pig > > > > F ðl 0 Þ e aj cjk ; k j; < j > F ðlk0 Þ > > > : F ðl 0 Þ aj cjk ; k 0 j: j Finally, we compute the ratio F ðlk0 Þ=F ðlj0 Þ. For lj0 B Z: 0 F ðlk0 Þ p sin pðlj 1Þ 2pi e 2piðlj gÞ 1 ak aj ½g ¼ 2pig ; ¼ ¼ 0 0 0 e 1 e 2pilj 1 ak ½g aj sin plj F ðlj Þ sin pg 0 where we have used the fact that ak ¼ 2pi and ak ½g ¼ e 2pig 1. For lj0 A Z: F ðlk0 Þ ¼ 1: F ðlj0 Þ In this last case, observe that ak ¼ aj ¼ 2pi and ak ½g ¼ aj ½g ¼ e 2pig 1. 432 Davide Guzzetti The above computations imply the statement of Proposition 10 when lk0 A N. r References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] Balser, W., Birkho¤ ’s reduction problem, Russian Math. Surveys, 59 (2004), 1047–1059. Balser, W., Jurkat, W. B. and Lutz, D. A., Birkho¤ Invariants and Stokes’ Multipliers for Meromorphic Linear Di¤erential Equations, J. Math. Anal. Appl., 71 (1979), 48–94. Balser, W., Jurkat, W. B. and Lutz, D. A., A General Theory of Invariants for Meromorphic Di¤erential Equations; Part II, Proper Invariants, Funkcial. Ekvac., 22 (1979), 257–283. Balser, W., Jurkat, W. B. and Lutz, D. A., On the reduction of connection problems for di¤erential equations with an irregular singular point to ones with only regular singularities, I, SIAM J. Math. Anal., 12 (1981), 691–721. Balser, W., Jurkat, W. B. and Lutz, D. A., Transfer of Connection Problem for Meromorphic Di¤erential Equations of Rank r b 2 and Representation of Solutions, J. Math. Anal. Appl., 85 (1982), 488–542. Birkho¤, G. D., Singular points of ordinary linear di¤erential equations, Trans. Amer. Math. Soc., 10 (1909), 436–470. Doetsch, G., Introduction to the Theory and Application of the Laplace Transformation, Springer-Verlag, New York-Heidelberg, 1974. Dubrovin, B., Geometry of 2D topological field theories, Lecture Notes in Math, 1620, Springer, Berlin, 1996, pp. 120–348. Dubrovin, B., Painlevé trascendents in two-dimensional topological field theory, CRM Ser. Math. Phys., Springer, New York, 1999, pp. 287–412. Dubrovin, B., On almost duality for Frobenius manifolds, Amer. Math. Soc. Transl. Ser. 2, 212, Amer. Math. Soc., Providence, RI, 2004, pp. 75–132. Guzzetti, D., arXiv:1407.1206 (2014). Écalle, J., Les Fonctions Résurgentes, I, II, III, Publications Mathématiques d’Orsay, Université de Paris-Sud, Départment de Mathématiques, Orsay, 1981–1985. Galkin, S., Golyshev, V. and Iritani, H., Gamma Classes and Quantum Cohomology of Fano Manifolds: Gamma Conjectures, Duke Math. J., 165 (2016), 2005–2077. Ince, E. L., Ordinary di¤erential equations, Dover Publications, New York, 1956. Loday-Richaud, M., Rank Reduction, Normal Forms and Stokes Matrices, Expo. Math., 19 (2001), 229–250. Loday-Richaud, M. and Remy, P., Resurgence, Stokes phenomenon and alien derivatives for level-one linear di¤erential systems, J. Di¤erential Equations, 250 (2011), 1591–1630. Lutz, D. A., On the reduction of rank of linear di¤erential systems, Pacific J. Math., 42 (1972), 153–164. Mazzocco, M., Painlevé sixth equation as isomonodromic deformations equation of an irregular system, CRM Proc. Lecture Notes, 32, Amer. Math. Soc., Providence, RI, 2002, pp. 219–238. Remy, P., Matrices de Stokes-Ramis et constantes de connexion pour les systèmes di¤érentiels linéaires de niveau unique, Ann. Fac. Sci. Toulouse Math. (6), 21 (2012), 93–150. Turrittin, H. L., Reducing the rank of ordinary di¤erential equations, Duke Math. J., 30 (1963), 271–274. Ugaglia, M., On a Poisson Structure on the Space of Stokes Matrices, Internat. Math. Res. Notices, 1999 (1999), 473–493. On Stokes Matrices in terms of Connection Coe‰cients 433 [22] Wasow, W., Asymptotic Expansions for Ordinary Di¤erential Equations, Pure and Applied Mathematics, Vol. XIV, Interscience Publishers John Wiley & Sons, Inc., New York-LondonSydney, 1965. [23] Cotti, G., Coalescence Phenomenon of Quantum Cohomology of Grassmannians and the Distribution of Prime Numbers, arXiv:1608.06868 (2016). nuna adreso: Scuola Internazionale Superiore di Studi Avanzati Via Bonomea 265 34136 Trieste Italy http://orcid.org/0000-0002-6103-6563 (Ricevita la 29-an de oktobro, 2014) (Reviziita la 4-an de aŭgusto, 2015)