Isomonodromic Laplace Transform with Coalescing Eigenvalues and Confluence of Fuchsian Singularities
IIsomonodromic Laplace Transform with CoalescingEigenvalues and Confluence of Fuchsian Singularities
Davide Guzzetti
SISSA, Via Bonomea, 265, 34136 Trieste – Italy. E-MAIL: [email protected]
Abstract:
We consider a Pfaffian system expressing isomonodromy of an irregular system of Okubo type,depending on complex deformation parameters u “ p u , ..., u n q , which are eigenvalues of the leading matrixat the irregular singuilarity. At the same time, we consider a Pfaffian system of non-normalized Schlesingertype expressing isomonodromy of a Fuchsian system, whose poles are the deformation parameters u , ..., u n .The parameters vary in a polydisc containing a coalescence locus for the eigenvalues of the leading matrix of theirregular system, corresponding to confluence of the Fuchsian singularities . We construct isomonodromic selectedand singular vector solutions of the Fuchsian Pfaffian system together with their isomonodromic connectioncoefficients , so extending a result of [4] and [20] to the isomonodromic case, including confluence of singularities.Then, we introduce an isomonodromic Laplace transform of the selected and singular vector solutions, allowingto obtain isomonodromic fundamental solutions for the irregular system, and their Stokes matrices expressed interms of connection coefficients. These facts, in addition to extending [4, 20] to the isomonodromic case (withcoalescences/confluences), allow to prove by means of Laplace transform the main result of [11], which is theanalytic theory of non-generic isomonodromic deformations of the irregular system with coalescing eigenvalues.Keywords: Non generic Isomonodromy Deformations, Schlesinger equations, Isomonodromic conflu-ence of singularities, Stokes phenomenon, Coalescence of eigenvalues, Resonant IrregularSingularity, Stokes matrices, Monodromy data Contents D p u c q and Vanishing Conditions 175 Selected Vector solutions depending on parameters u P D p u c q
196 Proof of Theorem 5.1 by steps 23 ~ Ψ i , part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.3 Singular Solutions ~ Ψ p sing q i , part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.4 Local behaviour at λ “ u i , i “ , ..., p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.5 Selected and Singular vectors solutions, part II. Completion of the proof of Th. 5.1 . . . . 366.6 Analogous proof for all coalescences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.7 Proof of Corollary 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 D p u c q a r X i v : . [ m a t h . C A ] J a n (Non) Uniqueness of the formal solution of (1.1) at u “ u c
499 Appendix A. Non-normalized Schlesinger System 5110 Appendix B. Proof of Proposition 3.1 5211 Appendix C 54
In this paper I answer a question asked when I presented the results of [11] and the related paper [21].Paper [11] deals with extension of the theory of isomonodromic deformations of the irregular differentialsystem (1.1) below, in presence of a coalescence phenomenon involving the eigenvalues of the leadingmatrix Λ. These eigenvalues are the deformation parameters. The question is if we can obtain someresults of [11] in terms of the Laplace transform relating system (1.1) to a Fuchsian one, such as system(1.3) below. The latter has simple poles at the eigenvalues of Λ, so that coalescence of eigenvalueswill correspond to confluence of Fuchsian singularities. So the question is if combining isomonodromicdeformations of Fuchsian systems, confluence of singularities and Laplace transform, we can obtain theresults of [11]. The positive answer to the question is the content of
Theorem 7.1 . of this paper.In order to achieve this, we extend to the case depending on deformation parameters, including theircoalescence, one main result of [4] and [20] concerning the existence of selected and singular vectorsolutions of a Pfaffian Fuchsian system associated with (1.3) (see the system (5.3) below), and theirconnection coefficients, which we will be isomonodromic. This will be obtained in
Theorem 5.1 andits
Corollary 5.1 .In [11] the isomonodromy deformation theory of an n dimensional differential system with Fuchsiansingularity at z “ z “ 8 of Poincaré rank 1 dYdz “ ˆ Λ p u q ` A p u q z ˙ Y, Λ p u q “ diag p u , ..., u n q , (1.1)has been considered , where u “ p u , ..., u n q varies in a polydisc where the matrix A p u q is holomorphic .One of the main results of [11] is the extension of the theory of isomonodromic deformations of (1.1) toa non-generic case , namely when Λ has coalescing eigenvalues. This means that the polydisc contains alocus of coalescence points such that u i “ u j for some 1 ď i ‰ j ď n . In this case, z “ 8 is sometiomescalled resonant irregular singularity . Theorem 1.1 and corollary 1.1 of [11] say that the extension ispossible if the entries of A p u q satisfies the vanishing conditions p A p u qq ij Ñ u tends to a coalescence point such that u i ´ u j Ñ . In this case, the following results (also summarized in Theorem 2.2 of Section 2.1 below) hold.(I) Fundamental matrix solutions in Levelt form at z “ z “ 8 are holomorphic of u in the polydisc. Also thecoefficients of the formal solution determining the asymptotics at are holomorphic. With the notation p A p u q for A p u q . Essential monodromy data , such as Stokes matrices, the central connection matrix, the formalmonodromy exponent at infinity and the Levelt exponents at z “ ν P Z ) satisfy the vanishing conditions p S ν q ij “ p S ν q ji “ , i ‰ j , if there is a coalescence point such that u i “ u j .(III) The above constant essential monodromy data can be computed restricting to the system at a fixedcoalescence point. In particular, if the constant diagonal entries of A do not differ by non-zerointegers, then there is no ambiguity in this computation, being the formal solution unique.The results above have been established in [11] by direct analysis of system (1.1), of its Stokesphenomenon and its isomonodromic deformations in a polydisc containing coalescence points.For future use, we denote by λ , . . . , λ n the diagonal entries of A p u q , and B : “ diag p A p u qq “ diag p λ , . . . , λ n q . We will see that these λ k are constant, in the isomonodromic case.From another perspective, if u is fixed and u i ‰ u j for i ‰ j , namely for a system (1.1) not dependingon parameters with pairwise distinct eigenvalues of Λ, it is well known that columns of fundamentalmatrix solutions with prescribed asymptotics in Stokes sectors at z “ 8 can be obtained by Laplace-type integrals of certain selected column-vector solutions of an n -dimensional Fuchsian system of thetype d Ψ dλ “ n ÿ k “ B k λ ´ u k Ψ , B k : “ ´ E k p A ` I q . (1.2)Here, E k is the elementary matrix whose entries are zero, except for p E k q kk “
1. These facts in generic cases are studied in the seminal paper [4]. By generic, we mean that in [4] it is assumed that thediagonal entries λ k of A are not integers. If we allow these entries to take any complex value, includingintegers, the analysis becomes more complicated, but richer and interesting. This general case, withoutassumptions on A , has been studied in [20], where the results of [4] have been extended.The purpose of the present paper is to introduce an isomonodromic Laplace transform relating (1.1)to an isomonodromic Fuchsian system d Ψ dλ “ n ÿ k “ B k p u q λ ´ u k Ψ , B k : “ ´ E k p A p u q ` I q . (1.3)when u , ..., u n vary in a polydisc containing a locus of coalescence points . The two main goals will be toconstruct isomonodromic selected solutions and singular solutions of (1.3), and to prove through theirisomonodromic Laplace transform the main statements of [11], as in (I), (II) and (III) above, concerningthe Stokes phenomenon, Stokes matrices, monodromy data and fundamental matrix solutions of (1.1).The main results of the paper are summarized in• Theorem 5.1 , which characterises selected vector solutions and singular vector solutions of (1.3),so extending the results of [4] and [20] to the case depending on isomonodromic deformationparameters, including confluence of Fuchsian singularities u , ..., u n .3 Theorem 7.1 , in which the Laplace transform of the vector solutions of Theorem 5.1 allows toobtain the main results of [11] in presence of coalescing eigenvalues u , ..., u n of Λ p u q .In detail, the results are as follows. ‚ First, in Proposition 3.1 we will establish the equivalence between strong isomonodoromic defor-mations (non-normalized Schlesinger deformations) of (1.3) and strong isomonodromic deformations of(1.1). In particular, we will show that A is isospectral and its diagonal entries are constant. ‚ Successively, we will study isomonodoromic deformations of (1.3) when u varies in a polydisc con-taining a locus where some of the poles u , ..., u n coalesce (confluence of singularities). The main result,in Theorem 5.1 , provides selected and singular vector solutions of (1.3), which are the isomonodromicanalogue of solutions introduced in [4, 20]. These will be denoted by ~ Ψ k p λ, u | ν q and ~ Ψ p sing q k p λ, u | ν q , k “ , ..., n , the latter being singular at λ “ u k . The integer ν P Z comes from the necessity to labelthe directions of branch cuts in the punctured λ -plane at the poles u , ..., u n , as will be explained later.These solutions allow to introduce connection coefficients c p ν q jk , defined by ~ Ψ k p λ, u | ν q “ ~ Ψ p sing q j p λ, u | ν q c p ν q jk ` holomorphic part at λ “ u j , @ j ‰ k. The above is the deformation parameters dependent analogue of the definition of connection coefficientsin [20]. ‚ In Corollary 5.1 , we will prove that the c p ν q jk are isomonodromic connection coefficients ,namely are independent of u , and satisfy c p ν q jk “ , for j ‰ k such that there is a coalescence u j “ u k at least at one point in the polydisc. ‚ In Theorem 7.1 , we will show that the Laplace transform of the vectors ~ Ψ k p λ, u | ν q or ~ Ψ p sing q k p λ, u | ν q yields the columns of isomonodromic fundamental matrix solutions Y ν p z, u q of (1.1), labelled by ν P Z ,uniquely determined by a prescribed asymptotic behaviour in certain u -independent sectors p S ν , of cen-tral opening angle greater than π . Analyticity properties for the matrices Y ν p z, u q will be proved, sore-obtaining the result (I) above.In order to describe the Stokes phenomenon, only three solutions Y ν p z, u q , Y ν ` µ p z, u q and Y ν ` µ p z, u q will suffice. The labelling will be explained later. The Stokes matrices S ν ` kµ , k “ ,
1, defined by arelation Y ν `p k ` q µ “ Y ν ` kµ S ν ` kµ in p S ν ` kµ X p S ν `p k ` q µ , will be expressed in terms of the coefficients c p ν q jk .This extends to the isomonodromic case, including coalescences, an analogous expression appearing in[4, 20]. Moreover, in this way we re-obtaining results (II) above. ‚ In Section 8, we will re-obtain the result (III), namely that system (1.1), "frozen" by fixing u equalto a coalescent point, admits a unique formal solution if and only if the (constant) diagonal entries of A do not differ by non-zero integers. This will be done showing that only in this case are uniquelydetermined the selected vector solutions of the Fuchsian system (1.3) at the fixed coalescence point,solutions needed to perform the Laplace transforms at the fixed coalescent point. On the other hand, ifthe diagonal entries of A differ by non-zero integers, we will show that at a coalescence point there is afamily of solutions of the Fuchsian system (1.3), depending on a finite number of parameters, and thisfacts is responsible, through the Laplace transform, of the existence of a family of formal solutions atthe coalescence point.In [16, 17], B. Dubrovin related system (1.1) to an isomonodromic system of type (1.3), in the specificcase when such systems respectively produce flat sections of the deformed connection of a semisimple4ubrovin-Frobenius manifold and flat sections of the intersection form (extended Gauss-Manin system).In [16, 17], the solutions of (1.1) are expressed by Laplace transform of the isomonodromic (1.3), butthe eigenvalues u , ..., u n are assumed to be pairwise distinct , varying in a sufficiently small domain(analogous to the polydisc D p u q to be introduced later). Moreover, A is skew-symmetric, so its diagonalelements are zero ( A is denoted by V and Λ by U in [16, 17]). By a Coxeter-type identity, the entriesof the monodromy matrices for special solutions of (1.3) (which are part of the monodromy of theDubrovin-Frobenius manifold ) are expressed in terms of entries of the Stokes matrices. See also [42, 18].In [19], the authors prove (I) above in proposition 2.5.1, when system (1.1) is associated with aDubrovin-Frobenius manifold with semisimple coalescence points, and A is skew-symmetric (in [19] theirregular singularity is at z “ Y p z, u q in (I) are obtainable, by Laplace transform, from the analytic properties of a fundamental matrixsolution Ψ p λ, u q of the Fuchsian Pfaffian system associated with (1.3) (see their lemma 2.5.3). The latteris a particular case of the Fuchsian Pfaffian systems studied in [44]. On the other hand, the analysis ofselected and singular vector solutions of the Fuchsian Pfaffian system, required in our paper to cover allpossible cases (all possible A ), is not necessary in [19], due to the skew-symmetry of A , and the specificform of their Pfaffian system (see their equation (2.5.2); their discussion is equivalent our case λ j “ ´ j “ , ..., n ). Moreover, points (II) an (III) are not discussed in [19] by means of the Laplacetransform.In the present paper, by an isomonodromic Laplace transform, we prove (I), (II) and (III) with noassumptions on A , and at the same time we generalise the results of [4, 20] to the isomonodromic casewith coalescences. This construction, to the best of our knowledge, cannot be found in the literature.The approach of the present paper may also be used to extend the results of [16, 17] described above,relating the deformed flat connection and the intersection form, namely Stokes matrices and monodromygroup of the Dubrovin-Frobenius manifold, in case of semisimple coalescent Frobenius structures studiedin [12].For further comments and reference on the use of Laplace transform and confluence of singularitiesand related topics, see the introduction of [20] and [9, 32, 34, 35, 38, 39, 40, 41, 29, 30, 31, 24] . Acknowledgements
I would like to thank professor H. Iritani for kindly drawing my attention to the proof of proposition 2.5.1of [19]. I also would like to thank doctor G. Cotti for several stimulating discussions in the backgroundof this paper. I remember with gratitude professor B. Dubrovin, who gave us the initial motivation toinvestigate the problem of non-generic isomonodromic deformations, contributing in our discussions withinsight and experience. The author is a member of the European Union’s H2020 research and innovationprogramme under the Marie Skłlodowska-Curie grant No. 778010
IPaDEGAN . This section contains known and essential material to motivate and understand our paper. For X atopological space, we denote by R p X q its universal covering. For α ă β P R , a sector is written asfollows S p α, β q : “ t z P R p C zt uq such that α ă arg z ă β u . .1 Background 1: Isomonodromy Deformations of (1.1) with coalescenceof eigenvalues. Here, we review results of [11, 21] (see also [13]). Consider a linear differential system (1.1) of dimension n ˆ n with matrix coefficient A p u q holomorphic in a polydisc D p u c q : “ t u P C n such that max ď j ď n | u j ´ u cj | ď (cid:15) u , (cid:15) ą . (2.1)The polydisc is centered at a coalescence point u c “ p u c , ..., u cn q , so called because u ci “ u cj for some i ‰ j. The eigenvalues of Λ p u q coalesce at u c and also along the following coalescence locus ∆ : “ D p u c q X ´ď i ‰ j t u i ´ u j “ u ¯ , We assume that D p u c q is sufficiently small so that u c is the most coalescent point . Namely, if u cj ‰ u ck for some j ‰ k , then u j ‰ u k for all u P D p u c q . More precise characterisation of the radius (cid:15) of thepolydisc will be given in Section 5. For u P D p u c qz ∆, let D p u q Ă p D p u c qz ∆ q be a (smaller) polydisc centered at u , not containing coalescence points. We will choose it more preciselylater. D p u q If D p u q is sufficiently small, the isomonodromic theory of Jimbo, Miwa and Ueno [28] assures thatthe essential monodromy data of (1.1) (see Definition 2.1 below) are constant over D p u q and can becomputed fixing u “ u .In order to give fundamental solutions with “canonical” form at z “ 8 , in R p C zt uq we introducethe Stokes rays of Λ p u q , defined by < pp u j ´ u k q z q “ , = pp u j ´ u k q z q ă , ď j ‰ k ď n. Let arg z “ τ p q (2.2)be a direction which does not coincide with any of the Stokes rays of Λ p u q , called admissible at u .Each sector of amplitude π , whose boundaries are not Stokes rays of Λ p u q , contains a certain number µ p q ě p u q , with angular directionsarg z “ τ , τ , ..., τ µ p q ´ , that we decide to label from 0 to µ p q ´
1. They are "basic" rays, since they generate all the other Stokesrays in R p C zt uq associated with Λ p u q , with the following directions τ ν : “ τ ν ` kπ, ď ν ď µ p q ´ , ν “ ν ` kµ p q , k P Z . (0)
0. Here, D is diagonal withinteger entries (called valuations), L has eigenvalues with real part lying in r , q , and D ` lim z Ñ z D Lz ´ D is a Jordan form of A . A central connection matrix C ν p u q is defined by Y ν p z, u q “ Y p q p z, u q C ν p u q . (2.13)A pair of Stokes matrices S ν , S ν ` µ p q , together with B , C ν and L are sufficient to calculate all theother S ν and C ν , for all ν P Z (see [1, 11]). The monodromy matrices at z “ M : “ e πiL and e πiB p S ν S ν ` µ p q q ´ “ C ´ ν M C ν for Y p q and Y ν respectively. Hence, it makes sense to give the following Definition 2.1.
Fixed a ν P Z , we call essential monodromy data the matrices S ν , S ν ` µ p q , B, C ν , L, D. The deformation u is strongly isomonodromic on D p u q , if the essential monodromy data are constanton D p u q . all the essential monodromy data , contrary to the case of "weak" isomonodromic deformations,which only preserve monodromy matrices of a certain fundamental matrix solution. For a deformationto be weakly isomonodromic it is necessary and sufficient that (1.1) is the z -component of a certainPfaffian system dY “ ω p z, u q Y , Frobenius integrable (i.e. dω “ ω ^ ω ). If ω is of very specific form, thedefomation becomes strongly isomonodromic, according to the following Theorem 2.1.
System (1.1) is strongly isomonodromic in D p u q if and only Y ν p z, u q , for every ν , and Y p q p z, u q , satisfy the Frobenius integrable Pfaffian system dY “ ω p z, u q Y, ω p z, u q “ ˆ Λ p u q ` A p u q z ˙ dz ` n ÿ k “ ω k p z, u q du k , (2.14) with the matrix coefficients (here F is in (2.8)) ω k p z, u q “ zE k ` ω k p u q , ω k p u q “ r F p u q , E k s . (2.15) Equivalently, (1.1) is strongly isomonodromic if and only if A satisfies dA “ n ÿ j “ ” ω k p u q , A ı du k . (2.16) If the deformation is strongly isomonodromic, then Y p q p z, u q in (2.12) is holomorphic on R p C zt uq ˆ D p u q , with holomorphic matrix coefficients Ψ j p u q , and the series is convergent uniformly w.r.t. u P D p u q . Moreover, G p q p u q is a holomorphic fundamental solution of the integrable Pfaffian system dG “ ´ n ÿ j “ ω k p u q du k ¯ G, (2.17) and A p u q is holomorphically similar to the Jordan form J “ G p q p u q ´ A p u q G p q p u q , so that its eigenval-ues are constant. The above theorem is analogous to the characterisation of isomonodromic deformations in [28], in-cluding also possible resonances in A (see [11] and Appendix B of [21]). D p u c q with coalescences When the polydics contains a coalescence locus ∆, the analysis presents problematic issues.• A fundamental matrix solution Y p z, u q holomorphic on R ` p C zt uq ˆ p D p u c qqz ∆ q ˘ , may be singularat ∆, namely the limit for u Ñ u ˚ P ∆ along any direction may diverge, and ∆ is in general a branching locus [33].• The monodromy data associated with a fundamental matrix solution ˚ Y p z q of dYdz “ ˆ Λ p u c q ` A p u c q z ˙ Y, (2.18)differ from those of any fundamental solution Y p z, u q of (1.1) at u R ∆ ([2], [3], [11]). Conditions (2.15) and (2.16) imply Frobenius integrability of (2.14), so that the deformation is strongly isomonodromic.Conversely, given (2.14) with ω k p z, u q holomorphic in C ˆ D p u q , with z “ 8 at most a pole, then the integrability dω p z, u q “ ω p z, u q ^ ω p z, u q , which is necessary condition for isomonodromicity, implies that ω k p z, u q “ zE k ` ω k p , u q and(2.16). Computations give that ω k p , u q “ r F p u q , E k s ` D k p u q , where D k p u q is an arbitrary diagonal holomorphic matrix.Imposing that Y p q p z, u q and all the Y ν p z, u q satisfy (2.14), then D k p u q “ ω k p , u q “ r F p u q , E k s . R p C zt uq , we introduce the Stokes rays of Λ p u c q < pp u ci ´ u ck q z q “ , = pp u ci ´ u ck q z q ă , u i ‰ u k , and an admissible direction at u c arg z “ τ, (2.19)such that none of the Stokes rays at u “ u c take this direction. Notice that τ is associated with u c ,differently form τ p q of Section 2.1.1. We choose µ basic Stokes rays of Λ p u c q . These are all and the onlyStokes rays lying in a sector of amplitude π , whose boundaries are not Stokes rays of Λ p u c q . Notice that µ is different from µ p q used in Section 2.1.1. We label their directions arg p z q as follows: τ ă τ ă ... ă τ µ ´ . The directions of all the other Stokes rays of Λ p u c q in R p C zt uq are consequently labelled by an integer ν P Z arg z “ τ ν : “ τ ν ` kπ, with ν P t , ..., µ ´ u and ν : “ ν ` kµ. (2.20)They satisfy τ ν ă τ ν ` .Analogously, at any other u P D p u c q , we define Stokes rays < pp u i ´ u j q z q “ = pp u i ´ u j q z q ă p u q . They behave differently form the case of D p u q . Indeed, if u varies in D p u c q , some Stokes rayscross the admissible directions arg z “ τ mod π , as follows. Let i, j, k be such that u ci “ u cj ‰ u ck . Then,as u moves away from u c , a Stokes ray of Λ p u c q characterized by < pp u ci ´ u ck q z q “ < pp u i ´ u k q z q “ < pp u j ´ u k q z q “
0. If D p u c q is sufficiently small (as in (5.1)below), they do not cross arg z “ τ mod π as u varies in D p u c q . The third ray is < pp u i ´ u j q z q “
0. Since u varying in D p u c q is allowed to make a complete loop around the locus t u P D p u c q | u i ´ u j “ u Ă ∆,along such a loop the above ray crosses arg z “ τ mod 2 π and arg z “ τ ´ π mod 2 π . This crossingphenomenon identifies a crossing locus X p τ q in D p u c q of points u such that there exists a Stokes ray ofΛ p u q (so infinitely many in R p C zt uq ) with direction τ mod π . Proposition 2.2 ([11]) . Each connected component of D p u c qzp ∆ Y X p τ qq is simply connected and home-omorphic to a ball, so it is a topological cell, called τ -cell . Thus, the choice of τ induces a cell decomposition of D p u c q . If u varies in the interior of a τ -cell, noStokes rays cross the admissible directions arg z “ τ mod π , but if u varies in the whole D p u c q , then X p τ q is crossed, and thus Proposition 2.1 does not hold.To overcome this difficulty, we first take a point u in a τ -cell, so that we can consider a polydisc D p u q contained in the τ -cell , satisfying the assumptions of sub-section 2.1.1. Accordingly, we can defineas before the sectors (of angular amplitude greater than π ) S ν ` kµ p u q and S ν ` kµ p D p u qq “ č u P D p u q S ν ` kµ p u q Ă t τ ν ` kµ ´ π ă arg z ă τ ν ` kµ ` u . Now we are using τ and µ in place of τ p q and µ p q . Namely, p u i ´ u j q ÞÑ p u i ´ u j q e πi . u varying in D p u q . Now, ω p z, u q in (2.14)-(2.15) has components ω k p u q “ ˆ A ij p δ ik ´ δ jk q u i ´ u j ˙ ni,j “ “ ¨˚˚˚˚˚˚˚˝ ´ A k u ´ u k ... A k u k ´ u ¨ ¨ ¨ ¨ ¨ ¨ A kn u k ´ u n ... ´ A nk u n ´ u k ˛‹‹‹‹‹‹‹‚ (2.21)Since A p u q is holomorphic in D p u q , then ω k p z, u q is holomorphic on D p u c qz ∆. Thus, the fundamental ma-trix solutions Y ν p z, u q , Y p q p z, u q of sub-section 2.1.1 extend analytically on R ` p C zt uq ˆ p D p u c qqz ∆ q ˘ ‰ R p C z zt uq ˆ p D p u c qqz ∆ q , and ∆ may be a branching locus for them.The extension of the theory of isomonodromy deformations on the whole D p u c q is given in [11] bythe following theorem, which is a detailed exposition of the points (I) and (II) of the Introduction, whilepoint (III) is expressed by Corollary 2.1 below. Theorem 2.2 ([11]) . Let A p u q be holomorphic on D p u c q . Assume that system (1.1) is strongly isomon-odromic on D p u q contained in a τ -cell of D p u c q , so that Theorem 2.1 holds. Part I.
The form ω p z, u q in (2.15) and (2.21) is holomorphic on the whole D p u c q if and only if A ij p u q “ O p u i ´ u j q Ñ whenever p u i ´ u j q Ñ for u approaching ∆ . (2.22) In this case, the following holds.(I,1) Y p q p z, u q and the Y ν p z, u q , ν P Z , have analytic continuation on R p C zt uq ˆ D p u c q , so they areholomorphic of u P D p u c q .The coalescence locus ∆ is neither a singularity locus nor a branching locus for the Y ν p z, u q .(I,2) The coefficients of Y F p z, u q are holomorphic of u P D p u c q .(I,3) The fundamental matrix solutions Y ν p z, u q have asymptotics Y ν p z, u q „ Y F p z, u q uniformly in u P D p u c q , for z Ñ 8 in a wide sector p S ν containing S ν p D p u qq , to be defined later in (7.3).(I,4) A p u q is holomorphically similar on D p u c q to a Jordan form J if and only if (2.22) holds. Similarityis realized by a fundamental matrix solution of (2.17), which exists holomorphic on the whole D p u c q . Part II.
Assume that A p u q satisfies the vanishing conditions (2.22). Then,(II,1) the essential monodromy data S ν , S ν ` µ , B “ diag p A p u c qq , C ν , L , D , initially defined on D p u q byrelations (2.11)-(2.13), are well defined and constant on the whole D p u c q . They satisfy S ν “ ˚ S ν , S ν ` µ “ ˚ S ν ` µ , L “ ˚ L, C ν “ ˚ C ν , D “ ˚ D, where(II,2) ˚ S ν , ˚ S ν ` µ are the Stokes matrices of fundamental solutions ˚ Y ν p z q , ˚ Y ν ` µ p z q , ˚ Y ν ` µ p z q of (2.18)having asymptotic behaviour ˚ Y F p z q “ Y F p z, u c q , for z Ñ 8 respectively on sectors τ ν ´ π ă arg z ă τ ν ` , τ ν ă arg z ă τ ν ` µ ` and τ ν ` µ ă arg z ă τ ν ` µ ` ; II,3) ˚ L , ˚ D are the exponents of a fundamental solution ˚ Y p z q “ ˚ G ´ I ` ř j “ ˚Ψ j z j ¯ z ˚ D z ˚ L of (2.18) inLevelt form;(II,4) ˚ C ν connects ˚ Y ν p z q “ ˚ Y p z q ˚ C ν .(II,5) The Stokes matrices satisfy the vanishing condition p S ν q ij “ p S ν q ji “ , p S ν ` µ q ij “ p S ν ` µ q ji “ @ ď i ‰ j ď n such that u ci “ u cj . Corollary 2.1 ([11]) . If A ii ´ A jj R Z zt u , then there the formal solution ˚ Y F p z q of (2.18) is unique andcoincides with “ Y F p z, u c q . By the above corollary and (II,1) , if A ii ´ A jj R Z zt u , in order to obtain the essential monodromydata of (1.1), it suffices to compute ˚ S ν , ˚ S ν ` µ , ˚ L , ˚ C ν and ˚ D for (2.18). Since A ij p u c q “ i, j suchthat u ci “ u cj , (2.18) is simper than (1.1). This may allow to explicitly compute monodromy data. Animportant example with algebro-geometric implications can be found in [12]. Remark 2.1.
The difficulty in proving Theorem 2.2 is the analysis of the Stokes phenomenon at z “ 8 .On the other hand, coalescences does not affect the analysis at the Fuchsian singularity z “
0, so it is notan issue for the proof of the statements concerning Y p q p z, u q , L , D and C ν (as far as the contributionof Y p q is concerned). See Proposition 17.1 of [11], and the proof of Theorem 4.9 in [21]. For this reason,in the present paper we will not deal with Y p q p z, u q , L , D , C ν and (II,3)-(II,4) above.In Theorem 7.1 we introduce an isomonodromic Laplace transform in order to prove the statementsof Theorem 2.2 above, concerning the Stokes phenomenon , namely (I,1), (I,2), (I,3) and (II,1), (II,2),(II,5) . Also point (I,4) will be proved in Section 4, Remark 4.2. In this section, we fix u P D p u c qz ∆. Accordinly, system (1.1) is to be considered as a system not dependingon deformation parameters , with leading matrix Λ having pairwise distinct eigenvalues , and system (1.3)is equivalent to (1.2), which does not depend on parameters. For simplicity of notations, let us fix forexample u “ u , as in Section 2.1.1 . Solutions Y ν p z q of (1.1) with canonical asymptotics Y F p z q ( u “ u fixed is not indicated) can beexpressed in terms of convergent Laplace-type integrals [5, 26], where the integrands are solutions of theFuchsian system p Λ ´ λ q d Ψ dλ “ p A ` I q Ψ , I : “ identity matrix (2.23)Indeed, let ~ Ψ p λ q be a vector valued function and define ~Y p z q “ ż γ e λz ~ Ψ p λ q dλ, The notation A and A is used in [20] for Λ and A . In [4] the notation for Λ is the same, while A is denoted by A .The notation λ , ..., λ n is used in [4, 21] for u , ..., u n . There is a misprint in the first page of [20] where it is said that A P GL p n, C q ; the correct statement is A P Mat p n, C q . γ is a suitable path. Then, substituting into (1.1), we have p z Λ ` A q ż γ e λz ~ Ψ p λ q dλ “ z ddz ż γ e λz ~ Ψ p λ q dλ “ z ż γ λe λz ~ Ψ p λ q dλ. This implies that A ż γ e λz ~ Ψ p λ q dλ “ ż γ d p e λz q dλ p λ ´ Λ q ~ Ψ p λ q dλ ““ e λz p λ ´ Λ q ~ Ψ p λ q ˇˇˇ γ ´ ż γ e λz « p λ ´ Λ q d~ Ψ p λ q dλ ` ~ Ψ p λ q ff dλ. (2.24)If γ is such that e λz p λ ´ Λ q ~ Ψ p λ q ˇˇˇ γ “
0, and if the function ~ Ψ p λ q solves (2.23), then ~Y p z q solves (1.1).Multiplying to the left by p Λ ´ λ q ´ , system (2.23) becomes (1.2), d Ψ dλ “ n ÿ k “ B k λ ´ u k Ψ , B k : “ ´ E k p A ` I q . (2.25)A fundamental matrix solution is multivalued in C zt u , ..., u n u . Following [4], we fix branch cuts L k “ L k p η p q q oriented from u k to L k p η p q q : “ t λ P R p C zt u , ..., u n uq | arg p λ ´ u k q “ η p q u , ď k ď n, where η p q P R is an admissible direction in the λ -plane (admissible for u ) η p q ‰ arg p u j ´ u k q mod π, for all 1 ď j, k ď n. The admissibility condition means that a cut L k does not contain another pole u j , j ‰ k . See figure 2.This construction selects a sheet of R p C zt u , ..., u n uq ), which is (notations as in [4] and [20]) P η p q : “ ! λ P R p C zt u , ..., u n uq | η p q ´ π ă arg p λ ´ u k q ă η p q , ď k ď n ) . (2.26)Stokes matrices for (1.1), for fixed and pairwise distinct u , ..., u n , can been expressed in termsofconnection coefficients of selected solutions of (2.25). The explicit relations have been obtained in[4] for the generic case when all λ , ..., λ n R Z ; and in [20] for the general case with no restrictions on λ , ..., λ n and A . Selected Vector Solutions
The Laplace transform involves three types of vector solutions or (2.25), denoted in [20] respectively by ~ Ψ k p λ q , ~ Ψ ˚ k p λ q and ~ Ψ p sing q k p λ q , for k “ , ..., n (in [4] the notation used is Y k and Y ˚ k , while Y p sing q k doesnot appear, since it reduces to Y k in the generic case λ k R Z ). We will not describe here the ~ Ψ ˚ k p λ q ,which play mostly a technical role. Let N “ t , , , ... u integers , Z ´ “ t´ , ´ , ´ , ... u negative integers ,~e k “ standard k -th unit column vector in C n . It is proved in [20] that there are at least n ´ λ “ u k . The remaining independent solution is singular at λ “ u k , except for some exceptional cases possibly13ccurring when λ k ď ´ n independent solutions holomorphic at λ “ u k ). The selected vector solutions ~ Ψ k are obtained as follows.• If λ k ď ´ u k ,then ~ Ψ k is the unique analytic solution with the following normalization: ~ Ψ k p λ q “ ˜ p´ q λ k p´ λ k ´ q ! ~e k ` ÿ l ě ~b p k q l p λ ´ u k q l ¸ p λ ´ u k q ´ λ k ´ . • In all other cases, there is a solution ~ Ψ p sing q k , singular at λ “ u k . This is determined up to amultiplicative factor and the addition of an arbitrary linear combination of the remaining n ´ λ “ u k solutions, denoted below with reg p λ ´ u k q . In [20], it has the following structure ~ Ψ p sing q k p λ q “ $’’’’’’&’’’’’’% ~ψ k p λ qp λ ´ u k q ´ λ k ´ ` reg p λ ´ u k q , λ k R Z ,~ψ k p λ q ln p λ ´ u k q ` reg p λ ´ u k q , λ k P Z ´ ,P k p λ qp λ ´ u k q λ k ` ` ~ψ k p λ q ln p λ ´ u k q ` reg p λ ´ u k q , λ k P N . (2.27)Here ~ψ k p λ q is analytic at u k and P k p λ q “ ř λ k l “ b p k q l p λ ´ u k q l is a polynomial of degree λ k . Wechoose the following normalization at λ “ u k $’’’’’’’&’’’’’’’% ~ψ k p λ q “ Γ p λ k ` q ~e k ` ř l ě ~b p k q l p λ ´ u k q l , λ k R Z ,~ψ k p λ q “ ˜ p´ q λ k p´ λ k ´ q ! ~e k ` ř l ě ~b p k q l p λ ´ u k q l ¸ p λ ´ u k q ´ λ k ´ λ k P Z ´ ,P k p λ q “ λ k ! ~e k ` O p λ ´ u k q λ k P N , The coefficients ~b p k q l P C n are uniquely determined by the normalization. Then the selected vectorsolutions ~ Ψ k are uniquely defined by ~ Ψ k p λ q : “ ~ψ k p λ qp λ ´ u k q ´ λ k ´ for λ k R Z ; ~ Ψ k p λ q : “ ~ψ k p λ q for λ k P Z . (2.28)In case λ k P N , depending on the system, it may exceptionally happen that ~ Ψ k : “ ~ψ k ” Connection Coefficients
Above, the behaviour of ~ Ψ k p λ q has been described at λ “ u k . The behaviour at any point λ “ u j , for j “ , ..., n , will be expressed by the connection relations ~ Ψ k p λ q “ ~ Ψ p sing q j p λ q c jk ` reg p λ ´ u j q . (2.29) c jk : “ , @ k “ , ..., n, when ~ Ψ p sing q j p λ q ” λ j P ´ N ´ . The above relations define the connection coefficients c jk . From the definition, we see that c kk “ λ k R Z , while c kk “ λ k P Z . In case λ k P N , if it happens that ~ Ψ k ”
0, then c jk “ j “ , .., n . Such cases never occur if none of the eigenvalues of A is a negative integer. The singular part of Ψ p sing q is uniquely determined by the normalization, but not Ψ p sing q itself, because the analyticadditive term reg p λ ´ u k q is an arbitrary linear combination of the remaining n ´ n
Let u “ u be fixed so that Λ p u q has pairwise distinct eigenvalues. Let η p q and τ p q “ π { ´ η p q be admissible for u in the λ -plane and z -plane respectively. Suppose that the labellingof Stokes rays is (2.3) and (2.31). Then, the Stokes matrices of system (1.1) are given in terms of theconnection coefficients c p ν q jk of system (2.25), according to the following formulae ` S ν ˘ jk “ $’’’’&’’’’% e πiλ k α k c p ν q jk for j ă k, for j “ k, for j ą k, ` S ´ ν ` µ p q ˘ jk “ $’’’’&’’’’% for j ă k, for j “ k, ´ e πi p λ k ´ λ j q α k c p ν q jk for j ą k. where, α k : “ p e ´ πiλ k ´ q if λ k R Z ; α k : “ πi if λ k P Z . l In the above discussion, the differential systems do not depend on parametersd ( u is fixed). Thepurpose of the present paper is to extend the description of Background 2 to the case depending on de-formation parameters and include coalescences in D p u c q , and then to obtain Theorem 2.2 of Background1 in terms of an isomonodromic Laplace transform. The first step in our construction is Proposition 3.1 below, establishing the equivalence between strongisomonodromy deformations of systems (1.1) and (1.3), for u varying in a τ -cell of D p u c q . In the specific The key point is the fact that ~ Ψ p sing q k in (7.5), or equivalently ~ Ψ k for λ , ..., λ n R Z , can be substituted by anotherset of vector solutions, denoted in [20] by ~ Ψ ˚ k p λ, u | ν q and in [4] by Y ˚ k . The effect of the change of the branch cut from η ν ` ă η ă η ν to η ν ` µ ` ă η ă η ν ` µ can be relatively easily analysed for the ~ Ψ ˚ k p λ, u | ν q , and yields a linear relation ~ Ψ ˚ k p λ, u | ν ` µ q “ ~ Ψ ˚ k p λ, u | ν q C ` ν , where the connection matrix C ` ν is expressed in terms of the connection coefficients c p ν q jk relative to ~ Ψ p sing q k p λ, u | ν q . The same can be done for the change of branch cut from η ν ` µ ` ă η ă η ν ` µ to η ν ` µ ` ă η ă η ν ` µ , yielding a relation ~ Ψ ˚ k p λ, u | ν ` µ q “ ~ Ψ ˚ k p λ, u | ν ` µ q C ´ ν (please, refer to [20] for notations anddetail, especially see section 7 there). Substituting these relations in the Laplace integrals, we obtain the statement, with S ν “ C ` ν and S ´ ν ` µ “ C ´ ν . D p u q contained ina τ -cell of D p u c q if and only if dA “ n ÿ j “ r ω j p u q , A s du j , ω j p u q “ r F p u q , E j s . (3.1)On the other hand, system (1.3) is strongly isomonodromic in D p u q , by definition, when fundamentalmatrix solutions in Levelt form at each pole λ “ u j , j “ , ..., n , have constant monodromy exponents and are related to each other by constant connection matrices (see [21] for this definition, especiallyAppendix A). From the results of [7, 8, 21], the necessary and sufficient condition for the deformationto be strongly isomonodromic (this can also be taken as the definition) is that (1.3) is the λ -componentof a Frobenius integrable Pfaffian system with the following structure d Ψ “ P p λ, u q Ψ , P p z, u q “ n ÿ k “ B k p u q λ ´ u k d p λ ´ u k q ` n ÿ k “ γ k p u q du k . (3.2)The integrability condition dP “ P ^ P is the non-normalized Schlesinger system (see Appendix A and[6, 7, 8, 21, 22, 44]) B i γ k ´ B k γ i “ γ i γ k ´ γ k γ i , (3.3) B i B k “ r B i , B k s u i ´ u k ` r γ i , B k s , i ‰ k (3.4) B i B i “ ´ ÿ k ‰ i r B i , B k s u i ´ u k ` r γ i , B i s (3.5) Proposition 3.1.
Let ω j p u q “ r F , E j s , j “ , .., n , where F p u q is given in (2.8). Then, (3.1) isequivalent to (3.3)-(3.5) if and only if γ j p u q “ ω j p u q , j “ , ..., n. Namely, (1.1) is stronlgy isomonodromic in a polydisc on D p u q contained in a τ -cell if and only if (1.3)is strongly isomonodromic.Proof. See Appendix B. D p u c q and Vanishing Conditions In this section, Proposition 4.1, we holomorphically extend to D p u c q the non-normalized Schlesingersystem associated with (1.3), when certain vanishing conditions (4.4) are satisfied. This is the secondstep to obtain the results of [11] by Laplace transform.To start the discussion, we do not need to require that B j “ ´ E j p A ` I q . Consider a matrix G p u q holomorphically invertible on a polydisc D p u q contained in a τ -cell. It is straightforward to see that γ j p u q “ B j G p u q ¨ G p u q ´ , j “ , ..., n, (4.1) As already mentioned when stating Theorem 2.1, equations dA “ r ω i p u q , A s and ω i p u q “ r F , E i s for i “ , ..., n areexactly the the Frobenius integrability conditions of (2.14) when (1.1) is strongly isomomodromic [11].
17s a solution of (3.3). Let B , ..., B n be solutions to the non-normalized Schlesinger system (3.4)-(3.5)on D p u q (or possibly on a smaller neighbourhood of u ), with the above γ j . We make the followingassumptions.(i) G p u q has analytic continuation, and is holomorphically invertible, on the whole D p u c q , so that the γ j p u q are analytic on the whole D p u c q . Equivalently, the Pfaffian system dG “ n ÿ j “ γ j p u q du j G (4.2)has coefficients γ j p u q holomorphic on D p u c q and is Frobenius integrable there (namely, equations(3.3) have holomorphic solution γ j on D p u c q ).(ii) B p u q , ..., B n p u q have analytic continuation on the whole D p u c q as holomorphic matrix valuedfunctions (we mean continuation as functions, not as solutions of (3.4)-(3.5)). Remark 4.1.
The equivalence in assumption (i) is proved as follows. If there is a G p u q holomorphicallyinvertible on the whole D p u c q and we define γ j by (4.1), so that (3.3) are automatically satisfied, then G p u q satisfies (4.2) by definition. Conversely, if (4.2) is given with holomorphic on D p u c q coefficients γ j satisfying (3.3), then both dG “ ř j γ j du j G and d p G ´ q “ ´ G ´ ř j γ j du j are integrable in D p u c q . Sincethey are linear Pfaffian systems with holomorphic coefficients, there is a fundamental matrix solution G p u q analytic on the whole D p u c q . Lemma 4.1.
With the assumptions (i), (ii) above, B ,..., B n are holomorphic solutions to (3.4)-(3.5)on the whole D p u c q if and only if r B i p u q , B j p u qs ÝÑ , whenever u i ´ u j Ñ in D p u c q . (4.3) Namely, (3.2) is Frobenius integrable with holomorphic coefficients on the whole D p u c q if and only if(4.3) holds.Proof. If B ,..., B n are holomorphic solutions to (3.4)-(3.5) on D p u c q , then in (3.4) the term r B i , B k s must holomorphically vanish at ∆ Ă D p u c q . Conversely, let B , ..., B n satisfy (3.4)-(3.5) on D p u q andbe holomorphic on D p u c q . If (4.3) holds, then (3.4)-(3.5) hold true holomorphically on D p u c q Now, we specify to the case when B j “ ´ E j p A ` I q . Lemma 4.2.
Let A p u q be holomorphic on D p u c q and B j p u q : “ ´ E j p A p u q ` I q , j “ , ..., n . Then (4.3)holds if and only if ` A p u q ˘ ij ÝÑ , for u i ´ u j Ñ in D p u c q . (4.4) Moreover, the matrices ω j p u q “ r F p u q , E j s are holomorphic on D p u c q if and only if (4.4) holds.Proof. Let u ˚ P ∆, so that for some i ‰ j it occurs that u ˚ i “ u ˚ j . Since B j “ ´ E j p A ` I q “ ¨˚˚˚˚˚˚˝ ... ... ... ´ A j ¨ ¨ ¨ ´ A j,j ´ ´ λ j ´ ´ A j,j ` ¨ ¨ ¨ ´ A jn ... ... ... ˛‹‹‹‹‹‹‚ . (4.5)18t is an elementary computation to check the equivalence between the relation r B i p u ˚ q , B j p u ˚ qs “ p A p u ˚ qq ij “
0. Also the statement regarding analyticity of r F p u q , E j s is straightforward. Proposition 4.1.
Consider a Frobenius integrable Pfaffian system (3.2) on D p u q with B j p u q “ ´ E j p A p u q ` I q and γ j p u q ” ω j p u q “ r F p u q , E j s . (4.6) Assume that A p u q is holomorphic on the whole D p u c q . Then, the system is Frobenius integrable on D p u c q with holomorphic matrix coefficients, namely the non-normalized Schlesinger system (3.3)-(3.5)has holomorphic solution of the form (4.6) on the whole D p u c q , if and only if the vanishing conditions(4.4) hold.Proof. Since A p u q is holomorphic on D p u c q , assumption (ii) holds. By assumption, the Pfaffian systemwith coefficients (4.6) satisfies Proposition 3.1, so that γ j “ ω j is solution of (3.3). Assumption (i) holdsif and only if the ω j p u q are holomorphic on D p u c q , and this in turn holds if and only if the conditions(4.4) hold, by Lemma 4.2. Therefore Lemma 4.1 holds. Remark 4.2 (Proof of point (I,4) of Theorem 2.2) . As a corollary of Lemma 4.1 we receive the following.With the assumptions (i), (ii), if conditions (4.3) hold, then ř nk “ B k p u q is holomorphically similar to aconstant Jordan form J on the whole D p u c q , the equivalence being realised by G p u q , namely G p u q ´ n ÿ k “ B k p u q G p u q “ J. Indeed, if γ j p u q “ B j G p u q ¨ G p u q ´ , then ř nk “ B k p u q is holomorphically equivalent to its Jordan formon D p u q , as it follows from (10.5)-(10.7) in the proof of Proposition 3.1 (see Appendix B). Moreover, G p u q is holomorphically invertible on D p u c q by assumption (i). If (ii) and if (4.3) hold, by Lemma 4.1 B , ..., B n extend as holomorphic solutions to (3.4)-(3.5) on D p u c q . Thus, proceeding as in (10.5)-(10.7),we see that G p u q ´ ř k B k p u q G p u q “ J on the whole D p u c q .If follows from the above, from Lemma 4.2 and Proposition 3.1 that if system (1.1) is strongly isomon-odromic on D p u q , and if A p u q is holomorphic on D p u c q , then A p u q “ ´ ř k B k ´ I is holomorphicallysimilar in D p u c q to a constant Jordan form if and only if (4.4) holds. The similarity is realised by afundamental matrix solution of dG “ p ř nj “ ω j p u q du j q G . This proves Proposition 19.2 of [11] and point (I,4) of Theorem 2.2. u P D p u c q In this section we prove one main result of the paper, Theorem 5.1 below. It introduces solutions ofthe the Pfaffian system (3.2), which are the isomonodromic analogue of the selected and singular vectorsolutions introduced in Background 2, Section 2.2, namely in [20]. This is the third step required toobtain the results of [11] by Laplace transform.Preliminary, we need to characterise the radius (cid:15) ą D p u c q in (2.1). The coalescencepoint u c “ p u c , ..., u cn q contains s ă n distinct values, say λ , ..., λ s , with algebraic multiplicities p , ..., p s respectively ( p ` ¨ ¨ ¨ ` p s “ n ). Suppose that arg z “ τ is an admissible direction at u c , as definedin (2.19), and let η “ π { ´ τ ↵
Remark 5.1.
For λ i R Z ´ , the singular solution ~ Ψ p sing q i is unique, identified by its singular behaviour at λ “ u i and the normalization (5.5)-(5.6) when λ i P C z Z , and by the normalization (5.11) when λ i P N .For λ i P Z ´ , a singular solution in (5.8) is not unique, but its singular behaviour (5.9) at λ “ u i isuniquely fixed by the normalization (5.5)-(5.6). There is a freedom due to the choice of the coefficients r m and the ~φ i in (5.8). See Remark 6.3 for more details.The singular behaviour of ~ Ψ k at λ “ u j is expressed by connection coefficients. Definition 5.1.
The connection coefficients are defined by ~ Ψ k p λ, u | ν q “ λ Ñ u j ~ Ψ p sing q j p λ, u | ν q c p ν q jk ` reg p λ ´ u j q , λ P P η , (5.12) and by c p ν q jk : “ , @ k “ , ..., n, when ~ Ψ p sing q j ” , possibly occurring for λ j P ´ N ´ . (5.13)The uniqueness of the singular behaviour of ~ Ψ p sing q j at λ “ u j implies that the c jk are uniquelydefined . From the definition, we see that ‚ If λ k R Z , c p ν q kk “ ‚ If λ k P Z , c p ν q kk “ ‚ If λ k P N and ~ Ψ k p λ, u | ν q ”
0, then c p ν q k “ c p ν q k “ ¨ ¨ ¨ “ c p ν q nk “ ‚ If λ j P ´ N ´ ~ Ψ p sing q j p λ, u | ν q ”
0, then c p ν q j “ c p ν q j “ ¨ ¨ ¨ “ c p ν q jn “ Corollary 5.1.
The coefficients in (5.12)-(5.13) are isomonodromic connection coefficients , namelythey are independent of u P D p u c q . They satisfy the vanishing relations c p ν q jk “ for j ‰ k such that u cj “ u ck . (5.14) Proof.
See Section 6.7.
Point (1) of the statement is straightforward, because Proposition 4.1 holds under the assumptions inthe theorem. We prove points (2) and (3), constructing the selected vector solutions.
Remark on notations:
We are dealing with functions, say f “ f p λ, u | ν q , defined on P η p u q ˆ ˆ D p u c q ,but for simplicity we will omit ν in all formulae, writing f “ f p λ, u q , and c jk in place of c p ν q jk . Without loss of generality, we order the eigenvalues so that u c “ ¨ ¨ ¨ “ u cp “ λ ; u cp ` “ ¨ ¨ ¨ “ u cp ` p “ λ ; (6.1) In this way, D p u c q “ D ˆ p ˆ ¨ ¨ ¨ ˆ D ˆ p s s , where D α “ t x P C | | x ´ λ α | ď (cid:15) u , α “ , ..., s . cp ` p ` “ ¨ ¨ ¨ “ u cp ` p ` p “ λ ; ..... up to u cp `¨¨¨` p s ´ ` “ ¨ ¨ ¨ “ u cp `¨¨¨` p s ´ ` p s “ λ s . (6.2)We will analyse first the coalescence of u , ..., u p to λ . Other cases are analogous. We changevariables p u , ..., u n , λ q ÞÑ p x , ..., x n ` q as follows x n ` “ λ ´ λ , x j “ λ ´ u j , ď j ď p ; u j ´ λ , p ` ď j ď n. The inverse transformation is λ “ x n ` ` λ , u j “ x n ` ´ x j ` λ , ď j ď p ,x j ` λ , p ` ď j ď n. Let x : “ p x , ..., x p loooomoooon p , x p ` , ...., x n loooooomoooooon n ´ p , x n ` q ” p x , ..., x p loooomoooon p , x , x n ` q , where x : “ p x p ` , ...., x n q . We are interested in the behaviour of solutions for x ÝÑ p , , ..., loooomoooon p , x , q , corresponding to u Ñ λ , . . . , u p Ñ λ , and λ Ñ λ namely u i ´ u j Ñ i ‰ j and λ ´ u i Ñ
0, for i, j
P t , ..., p u . The Pfaffian system (5.3) in variables x , with Fuchsian singularities at x “ , . . . , x p “
0, becomes d Ψ “ P p x q Ψ , P p x q “ p ÿ j “ P j p x q x j dx j ` n ` ÿ j “ p ` p P j p x q dx j (6.3)where P j p x q x j “ B j p x q x j ´ γ j p x q , ď j ď p , p P j p x q “ B j p x q x j ´ x n ` ` γ j p x q , p ` ď j ď n, p P n ` p x q “ n ÿ j “ p ` B j p x q x n ` ´ x j ` p ÿ j “ γ j p x q Since Proposition 4.1 holds, the Pfaffian system is integrable with holomorphic in D p u c q coefficients B p u q , . . . , B n p u q and γ p u q , . . . , γ n p u q . Therefore P p x q , . . . , P p p x q and p P p ` p x q , . . . , p P n ` p x q are holo-morphic at p , . . . , loomoon p , x , q , for x varying as u p ` , . . . , u n vary in D p u c q . Remark 6.1.
The commutation relations (4.3) at u “ p λ , . . . , λ loooomoooon p , u q , where u : “ p u p ` , . . . , u n q , are r B i p λ , . . . , λ , u q , B j p λ , . . . , λ , u qs “ , ď i ‰ j ď p . (6.4)They also follow from the integrability condition dP p x q “ P p x q ^ P p x q of (6.3), which implies BB x i ˆ P j x j ˙ ´ BB x j ˆ P i x i ˙ ´ P i P j ´ P j P i x i x j “ , ď i ‰ j ď p . k “ p k , ..., k p q , and write ˆ l ď ˆ k if k i ď l i for all i P t , ..., p u . We write a Taylor convergent series P i p x q “ ÿ k `¨¨¨` k p ě P i, ˆ k p x , x n ` q x k ¨ ¨ ¨ x k p p , with coefficients P i, k p x , x n ` q holomorphic of x , x n ` . The integrability condition becomes [44] k j P i, ˆ k ´ k i P j, ˆ k ` ÿ ď ˆ l ď ˆ k r P i, ˆ l , P j, ˆ k ´ ˆ l s “ , ď i ‰ j ď p . (6.5)In particular, for ˆ k “ ˆ , we have that P i, ˆ p x , x n ` q “ B i p λ , . . . , λ loooomoooon p , u q , so that (6.5) reduces to (6.4). Lemma 6.1.
Let assumptions (i), (ii) of Lemma 4.1 hold and let the vanishing conditions (4.3) hold,so that the γ j and B j , j “ , ..., n , are holomorphic solutions of the non-normalized Schlesinger systemon the whole D p u c q . Then, the following holds.1) Every B j p u q is holomorphically similar to a constant Jordan form on D p u c q , namely there is aholomorphically invertible matrix G p j q p u q such that ` G p j q p u q ˘ ´ B j p u q G p j q p u q is Jordan and con-stant.2) If u ˚ P ∆ is such that u ˚ i “ u ˚ j for some i ‰ j , the corresponding B i p u ˚ q and B j p u ˚ q are simulta-neously reducible to triangular form,3) In case B j p u q “ ´ E j p A p u q ` I q , ď j ď n , the Jordan form at item 1) is ` G p j q p u q ˘ ´ B j p u q G p j q p u q “ p T p j q : “ diag p , . . . , , ´ ´ λ j , , . . . , q , λ j ‰ ´ ,J p j q “ Jordan form (6.9) , λ j “ ´ , (6.6) In diag p , . . . , , ´ ´ λ j , , . . . , q all entries are zero, except for the entries ´ ´ λ j in position j .In J p j q all entries are zero, except for one entry equal to 1, that can be taken to be on the j -th rowand on a column at position m j ě j ` .The simultaneous triangolar forms of B i p u ˚ q and B j p u ˚ q at item 2) coincide with p T p i q and p T p j q .Proof.
1) For every j “ , ..., n , the Schlesinger system (3.3)-(3.5) implies the Frobenius integrability on D p u q of the the linear Pfaffian system (see Corollary 9.1, Appendix A) B G p j q B u k “ ˆ B k u k ´ u j ` γ k ˙ G p j q , k ‰ j, B G p j q B u j “ ´ ÿ k ‰ j ˆ B k u k ´ u j ` γ k ˙ G p j q (6.7)From (3.4)-(3.5) and the above we receive B k ` p G p j q q ´ B j G p j q ˘ “ k “ , ..., n , for a fundamentalmatrix solution G p j q p u q . Thus, up multiplication G p j q ÞÑ G p j q G p j q , G p j q P GL p n, C q , we can choose G p j q p u q which holomorphically puts B j in constant Jordan form. If moreover (4.3) holds, the solutionsto the Schlesinger system B j p u q extend analytically on D p u c q , the coefficients of the linear system (6.7)are holomorphic on D p u c q , and so is for G p j q p u q .Simultaneous triangularization in item 2) for commuting matrices is a standard result.25f we consider each B j separately, it is straightforward that the Jordan forms are p T p j q in item 3). It remains to show that the simultaneous reduction to triangular form is again realized by the matrices p T p j q . Without loss in generality, let u ˚ “ p λ , . . . , λ loooomoooon p , u q (here u “ p u p ` , ..., u n q is allowed to vary).An elementary computation shows that B p u ˚ q , ..., B p p u ˚ q are reducible to p T p q , ...., p T p q simultaneously,because only the j -th row of B j p u ˚ q is non-zero, and by (4.3) the first p entries of this row are zero,except for the p j, j q -entry equal to ´ λ j ´ Namely, B j p u ˚ q “ ¨˚˚˚˚˚˚˝ ¨ ¨ ¨ ... ... ´ λ j ´ ´ A p j q j,p ` p u ˚ q ¨ ¨ ¨ ´ A j,n p u ˚ q ... ¨ ¨ ¨ ˛‹‹‹‹‹‹‚ ÐÝ row j. Corollary 6.1.
In Lemma 6.1, point 3), if u ˚ “ u c , then B p u c q , ..., B p p u c q are reducible simultaneouslyto their respective Jordan forms (6.6), B p ` p u c q , ..., B p ` p p u c q are reducible simultaneously to theirrespective Jordan forms, and so on up to B p ` ... ` p s ´ ` p u c q , ..., B p ` ... ` p s p u c q . Recall that we are considering coalescence of u , ..., u p to λ . We can label u , ..., u p so that λ j P C z Z , for 1 ď j ď q , λ j P Z , for q ` ď j ď p . If all λ j P Z , then q “
0, if all λ j R Z , then q “ p . By the above corollary at u ˚ “ u c , we simultaneouslyreduce B p u c q , ..., B p p u c q to the forms p T p j q , with p T p j q “ diag p , . . . , , ´ ´ λ j looomooon position j, , . . . , q , for λ j ‰ ´ . (6.8) p T p j q “ J p j q : “ ¨˚˚˚˚˚˚˚˚˝ ¨ ¨ ¨ ... . . . ... ¨ ¨ ¨ ¨ ¨ ¨ r p j q m j ... . . . ... ¨ ¨ ¨ ˛‹‹‹‹‹‹‹‹‚ ÐÝ row j, for λ j “ ´ , (6.9) r p j q m j : “ , is the only non-zero entry in position p j, m j q , with m j ě p ` . We will put the non-zero entry r p j q m j “ m j -th column, with m j ě p `
1, differently from theusual convention to put it in the column j ` p : “ p , ..., p q . The first and fundamental step to achieve Theorem 5.1 is the following It is also elementary to find a holomorphic G p k q explicitly. For example, if all B k p u q are diagonalizable (i.e λ k ‰ ´ p G p k q p u qq ´ B k p u q G p k q p u q “ p T p k q , k “ , , ..., n, , where the columns of G p k q areas follows: k -th column is multiple of ~e k P C n ; l -th column, l ‰ k , is multiple of ~e l ´ A kl p u q λ k ` ~e k . For example, in case of the previous footnote, the simultaneous reductuion to Jordan form is realized by G p q p u ˚ q ¨ ¨ ¨ G p p q p u ˚ q , which depends holomorphically on u heorem 6.1. The Paffian system (5.3) admits the following fundamental matrix solution Ψ p p q p λ, u q “ G p p q U p p q p λ, u q ¨ p ź l “ p λ ´ u l q p T p l q ¨ p ź j “ q ` p λ ´ u j q p R p j q , (6.10) where G p p q is a constant invertible matrix simultaneously reducing B p u c q , ..., B p p u c q to p T p q , ..., p T p p q in (6.8)-(6.9), and U p p q p λ, u q “ I `` ÿ k ą , k ` ... ` k p ě ” U p p q k ¨ p u p ` ´ u cp ` q k p ` ¨ ¨ ¨ p u n ´ u cn q k n p λ ´ λ q k n ` ı p λ ´ u q k ¨ ¨ ¨ p λ ´ u p q k p , is a matrix function holomorphic in D ˆ D p u c q . Here k : “ p k , ..., k n , k n ` q , k j ě , and k ą means that at least one k j ą ( j “ , ..., n ` ). The matrices U p p q k are constant. The exponents p R p q ` q , . . . , p R p p q are the following constant nilpotent matrices. • If λ j “ ´ , then p R p j q “ . (6.11)• If λ j P N “ t , , , ... u , only the entries p R p j q mj “ : r p j q m , for m “ , ..., n and m ‰ j , are possibly nonzero, namely p R p j q “ « ~ ˇˇˇˇˇ ¨ ¨ ¨ ˇˇˇˇˇ ~ ˇˇˇˇˇ n ÿ m ‰ j,m “ r p j q m ~e m ˇˇˇˇˇ ~ ˇˇˇˇˇ ¨ ¨ ¨ ˇˇˇˇˇ ~ ff , (6.12) where the possibly non-zero entries are on the j -th column. • If λ j P ´ N ´ “ t´ , ´ , ... u , only the entries p R p j q jm “ : r p j q m , for m “ , ..., n and m ‰ j , arepossibly non zero, namely p R p j q “ ¨˚˚˚˚˚˚˝ ¨ ¨ ¨ ¨ ¨ ¨ ... ...r p j q ¨ ¨ ¨ r p j q j ´ r p j q j ` ¨ ¨ ¨ r p j q n ... ... ¨ ¨ ¨ ¨ ¨ ¨ ˛‹‹‹‹‹‹‚ ÐÝ row j is possibly non zero . (6.13) The exponents p T p l q and R p j q satisfy the following commutation relations r p T p i q , p T p j q s “ , i, j “ , ..., p ; (6.14) r p R p j q , p R p k q s “ , r p T p i q , p R p j q s “ , i “ , ..., p , i ‰ j, j, k “ q ` , ..., p . (6.15) By analytic continuation, Ψ p p q p λ, u q defines an analyic function on the universal covering of P η p u q ˆ ˆ D p u c q . Remark 6.2.
Relations (6.14)-(6.15) imply that some entries of p R p j q must be zero, as in (6.26)-(6.27)below, and the constraints (6.28). Another representaion of (6.10) will be given in (6.25), with exponents(6.21)-(6.22). 27 roof. We apply the results of [44] at the point x “ x c : “ p , , ..., loooomoooon p , x c , q , with x c : “ p x cp ` , ...., x cn q ,corresponding to u “ u c and λ “ λ , where x cj “ u cj ´ λ , j “ p ` , ..., n . By Theorem 7 of [44], thePfaffian system (6.3) admits a fundamental matrix solutionΨ p p q p λ, u q “ U U p x q Z p x q , Z p x q “ p ź j “ x A j l p ź j “ x Q j l , det U ‰ , (6.16)for certain matrices A j which are simultaneous triangular forms of B p u c q , ..., B p p u c q . While in [44] alower triangular form is considered, we equivalently use the upper triangular one. The matrices Q j willbe described below. The matrix U p x q “ V p x q ¨ W p x q has structure V p x q “ I ` ÿ k ą , k p ` ` ... ` k n ` ą V k x k ¨ ¨ ¨ x k p p p x p ` ´ x cp ` q k p ` ¨ ¨ ¨ p x n ´ x cn q k n ¨ x k n ` n ` W p x q “ I ` ÿ k ` ... ` k p ą W k ,...,k p x k ¨ ¨ ¨ x k p p . The constant matrix coefficients V k , W k ,...,k p can be determined [44] from the constant matrix coeffi-cients P i, k in the Taylor expansion of the P j p x q and p P j p x q . Recall that x j “ λ ´ u j , 1 ď j ď p , and x n ` “ λ ´ λ . Moreover, for p ` ď j ď n , we have x j ´ x cj “ p u j ´ λ q ´ p u cj ´ λ q “ u j ´ u cj . Thus,restoring variables p λ, u q , we have V p λ, u q “ I `` ÿ k p ` ` ... ` k n ` ą ” V k p u p ` ´ u cp ` q k p ` ¨ ... ¨ p u n ´ u cn q k n ¨ p λ ´ λ q k n ` ı p λ ´ u q k ¨ ¨ ¨ p λ ´ u p q k p ,W p λ, u , ..., u p q “ I ` ÿ k ` ... ` k p ą W k ,...,k p p λ ´ u q k ¨ ... ¨ p λ ´ u p q k p . Therefore, the matrices appearing in the statement are G p p q : “ U and U p p q p λ, u q : “ V p λ, u q W p λ, u q ,which is holomorphic for p λ, u q P D ˆ D p u c q .We show that the exponents A j and Q j are respectively p T p j q in (6.8)-(6.9) and p R p j q in (6.11)-(6.12)-(6.13). According to [44] (see theorems 2 and 5), the matrix function G p p q ¨ U p p q p λ, u q in (6.10) providesthe gauge transformationΨ “ G p p q ¨ U p p q p λ, u q Z ” in notation of [44] U U p x q Z, which brings (6.3) to the reduced form (being "reduced" is defined in [44]) dZ “ p ÿ j “ Q j p x q x j Z, Q j p x q “ A j ` ÿ p k ą Q p k ,j x k ¨ ¨ ¨ x k p p , Here and below we use the notation p k “ p k , ..., k p q ą
0, meaning least one k l ą
0. From [44], we havethe following. P i p x q “ ÿ k `¨¨¨` k n ` ě P i, k x k ¨ ¨ ¨ x k p p ¨ p x p ` ´ x cp ` q k p ` ¨ ¨ ¨ p x n ´ x cn q k n ¨ x k n ` n ` . and analogous for p P j p x q The A j are simultaneous triangular forms of B p u c q , ..., B p p u c q . Thus, by Lemma 6.1, they can betaken to be A j “ p T p j q as in (6.8)-(6.9), j “ , ..., p . ‚ The Q p k ,j satisfy diag p Q k ,j q “
0, while the entry p α, β q for α ‰ β satisfies p Q p k ,j q αβ ‰ p p T p j q q αα ´ p p T p j q q ββ “ k j ě , for all j “ , ..., p . Taking into account the particular structure (6.8)-(6.9), the above condition can be satisfied only for p k “ p , ..., loomoon q , , ..., , k j , , ..., looooooooomooooooooon p ´ q q , k j “ | λ j ` | ě j, because p p T p j q q αα ´ p p T p j q q ββ “ ´ λ j ´ ě λ j P ´ N ´ α “ j p β ‰ j q , (6.17) p p T p j q q αα ´ p p T p j q q ββ “ λ j ` ě λ j P N and β “ j p α ‰ j q . (6.18)This can occur only for j “ q ` , ..., p . Thus Q p k ,j “ , j “ , ..., q , Q p k ,j “ p R p j q in (6.11)-(6.12)-(6.13) , j “ q ` , ..., p . (6.19)In conclusion, the reduced form turns out to be dZ “ « p ÿ j “ ˜ p T p j q ` p R p j q x k j x j ¸ff Z, p R p q “ ¨ ¨ ¨ “ p R p q q “ . (6.20)Its integrability implies the commutation relations. Indeed, the compatibility B i B j Z “ B j B i Z , i ‰ j ,holds if and only if r p T p j q , p T p i q s x i x j ` r p R p j q , p R p i q s x k i ´ i x k j ´ j ` r p T p j q , p R p i q s x k i ´ i ` r p R p j q , p T p i q s x k j ´ j “ , ď i ‰ j ď p . Keeping into account that p R p q “ ¨ ¨ ¨ “ p R p q q “
0, the above holds if and only if (6.14)-(6.15) hold.The last to be checked is that a fundamental matrix of (6.20) is Z p x q in (6.16), namely Z p x q “ p ź l “ x p T p l q l p ź j “ q ` x p R p j q l . It suffices to verify this by differentiating Z p x q , keeping into account the commutation relations (6.14)-(6.15) and the formula B i x Mi “ p M { x i q x Mi , for a constant matrix M . For i “ , ..., q we receive BB x i Z p x q “ p T p i q x i Z p x q . For i “ q ` , ..., p we receive BB x i Z p x q “ T p i q x i Z p x q ` ´ p ź l “ x p T p l q l ¯ p R p i q x i ´ p ź j “ q ` x p R p j q l ¯ “ p T p i q x i Z p x q ` ´ i ´ ź l “ x p T p l q l ¯ x p T p i q i p R p i q x i ´ p ź l “ i ` x p T p l q l ¯´ p ź j “ q ` x p R p j q l ¯ “ p˚˚q . k i “ | λ i ` | and (6.17)-(6.18), we see that x p T p i q i p R p i q x ´ p T p i q i “ p R p i q x k i i . Therefore, p˚˚q “ p T p i q x i Z p x q ` p R p i q x k i i x i ´ p ź l “ x p T p l q l ¯´ p ź j “ q ` x p R p j q l ¯ “ p T p i q ` p R p i q x k i i x i Z p x q , as we wanted to prove.Finally, the fact that Ψ p p q p λ, u q has analytic continuation on the universal covering of P η p u q ˆ ˆ D p u c q follows from general results in the theory of linear Pfaffian systems [23, 27, 44].It is convenient to introduce a slight change of the exponents. Without loss in generality, we canlabel u , ..., u p in such a way that, for some q , c ě A holds: λ , . . . , λ q P C z Z , λ q ` , . . . , λ q ` c P Z ´ , λ q ` c ` , . . . , λ p P N . Clearly, 0 ď q ď p , 0 ď c ď p and 0 ď q ` c ď p . We define new exponents.• For λ j ‰ ´ T p j q : “ p T p j q , j “ , ..., p ; R p j q : “ p R p j q , j “ q ` , ..., p . (6.21)• For λ j “ ´ j P t q ` , ..., q ` c u ), T p j q : “ , R p j q : “ J p j q loomoon in (6.9) “ ¨˚˚˚˚˚˚˚˚˝ ¨ ¨ ¨ ... . . . ... ¨ ¨ ¨ ¨ ¨ ¨ r p j q m j ... . . . ... ¨ ¨ ¨ ˛‹‹‹‹‹‹‹‹‚ ÐÝ row j, r p j q m j “ . (6.22)Recall that m j ě p ` λ j P ´ N ´ λ j “ ´ Lemma 6.2.
With the definition (6.21)-(6.22), the following relations hold. r T p i q , T p j q s “ , i, j “ , ..., p ; (6.23) r R p j q , R p k q s “ , r T p i q , R p j q s “ , i “ , ..., p , i ‰ j, j, k “ q ` , ..., p , (6.24) Proof.
The equivalence between (6.14)-(6.15) and (6.23)-(6.24) is straightforward.
Corollary 6.2.
In Theorem 6.1, the fundamental matrix solution (6.10) is Ψ p p q p λ, u q “ G p p q ¨ U p p q p λ, u q ¨ p ź l “ p λ ´ u l q T p l q ¨ p ź j “ q ` p λ ´ u j q R p j q , (6.25) where the exponents are defined in (6.21)-(6.22). roof. It is an immediate consequence of the commutation relations being satisfied, that the represen-tation (6.10) for Ψ p p q still holds with the definition (6.21)-(6.22).The commutation relations impose a simplification on the structure of the matrices R p j q . Let thenew convention (6.21)-(6.22) be used. The relations r T p i q , R p j q s “ i “ , ..., p and j “ q ` , ..., p , j ‰ i , imply the vanishing of the first p non-trivial entries of R p j q , so that (by (6.12), (6.13) and (6.22)) R p j q “ ¨˚˚˚˚˚˚˝ ¨ ¨ ¨ ¨ ¨ ¨ ... ... ¨ ¨ ¨ r p j q p ` ¨ ¨ ¨ r p j q n ... ... ¨ ¨ ¨ ¨ ¨ ¨ ˛‹‹‹‹‹‹‚ ÐÝ row j, λ j P Z ´ ; (6.26) R p j q “ « ~ ˇˇˇˇˇ ¨ ¨ ¨ ˇˇˇˇˇ ~ ˇˇˇˇˇ n ÿ m “ p ` r p j q m ~e m ˇˇˇˇˇ ~ ˇˇˇˇˇ ¨ ¨ ¨ ˇˇˇˇˇ ~ ff , λ j P N . (6.27)The relations r R p j q , R p k q s “ j, k P t q ` , . . . , q ` c u or j, k P t q ` c ` , . . . , p u areautomatically satisfied. On the other hand, the commutators r R p j q , R p k q s “ j P t q ` , . . . , q ` c u and k P t q ` c ` , . . . , p u imply the further (quadratic) relations n ÿ m “ p ` r p j q m r p k q m “ . (6.28)In particular, if λ j “ ´ R p j q is (6.22), all the above conditions can be satisfied, provided that wetake m j ě p `
1, as we have agreed from the beginning. ~ Ψ i , part I Remark on notations.
For the sake of the proof, it is more convenient to use a slightly differentnotation with respect to the statement of the theorem. The identifications between objects in the proofand objects in the statement is ~ϕ i ÞÝÑ ~ψ i , r p m q i { r p i q k ÞÝÑ r m and ~ϕ k { r p i q k ÞÝÑ φ i .The selected vector solutions in the statement of Theorem 5.1 are obtained form columns, or certainlinear combinations of columns of the fundamental matrix Ψ p p q in (6.25).The i -th column of an n ˆ n matrix M is M ¨ ~e i (rows by columns multiplication), where ~e i isthe standard unit basic vector in C n . Taking into account (6.23)-(6.24), and (6.26)-(6.27)-(6.28), acomputation yields p ź l “ p λ ´ u l q T p l q ¨ p ź j “ q ` p λ ´ u j q R p j q ¨ ~e i ““ $’’’&’’’% p λ ´ u i q ´ λ i ´ ~e i , i “ , ..., q ` c , λ i P C z N ; p λ ´ u i q ´ λ i ´ ~e i ` ´ř nm “ p ` r p i q m ~e m ¯ ln p λ ´ u i q , i “ q ` c ` , ..., p , λ i P N ; ~e i ` ř q ` c m “ q ` ~e m r p m q i p λ ´ u m q ´ λ m ´ ln p λ ´ u m q , i “ p ` , ..., n. (6.29)31 efinition 6.1. For i “ , ..., n , we define column-vector valued functions ~ϕ i p λ, u q : “ G p p q U p λ, u q ¨ ~e i , i “ , ..., n, (6.30) holomorphic for p λ, u q P D ˆ D p u c q . For i “ , ..., p , we define vector valued functions ~ Ψ i p λ, u q : “ $&% ~ϕ i p λ, u qp λ ´ u i q ´ λ i ´ , i “ , ..., q ` c , λ i P C z N ; ř nk “ p ` r p i q k ~ϕ k p λ, u q , i “ q ` c ` , ..., p , λ i P N . (6.31)They have the following properties.• For i “ , ..., q , ~ Ψ i p λ, u q has a logarithmic singularity at λ “ u i and is regular at the remainingpoints λ “ u j , j “ , ..., p , j ‰ i .• For i “ q ` , ..., q ` c , ~ Ψ i p λ, u q is holomorphic in D ˆ D p u c q , and vanishes at λ “ u i when λ j ď ´ i “ q ` c ` , ..., p , ~ Ψ i p λ, u q is holomorphic in D ˆ D p u c q . It may exceptionally be identicallyzero, namely ~ Ψ i p λ, u q ” , λ i P N , (6.32)if for all k “ p ` , ..., n it happens that r p i q k “ i “ , ..., n , the i -th column of Ψ p p q p λ, u q isΨ p p q p λ, u q ¨ ~e i “ ~ Ψ i p λ, u q , i “ , ..., q ` c , (6.33) “ ~ Ψ i p λ, u q ln p λ ´ u i q ` ~ϕ i p λ, u qp λ ´ u i q λ i ` , i “ q ` c ` , ..., p , (6.34) “ ϕ i p λ, u q ` q ` c ÿ m “ q ` r p m q i ~ Ψ m p λ, u q ln p λ ´ u m q , i “ p ` , ..., n. (6.35) Proposition 6.1.
The ~ Ψ i p λ, u q in (6.31), for i “ , ..., p , are vector solutions (called selected ) of thePfaffian system (5.3). They are linear combinations of columns of Ψ p p q p λ, u q , as follows. ~ Ψ i p λ, u q “ $&% Ψ p p q p λ, u q ¨ ~e i , i “ , ..., q ` c , namely λ i P C z N ;Ψ p p q p λ, u q ¨ ř nk “ p ` r p i q k ~e k , i “ q ` c ` , ..., p , namely λ i P N . (6.36) Those ~ Ψ i p λ, u q which are not identically zero are linearly independent.Proof. For i “ , ..., q ` c , (6.36) is just (6.33), so it is a vector solution of (5.3). In case i “ q ` c ` , ..., p , we claim that ~ Ψ i p λ, u q is the following linear combination ~ Ψ i p λ, u q “ n ÿ k “ p ` r p i q k ´ Ψ p p q p λ, u q ¨ ~e k ¯ , i “ q ` c ` , ..., p , of the vector solutions (6.35), so it is a vector solution of (5.3). Indeed, the above combination is n ÿ k “ p ` r p i q k ´ Ψ p p q p λ, u q ¨ ~e k ¯ “ n ÿ k “ p ` r p i q k ˜ ϕ k p λ, u q ` q ` c ÿ m “ q ` r p m q k ~ Ψ m p λ, u q ln p λ ´ u m q ¸ “ p . q ~ Ψ i p λ, u q ` q ` c ÿ m “ q ` ˜ n ÿ k “ p ` r p i q k r p m q k ¸ ~ Ψ m p λ, u q ln p λ ´ u m q . ř nk “ p ` r p i q k r p m q k “
0, so proving the claim and the expressions (6.36).Linear independence follows form (6.36). ~ Ψ p sing q i , part I Using the previous results, we define singular vector solutions of the Pfaffian system.• For λ i R Z , i.e. i “ , ..., q , ~ Ψ p sing q i p λ, u q : “ ~ Ψ i p λ, u q ” Ψ p p q p λ, u q ¨ ~e i • For λ i P N , i.e. i “ q ` c ` , ..., p , ~ Ψ p sing q i p λ, u q : “ ~ Ψ i p λ, u q ln p λ ´ u i q ` ~ϕ i p λ, u qp λ ´ u i q λ i ` ” Ψ p p q p λ, u q ¨ ~e i . • For λ i P Z ´ , i.e. i “ q ` , ..., q ` c , we distinguish three subcases.i) If λ i ď ´ r p i q k ‰ k P t p ` , ..., n u , from (6.35) (change notation i ÞÑ k ) ~ Ψ p sing q i p λ, u q : “ r p i q k ϕ k p λ, u q ` q ` c ÿ m “ q ` r p m q k ~ Ψ m p λ, u q ln p λ ´ u m q + ” r p i q k Ψ p p q p λ, u q ¨ ~e k . ii) If λ i ď ´ r p i q k “ k P t p ` , ..., n u , ~ Ψ p sing q i p λ, u q : “ λ i “ ´
1, then r p i q m i “ k “ m i , so that ~ Ψ p sing q i p λ, u q “ ~ϕ m i p λ, u q ` ~ Ψ i p λ, u q ln p λ ´ u i q ` q ` c ÿ m ‰ i, m “ q ` r p m q m i ~ Ψ m p λ, u q ln p λ ´ u m q . “ Ψ p p q p λ, u q ¨ ~e m i , m i ě p ` . The above ~ Ψ p sing q i p λ, u q in i) and iii) is singular at u i , but possibly also at u q ` , . . . , u q ` c corre-sponding to λ m P Z ´ . The definition gives the following local behaviour as λ Ñ u i : ~ Ψ p sing q i p λ, u q “ λ Ñ u i ~ Ψ i p λ, u q ln p λ ´ u i q ` reg p λ ´ u i q , i “ q ` , ..., q ` c , (6.37) Remark 6.3.
The definition in i) contains the freedom of choosing k P t p ` , ..., n u , which changes ϕ k p λ, u q and the ratios r p m q k { r p i q k (in formula (5.8), ϕ k { r p i q k is denoted by φ i and r p m q k { r p i q k is r m ).Whatever is the choice of k , provided that r p i q k ‰
0, the behaviour at λ “ u i of the corresponding ~ Ψ p sing q i is always (6.37), so it is uniquely fixed if we fix the normalization of ~ Ψ i p λ, u q .As a consequence of the above definitions and Section 6.2, we receive the following Proposition 6.2.
The ~ Ψ p sing q i p λ, u q defined above, i “ , ..., p , when not identically zero, are linearlyindependent. They are represented as follows ~ Ψ p sing q i p λ, u q “ $’’’&’’’% Ψ p p q p λ, u q ¨ ~e i , λ i P C z Z ´ , Ψ p p q p λ, u q ¨ ~e k r p i q k , λ i P Z ´ , for some k P t p ` , ..., n u such that r p i q k ‰ , λ i P ´ N ´ , if r p i q k “ for all k P t p ` , ..., n u . .4 Local behaviour at λ “ u i , i “ , ..., p In order to proceed in the proof, and in view of the Laplace transform to come, we need local behaviourat λ “ u i . Lemma 6.3.
The following Taylor expansion holds at λ “ u i . ~ Ψ i p λ, u q “ ÿ l “ ~d p i q l p u qp λ ´ u i q l , λ i P N , i.e. i “ q ` c ` , ..., p , with certain vector coefficients d p i q l p u q holomorphic in D p u c q .Proof. By definition in (6.31) we have ~ Ψ i p λ, u q “ G p p q U p λ, u q ¨ p ř nm “ p ` r p i q m ~e m q , so it is holomorphicon D ˆ D p u c q . From this we conclude.The coefficients d p i q l p u q will be fixed by the normalization for ~ϕ i in (6.34), as in the following lemma. Lemma 6.4.
The following Taylor expansions hold at λ “ u i , uniformly convergent for u P D p u c q . λ i R N , i.e. i “ , ..., q ` c : ~ Ψ i p λ, u q λ i P N , i.e. q ` c ` , ..., p : ~ϕ i p λ, u qp λ ´ u i q λ i ` ,//.//- “ λ Ñ u i ´ f i ~e i ` ÿ l “ ~b p i q l p u qp λ ´ u i q l ¯ p λ ´ u i q ´ λ i ´ , with certain vector coefficients ~b p i q l p u q holomorphic in D p u c q , and constant leading term f i “ $’’’’&’’’’% Γ p λ i ` q , λ i P C z Z , i “ , ..., q , p´ q λ i p´ λ i ´ q ! , λ i P Z ´ , i “ q ` , ..., q ` c ,λ i ! ” Γ p λ i ` q , λ i P N , i “ q ` c ` , ..., p . (6.38) Proof.
We follow a few steps. ‚ The solution Ψ p p q p λ, u q , when restricted to a polydisc D p u q contained in a τ -cell of D p u c q , is afundamental matrix solution of the Fuchsian system (1.3) in the Levelt form (6.39) below at λ “ u i , i “ , ..., p . Indeed, by (6.24) it can be written asΨ p p q p λ, u q “ ! G p p q U p p q p λ, u q p ź l “ l ‰ i p λ ´ u l q T p l q p ź j “ q ` j ‰ i p λ ´ u j q R p j q ) ¨ p λ ´ u i q T p i q p λ ´ u i q R p i q , where it is understood that R p i q “ i “ , ..., q . We have U p p q p λ, u q “ I ` F i p u q ` O p λ ´ u i q , λ Ñ u i , F i p u q : “ U p p q p u i , u q , and O p λ ´ u i q represent vanishing terms at λ “ u i , holomorphic in D ˆ D p u c q . Next, we expand at λ “ u i the factors p λ ´ u l q T p i q and p λ ´ u j q R p j q , for l, j ‰ i , obtaining the Levelt form Ψ p p q p λ, u q “ λ Ñ u i G p i ; p q p u q ´ I ` O p λ ´ u i q ¯ p λ ´ u i q T p i q p λ ´ u i q R p i q , (6.39)34here O p λ ´ u i q are higher order terms, provided that u P D p u q (they contain negative powers p u i ´ u k q ´ m ), and G p i ; p q p u q : “ G p p q p I ` F i p u qq p ź l “ l ‰ i p u i ´ u l q T p l q p ź j “ q ` j ‰ i p u i ´ u j q R p j q . The matrix G p i ; p q p u q is holomorphically invertible if restricted to a polydisc D p u q contained in a τ -cell,but it is branched at the coalescence locus ∆ on the whole D p u c q .We show that the i -th column of G p i ; p q p u q , for i “ , ..., p , is holomorphic on the whole D p u c q , andit is actually constant there. First, it follows from (6.39) and the standard isomonodromic theory of [28]that G p i ; p q p u q holomorphically in D p u q reduces B i p u q to the diagonal form T p i q , when λ i ‰ ´ ´ G p i ; p q p u q ¯ ´ B i p u q G p i ; p q p u q “ T p i q , or to non-diagonal Jordan form when λ i “ ´ ´ G p i ; p q p u q ¯ ´ B i p u q G p i ; p q p u q “ R p i q ” J p i q , λ i “ ´ . For this reason, the i -th row is proportional to the eigenvector ~e i of B i p u q with eigenvalue ´ λ i ´ f i p u q , G p i ; p q p u q ~e i “ f i p u q ~e i . This is obvious for λ i ‰ ´
1, namely for diagonalizable B i . If λ i “ ´
1, the eigenvalue 0 of B i appearingin J p i q at entry p i, i q is associated with the eigenvector f i p u q ~e i . Moreover, for every invertible matrix G “ r˚| ¨ ¨ ¨ | ˚ | ~e i | ˚ | ¨ ¨ ¨ |˚s , where ~e i occupies the k -th column, then G ´ B i p u q G is zero eveywhere, exceptfor the k -th row. Now, since R p i q “ J p i q has only one non-zero entry on the i -th row, it follows that theeigenvector f i p u q ~e i must occupy the i -th column of G p i ; p q p u q . ‚ f i p u q is holomorphic on D p u c q . Indeed, by (6.29), when i “ , ..., p we have p ź l “ l ‰ i p u i ´ u l q T p l q p ź j “ q ` j ‰ i p u i ´ u j q R p j q ¨ ~e i “ ~e i . Therefore f i p u q ~e i “ G p i ; p q p u q ~e i ” G p p q p I ` F i p u qq ~e i , and F i p u q is holomorphic on D p u c q . ‚ f i is constant on D p u c q . Indeed, since Ψ p p q p λ, u q is an isomonodromic solution in D p u q , the matrix G p i ; p q p u q must satisfy the Pfaffian system (see Appendix A, identify G p i ; p q with G p i q in Corollary 9.1) B G p i ; p q B u j “ ˆ B j u j ´ u i ` γ j ˙ G p i ; p q , j ‰ i ; B G p i ; p q B u i “ ÿ j ‰ i ˆ B j u i ´ u j ` γ j ˙ G p i ; p q . (6.40)Here, γ j “ ω j “ r F , E j s as in (3.1). From the structure (2.21) and (4.5), we see that the i -th column of B j u j ´ u i ` γ j is null. Hence, the i -th column of G p i ; p q satisfies BB u j ´ G p i ; p q ~e i ¯ “ , @ j ‰ i. ř nj “ B j G p i ; p q “ . We conclude that the i -thcolumn of G p i ; p q is constant on D p u q , and being holomorphic on D p u c q , it is constant on the whole D p u c q . Namely, f i is constant, so that we can choose it as in (6.38).From (6.29) and definitions (6.30)-(6.31), we conclude. The above discussion provides the following list of behaviours for the selected solutions ~ Ψ i and thesingular solutions ~ Ψ p sing q i , with i “ , ..., p .• Case λ i P C z Z (i.e. i “ , ..., q ). We have the singular solutionΨ p p q p λ, u q ¨ ~e i “ ~ Ψ p sing q i p λ, u q ” ~ Ψ i p λ, u q“ λ Ñ u i ´ Γ p λ i ` q ~e i ` ÿ l “ ~b p i q l p u qp λ ´ u i q l ¯ p λ ´ u i q ´ λ i ´ , • Case λ i P Z ´ (i.e. i “ q ` , ..., q ` c ). We have the regular solutionΨ p p q p λ, u q ¨ ~e i “ ~ Ψ i p λ, u q“ λ Ñ u i ˜ p´ q λ i p´ λ i ´ q ! ~e i ` ÿ l “ ~b p i q l p u qp λ ´ u i q l ¸ p λ ´ u i q ´ λ i ´ , If λ i P ´ N ´ r p i q k ‰ k “ p ` , ..., n , we have the singular solutionΨ p p q p λ, u q ¨ ~e k r p i q k “ ~ Ψ p sing q i p λ, u q “ λ Ñ u i ~ Ψ i p λ, u q ln p λ ´ u i q ` reg p λ ´ u i q“ ˜ p´ q λ i p´ λ i ´ q ! ~e i ` ÿ l “ ~b p i q l p u qp λ ´ u i q l ¸ p λ ´ u i q ´ λ i ´ ln p λ ´ u i q ` reg p λ ´ u i q . Otherwise, if r p i q k “ k , ~ Ψ p sing q i p λ, u q ” . If λ i “ ´
1, we always have a non-trivial singular solutionΨ p p q p λ, u q ¨ ~e m i “ ~ Ψ p sing q i p λ, u q “ λ Ñ u i ~ Ψ i p λ, u q ln p λ ´ u i q ` reg p λ ´ u i q“ ´ ´ ~e i ` ÿ l “ ~b p i q l p u qp λ ´ u i q l ¯ ln p λ ´ u i q ` reg p λ ´ u i q . • Case λ i P N (i.e. i “ q ` c ` , ..., p ). We have the regular solutionΨ p p q p λ, u q ¨ n ÿ k “ p ` r p i q k ~e k “ ~ Ψ i p λ, u q “ λ Ñ u i ÿ l “ ~d p i q l p u qp λ ´ u i q l ,
36n some cases when all r p i q k “ ~ Ψ i p λ, u q ” . Moreover, we have the singular solutionΨ p p q p λ, u q ¨ ~e i “ ~ Ψ p sing q i p λ, u q “ ~ϕ i p λ, u qp λ ´ u i q λ i ` ` ~ Ψ i p λ, u q ln p λ ´ u i q“ λ Ñ u i λ i ! ~e i ` ř ´ λ i l “ ~b p i q l p u qp λ ´ u i q l p λ ´ u i q λ i ` ` ´ ÿ l “ d p i q l p u qp λ ´ u i q l ¯ ln p λ ´ u i q ` reg p λ ´ u i q . In conclusion, Theorem 5.1 is proved for i “ , ..., p , with some obvious identifications betweenobjects in the proof and objects in the statement, namely ~ϕ i ÞÝÑ ~ψ i , r p m q i { r p i q k ÞÝÑ r m and ~ϕ k { r p i q k ÞÝÑ φ i . With the labelling (6.1)-(6.2), the same strategy above holds for every coalescence p u p ` ... ` p α ´ ` , ..., u p ` ... ` p α q ÝÑ p λ α , ..., λ α q , α “ , ..., s. We find corresponding isomondromic fundamental matrices for the Pfaffian system (with self-explainingnotations)Ψ p p α q p λ, u q “ G p p α q ¨ U p p α q p λ, u q ¨ p ` ... ` p α ź l “ p ` ... ` p α ´ ` p λ ´ u l q T p l q p ` ... ` p α ź j “p p ` ... ` p α ´ ` q` q α p λ ´ u j q R p j q . where p α “ p p ` ... ` p α ´ ` , . . . , p ` ... ` p α q . Then, we proceed in the same way, constructing thesolutions ~ Ψ i and ~ Ψ p sing q i , with p ` ... ` p α ´ ` ď i ď p ` ... ` p α . l Proof.
Connection coefficients c p ν q jk “ c p ν q jk p u q are defined in (5.12)-(5.13). Here we omit ν for simplicity.It follows form the very definitions of the ~ Ψ k and ~ Ψ p sing q j that c jk “ u cj “ u ck . In order to prove independence of u , we express in terms of the coefficients the monodromy of thematrix Ψ p λ, u q : “ r ~ Ψ p λ, u q | ¨ ¨ ¨ | ~ Ψ n p λ, u qs , From the definition, we have (using the notations in the statement of Theorem 5.1) ~ Ψ k p λ, u q “ $’’’’’’’&’’’’’’’% ~ Ψ j p λ, u q c jk ` reg p λ ´ u j q , λ j R Z ~ Ψ j p λ, u q ln p λ ´ u j q c jk ` reg p λ ´ u j q , λ j P Z ´ ˜ ~ Ψ j p λ, u q ln p λ ´ u j q ` ψ j p λ, u qp λ ´ u j q λ j ` ¸ c jk ` reg p λ ´ u j q , λ j P N (6.41)37or u R ∆ and a small loop p λ ´ u k q ÞÑ p λ ´ u k q e πi we obtain from Theorem 5.1 ~ Ψ k p λ, u q ÞÝÑ ~ Ψ k p λ, u q e ´ πiλ k , which includes also the case λ k P Z , with e ´ πiλ k “ , while for a small loop p λ ´ u j q ÞÑ p λ ´ u j q e πi , j ‰ k , we obtain from Theorem 5.1 and (6.41) thefollowing transformations. ~ Ψ k ÞÝÑ ~ Ψ j e ´ πiλ j c jk ` reg p λ ´ u j q looooomooooon ~ Ψ k ´ ~ Ψ j c jk “ ~ Ψ k ` p e ´ πiλ j ´ q c jk ~ Ψ j for λ j R Z ~ Ψ k ÞÝÑ ~ Ψ j ´ ln p λ ´ u j q ` πi ¯ c jk ` reg p λ ´ u j q “ ~ Ψ k ` πic jk ~ Ψ j , for λ j P Z ´ ~ Ψ k ÞÝÑ ˜ ~ Ψ j ´ ln p λ ´ u j q ` πi ¯ ` ψ j p λ, u qp λ ´ u j q λ j ` ¸ c jk ` reg p λ ´ u j q “ ~ Ψ k ` πic jk ~ Ψ j , for λ j P N . Therefore, for u R ∆ and a small loop γ k : p λ ´ u k q ÞÑ p λ ´ u k q e πi not encircling other points u j (wedenote the loop by λ ÞÑ γ k λ ), we receiveΨ p λ, u q ÞÝÑ Ψ p γ k λ, u q “ Ψ p λ, u q M k p u q , where p M k q jj “ j ‰ k, p M k q kk “ e ´ πiλ k ; p M k q kj “ α k c kj , j ‰ k ; p M k q ij “ . and α k : “ p e ´ πiλ k ´ q , if λ k R Z ; α k : “ πi, if λ k P Z . We proceed by first analyzing the generic case, and then the general case.
Generic case.
Suppose that A p u q has no integer eigenvalues (recall that eigenvalues do not dependon u ). Let us fix u in a τ -cell. By Proposition 2.3, Ψ p λ, u q is a fundamental matrix solution of (1.3) forthe fixed u , and C “ p c jk q is invertible. Thus M k p u q “ Ψ p γ k λ, u q Ψ p λ, u q ´ . The above makes sense for every u in the considered τ -cell, being Ψ p λ, u q invertible at such an u . ButΨ p λ, u q and Ψ p γ k λ, u q are holomorphic on P η p u q ˆ ˆ D p u c q , so that the matrix M k p u q is holomorphic onthe τ -cell. Repeating the above argument for another τ -cell, we conclude that M k p u q is holomorphic oneach τ -cell. Now, on a τ -cell, we have d Ψ p γ k λ, u q “ P p λ, u q Ψ p γ k λ, u q “ P p λ, u q Ψ p λ, u q M k , and at the same time d Ψ p γ k λ, u q “ d ´ Ψ p λ, u q M k ¯ “ d Ψ p λ, u q M k ` Ψ p λ, u q dM k “ P p λ, u q Ψ p λ, u q M k ` Ψ p λ, u q dM k . The two expressions are equal if and only in dM k “
0, because Ψ p λ, u q is invertible on a τ -cell. Noticeanyway that τ -cells are disconnected from each other, so that separately on each cell, M k is constant,and so the connection coefficients are constant separately on each cell .38e further suppose that none of the λ j is integer . In this case, ~ Ψ p sing q j “ ~ Ψ j for all j “ , ..., n , sothat from (6.41) for u ck ‰ u cj (otherwise c jk “ ~ Ψ k p λ, u q “ λ Ñ u j ~ Ψ j p λ, u q c jk ` reg p λ ´ u j q . (6.42)Using the labelling (6.1)-(6.2), form the proof of Theorem 6.1 we have the fundamental matrix solutionΨ p p q p λ, u q “ ” ~ Ψ p λ, u q ˇˇˇ ¨ ¨ ¨ ˇˇˇ ~ Ψ p p λ, u q ˇˇˇ ~ϕ p q p ` p λ, u q ˇˇˇ ¨ ¨ ¨ ˇˇˇ ~ϕ p q n p λ, u q ı and in general at each λ α , α “ , ..., s (with ř α ´ j “ p j “ α “
1) we haveΨ p p α q p λ, u q ““ ” ~ϕ p α q p λ, u q ˇˇˇ ¨ ¨ ¨ ˇˇˇ ~ϕ p α q ř α ´ j “ p j p λ, u q ˇˇˇ ~ Ψ ř α ´ j “ p j ` p λ, u q ˇˇˇ ~ Ψ ř α ´ j “ p j ` p λ, u q ˇˇˇ ¨ ¨ ¨ ˇˇˇ ~ Ψ ř αj “ p j p λ, u q ˇˇˇˇˇˇ ~ϕ p α q ř αj “ p j ` p λ, u q ˇˇˇ ¨ ¨ ¨ | ~ϕ p α q n p λ, u q ı where ~ Ψ m p λ, u q “ ~ψ m p λ, u qp λ ´ u m q ´ λ m ´ , m “ α ´ ÿ j “ p j ` , . . . , α ÿ j “ p j , and the ~ψ m p λ, u q and ~ϕ p α q r p λ, u q are holomorphic in the corresponding D α ˆ D p u c q . The above allowsus to explicitly rewrite (6.42), for j such that u cj “ λ α , as ~ Ψ k p λ, u q “ p `¨¨¨` p α ÿ m “ p `¨¨¨` p α ´ ` c mk ~ψ m p λ, u qp λ ´ u m q ´ λ m ´ ` ÿ r Rt p `¨¨¨` p α ´ ` ,...,p `¨¨¨` p α u h r ~ϕ p α q r p λ, u q , (6.43)for suitable constant coefficients h r . Here one of the c mk is c jk of (6.42).Recall that each u m , with m “ p ` ¨ ¨ ¨ ` p α ´ ` , ..., p ` ¨ ¨ ¨ ` p α , varies in D α . Firstly, we canfix λ “ λ α in (6.43) consider the branch cut L α from λ α to infinity, in direction η (see Figure 3), andlet u vary in such a way that each component u p `¨¨¨` p α ´ ` , ..., u p `¨¨¨` p α vary in D α z L α , so that inthe r.h.s. of (6.43) all the ~ψ m p λ α , u qp λ α ´ u m q ´ λ m ´ and ~ϕ p α q r p λ α , u q are holomorphic with respectto u , providet that u m ‰ λ α . Despite of the fact that each u m is constrained to stay in D α z L α , wecan anyway reach every τ -cell of D p u c q starting from an initial point in one specific cell. This proves,by u -analytic continuation of (6.43) with fixed λ “ λ α , that the coefficients c mk are constant in p D α z L α q ˆ p α ˆ ´Ś β ‰ α D ˆ p β β ¯ Ă D p u c q .Now, we can slightly vary η in η ν ` ă η ă η ν , so that the cut L α is irrelevant . Thus, we concludethat the c mk are constant on (cid:32) u P D p u c q | u p ` ... ` p α ´ ` ‰ λ α , . . . , u p ` ... ` p α ‰ λ α ( .Finally, we can fix another value λ “ λ ˚ P D α in (6.43), and repeat the above discussion with the cut L α issuing from λ ˚ , so that all the c mk are constant on (cid:32) u P D p u c q | u p ` ... ` p α ´ ` ‰ λ ˚ , . . . , u p ` ... ` p α ‰ λ ˚ ( .This proves constancy of the c mk , m associated with λ α , on the whole D p u c q . Then, we repeat this forall α “ , .., s , proving constancy of the c jk for all j “ , ..., n . Hence, Corollary 5.1 is proved in thegeneric case. General case of any A p u q . If some of the diagonal entries λ , ..., λ n of A are integers, or someeigenvalues are integers, there exists a sufficiently small γ ą ă γ ă γ , A ´ γI Recall that D p u c q “ Ś sβ “ D ˆ p β β . The crossing locus X p τ q , τ “ π { ´ η , is as arbitrary as is the choice of τ in the range τ ν ă τ ă τ ν ` . λ ´ γ, ..., λ n ´ γ and no integer eigenvalues. Take such a γ , and forany 0 ă γ ă γ consider p Λ ´ λ q ddλ p γ Ψ q “ ´ p A p u q ´ γI q ` I ¯ γ Ψ . (6.44)namely ddλ p γ Ψ q “ n ÿ k “ B k r γ sp u q λ ´ u k γ Ψ , B k r γ sp u q : “ ´ E k ´ A p u q ` p ´ γ q I ¯ . (6.45) Lemma 6.5.
The above system (6.45) is strongly isomonodromic in D p u q contained in a τ -cell, and λ -component of the integrable Pfaffian system d γ Ψ “ P r γ s p λ, u q γ Ψ , P r γ s p λ, u q “ n ÿ k “ B k r γ sp u q λ ´ u k d p λ ´ u k q ` n ÿ j “ r F p u q , E j s du j . (6.46) where F p u q is defined as in (2.8), p F q ij “ A ij u j ´ u i , i ‰ j , and r F p u q , E j s is (2.21).Proof. We do a gauge trasformation γ Y p z q : “ z ´ γ Y p z q , γ P C , (6.47)which transforms (1.1) into d p γ Y q dz “ ˆ Λ ` A ´ γIz ˙ γ Y (6.48)For u P D p u q contained in a τ -cell, we write the unique formal solution γ Y F p z, u q “ z ´ γ Y F p z, u q , (6.49)where Y F p z, u q is (2.4), so that γ Y F p z, u q “ F p z, u q z B ´ γI e Λ z , B ´ γI “ diag p A ´ γ q “ diag p λ ´ γ, ... , λ n ´ γ q . The crucial point is that F p z, u q is the same as (2.5), so all the F k p u q are independent of γ . Thefundamental matrix solutions γ Y ν p z, u q : “ z ´ γ Y ν p z, u q , are uniquely defined by their asymptotics γ Y F p z, u q in S ν p D p u qq . Their Stokes matrices do not dependon γ because γ Y ν `p k ` q µ p z, u q “ γ Y ν ` kµ p z, u q S ν ` kµ ðñ Y ν `p k ` q µ p z, u q “ Y ν ` kµ p z, u q S ν ` kµ . The system (6.48) is thus strongly isomonodromic. By Proposition 3.1 we conclude.
Corollary 6.3.
Let the assumptions of Theorem 5.1 hold. Then Theorem 5.1 holds also for (6.46).
By Theorem 5.1 applied to (6.46), we receive independent vector solutions γ ~ Ψ k p λ, u q ” γ ~ Ψ p sing q k p λ, u q , k “ , ..., n , which form a fundamental matrix solution γ Ψ p λ, u q : “ r γ ~ Ψ p λ, u q | ¨ ¨ ¨ | γ ~ Ψ n p λ, u qs . For system (6.46) the results already proved in the generic case hold. Therefore, the connection coeffi-cients c p ν q jk r γ s defined by γ ~ Ψ k p λ, u | ν q “ γ ~ Ψ j p λ, u | ν q c p ν q jk r γ s ` reg p λ ´ u j q , λ P P η , (6.50)are constant on D p u c q . They depend on γ , but not on u P D p u c q .40 emark 6.4. It is explained in section 8 of [20] what is the relation between ~ Ψ p sing q k and γ ~ Ψ k , by meansof their primitives, and that in general both lim γ Ñ γ ~ Ψ k and lim γ Ñ c p ν q jk r γ s are divergent.Now, we invoke Proposition 10 of [20], which holds with no assumptions on eigenvalues and diagonalentries of A p u q . This result, adapted to our case, reads as follows.
Proposition 6.3.
Let u be fixed in a τ -cell. Let γ ą be small enough such that for any ă γ ă γ the matrix A ´ γI has no integer eigenvalues, and its diagonal part no integer entries. Let c p ν q jk be theconnection coefficients of the Fuchsian system (1.3) a the fixed u , as in Definition 5.1. Let c p ν q jk r γ s be theconnection coefficients in (6.50). Let α k : “ e ´ πiλ k ´ , λ k R Z πi, λ k P Z ; α k r γ s : “ e ´ πi p λ k ´ γ q ´ Then, the following equalities hold α k c p ν q jk “ e ´ πiγ α k r γ s c p ν q jk r γ s , if k ą j ; α k c p ν q jk “ α k r γ s c p ν q jk r γ s , if k ă j ; (6.51) where the ordering relation j ă k means, for the fixed u , that < p z p u j ´ u k qq ă for arg z “ τ “ π { ´ η satisfying (5.2). We use Proposition 6.3 to concude the proof of Corollary 5.1 in the general case. Indeed, Corollary5.1 is already proved in the generic case, so it holds for the c p ν q jk r γ s . Therefore, they are constant on thewhole D p u c q . Equalities (6.51) hold at any fixed u in τ -cell, so that each c p ν q jk is constant on a τ -cell, andsuch constant is the same in each τ -cell. With a slight variation of η in p η ν ` , η ν q , equalities (6.51) holdalso at the crossing locus X p τ q . They analytically extend at ∆, which is a complex braid arrangement. D p u c q By means of the Laplace transform with deformation parameters, we prove points (I1),(I2), (I3), (II1),(II2) and (II5) of Theorem 2.2, concerning the Stokes solutions Y ν on D p u c q and the Stokes matrices(while (I.4) has been proved in Section 4). Stokes matrices will be expressed in terms of the isomon-odromic (constant) connection coefficients satisfying Corollary 5.1. This is achieved in in Theorem 7.1below, which is the last step of our construction.Let τ be the chosen direction in the z -plane admissible at u c , and η “ π { ´ τ in the λ -plane. TheStokes rays of Λ p u c q will be labelled as in (2.20), so that (5.2) holds for a certain ν P Z . We define thesectors S ν “ t z P R p C zt uq such that τ ν ´ π ă arg z ă τ ν ` u . (7.1)If u only varies in D p u q contained in a τ -cell, then none of the Stokes rays associated with Λ p u q crossarg z “ τ mod π . If u varies in D p u c q , some Stokes rays associated with Λ p u q necessarily cross arg z “ τ mod π (see Section 2.1.2). The proof in [20] is laborious, because it is necessary to take into account all possible values of the diagonal entries λ k of A , including integer values. In [4] the proof is given only for non-integer values. Recall that eigenvalues and diagonal entries do not depend on u , in the isomonodromic case.
41n order to identify the Stokes rays which do not cross arg z “ τ mod π as u varies in D p u c q ,we take the radius (cid:15) as in (5.1). Consider the subset of the set of Stokes rays containing only rays t z P R | < p z p u j ´ u k qq “ u associated with pairs p u j , u k q such that u j P D α and u k P D β , α ‰ β ,namely u cj ‰ u ck . Following [11], we denote this subset by R p u q . If u varies in D p u c q , the rays in R p u q continuously rotate, but by the definition of (cid:15) they never cross any admissible rays arg z “ τ ` hπ ,where τ ν ` hµ ă τ ` hπ ă τ ν ` hµ ` , h P Z , (7.2)The above allows to define p S ν ` hµ p u q to be the unique sector containing S ` τ ` p h ´ q π, τ ` hπ ˘ andextending up to the nearest Stokes rays in R p u q . Then, let p S ν ` hµ : “ č u P D p u c q p S ν ` hµ p u q . (7.3)It has angular amplitude greater than π . The reason for the labeling is that p S ν ` hµ p u c q “ S ν ` hµ in (7.1).In the λ -plane, the admissible directions η ´ hπ correspond to τ ` hπ , with η ν ` hµ ` ă η ´ hπ ă η ν ` hµ . (7.4)Suppose that u is fixed in a τ -cell. Let us consider the matrix Y ν ` hµ p z, u q “ ” ~Y p z, u | ν ` hµ q ˇˇˇ . . . ˇˇˇ ~Y n p z, u | ν ` hµ q ı , fixed u, defined by ~Y k p z, u | ν ` hµ q “ πi ż γ k p η ´ hπ q e zλ ~ Ψ p sing q k p λ, u | ν ` hµ q dλ, for λ k R Z ´ , (7.5) ~Y k p z, u | ν ` hµ q “ ż L k p η ´ hπ q e zλ ~ Ψ k p λ, u | ν ` hµ q dλ, for λ k P Z ´ . (7.6)Here, ~ Ψ k p λ, u | ν ` hµ q , ~ Ψ p sing q k p λ, u | ν ` hµ q are the vector solutions of Theorem 5.1 for λ P P η ´ hπ p u q ,with u fixed in a τ -cell. L k p η ´ hπ q is the cut in direction η ´ hπ , oriented from u k to , and γ k p η ´ hπ q is the path coming from along the left side of L k p η ´ hπ q , encircling u k with a small loop excludingall the other poles, and going back to along the right side of L k p η ´ hπ q . The label ν ` hµ keeps trackof (5.2) and (7.2)-(7.4). Theorem 7.1.
Consider the matrices Y ν ` hµ p z, u q obtained by Laplace transform (7.5)-(7.6) at a fixed u P D p u q contained in a τ -cell. Then The Y ν ` hµ p z, u q define holomorphic matrix valued functions of p λ, u q P R p C zt uq ˆ D p u c q , whichare fundamental matrix solutions of (1.1). They have structure Y ν ` hµ p z, u q “ p Y ν ` hµ p z, u q z B e z Λ , B “ diag p λ , ..., λ n q , with asymptotic behaviour, uniform in u P D p u c q , p Y ν ` hµ p z, u q „ F p z, u q “ I ` ÿ l “ F l p u q z l , z Ñ 8 in p S ν ` hµ , given by the formal solution Y F p z, u q “ F p z, u q z B e z Λ . The coefficients F l p u q are holomorphic in D p u c q . Their explicit expressions are in formulae (7.12), (7.13), (7.15) (or (7.16)) and (7.17). ) Stokes matrices defined by Y ν `p h ` q µ p z, u q “ Y ν ` hµ p z, u q S ν ` hµ , z P p S ν ` hµ X p S ν `p h ` q µ , (7.7) are constant in the whole D p u c q and satisfy p S ν ` hµ q ab “ p S ν ` hµ q ba “ for a ‰ b such that u ca “ u cb . (7.8) The following representation in terms of the constant connection coefficients c p ν q jk of Corollary 5.1holds on D p u c q : p S ν q jk “ $’’’’’&’’’’’% e πiλ k α k c p ν q jk , j ă k , u cj ‰ u ck , j “ k, j ą k , u cj ‰ u ck , j ‰ k , u cj “ u ck , ; p S ´ ν ` µ q jk “ $’’’’’&’’’’’% j ‰ k , u cj “ u ck , j ă k , u cj ‰ u ck , j “ k, ´ e πi p λ k ´ λ j q α k c p ν q jk j ą k , u cj ‰ u ck , (7.9) where the relation j ă k is defined for j ‰ k such that u cj ‰ u ck and means that < p z p u cj ´ u ck qq ă when arg z “ τ . Remark 7.1.
The above (7.9) generalises Theorem 2.3 in presence of isomonodromic deformationparameters, including coalescences. Notice that the ordering relation ă here is referred to u c , while inTheorem 2.3 it refers to u . Proof.
We use the labelling (6.1)-(6.2). a) Case λ k R Z . ‚ Construction of ~Y k p z, u | ν q . We have ~ Ψ p sing q k p λ, u | ν q “ ~ Ψ k p λ, u | ν q . For every fixed u P D p u c q ,define ~Y k p z, u | ν q : “ πi ż γ k p η q e zλ ~ Ψ k p λ, u | ν q dλ (7.10)Since ~ Ψ k p λ, u | ν q grows at infinity no faster than some power of λ , the integral converges in a sector ofamplitude at most π . Now, ~ Ψ k p λ, u | ν q satisfies Theorem 5.1, hence if u varies in D p u c q the followingfacts hold.1. ~ Ψ k p λ, u | ν q is branched at λ “ u k and possibly at other poles u l such that u cl ‰ u ck .2. ~ Ψ k p λ, u | ν q is holomorphic at all λ “ u j such that u cj “ u ck , j ‰ k .It follows from 1. and 2. that the path of integration can be modified: for α such that u ck “ λ α , we have ~Y k p z, u | ν q “ πi ż Γ α p η q e zλ ~ Ψ k p λ, u | ν q dλ, (7.11)where Γ α p η q is the path which comes from in direction η ´ π , encircles λ α along B D α anti-clockwiseand goes to in direction η . This path encloses all the u j such that u cj “ λ α , end excludes the others.See figure 4. We conclude that u can vary in D p u c q and the integral (7.11) converges for z in the sector S p η q : “ ! z P R p C zt uq such that π ´ η ă arg z ă π ´ η ) ,
0, due to the exponential factor. By (2.24), the vectorsolutions ~Y k p z, u | ν q satisfies the system (1.1). ‚ Asymptotic behaviour . From (5.4)-(5.5), we write (7.11) as ~Y k p z, u | ν q “ πi ż Γ α p η q e zλ ´ Γ p λ j ` q ~e j ` ÿ l ě ~b p k q l p u qp λ ´ u k q l ¯ p λ ´ u k q ´ λ k ´ , dλ. with holomorphic ~b p k q l p u q on D p u c q . We split the series as ř l ě “ ř N l “ ` ř l ě N ` , and recall thestandard formula (see [15]) ż Γ α p η q p λ ´ λ k q a e zλ dλ “ ż γ k p η q p λ ´ λ k q a e zλ dλ “ z ´ a ´ e λ k z Γ p´ a q
44o that ~Y k p z, u | ν q “ ˜ ~e k ` N ÿ l “ ~b p k q l p u q Γ p λ k ` ´ l q z ´ l ` R N p z q ¸ z λ k e λ k z , with remainder R N p z q “ ¿ Γ p η q ÿ l ě N ~b p k q l p u q z l e x x l ´ λ k ´ dx “ O p z ´ N ` q . The integral is along a path Γ p η q , coming from along the left part of the half line oriented form0 to in direction η ` arg z , going around 0, and back to along the right part. The last estimate O p z ´ N ` q is standard. We conclude that ~Y k p z, u | ν q ´ z λ k e λ k z ¯ ´ „ ~e k ` ÿ l “ ~b p k q l p u q Γ p λ k ` ´ l q z ´ l ” ~e k ` ÿ l “ ~f p k q l p u q z ´ l , z Ñ 8 in p S ν with ~f p k q l p u q : “ ~b p k q l p u q Γ p λ k ` ´ l q . (7.12) b) Case λ k P N “ t , , , ... u . ‚ Construction of ~Y k p z, u | ν q . We define ~Y k p z, u | ν q : “ πi ż γ k p η q e zλ ~ Ψ p sing q k p λ, u | ν q dλ “ p . q πi ż γ k p η q e zλ ˜ ~ψ k p λ, u | ν qp λ ´ u k q λ k ` ` ~ Ψ k p λ, u | ν q ln p λ ´ u k q ¸ dλ. The same facts 1. and 2. of the previous case are now applied to ~ Ψ k p λ, u | ν q and ~ψ k p λ, u | ν q , based onthe analytic properties in Theorem 5.1, and allow to rewrite ~Y k p z, u | ν q “ πi ż Γ α p η q e zλ ˜ ~ψ k p λ, u | ν qp λ ´ u k q λ k ` ` ~ Ψ k p λ, u | ν q ln p λ ´ u k q ¸ dλ “ πi ż Γ α p η q e zλ ~ Ψ p sing q k p λ, u | ν q dλ. Analogously to the previous case, we conclude that ~Y k p z, u | ν q is analytic on p S ν ˆ D p u c q . Moreover, e λz p λ ´ Λ q ~ Ψ p sing q k p λ, u | ν q ˇˇˇ Γ p α q “
0, due to the exponential factor. By (2.24), the vector solution ~Y k p z, u | ν q satisfies the system (1.1). ‚ Asymptotic behaviour . By (5.7) and (5.11), and the fact that ~ψ k has no singularities at u j P D α , j ‰ k , so that the terms ř l ě ` λ k ~b p k q l p u qp λ ´ u k q l in ~ψ k p λ, u | ν q do not contribute to the integration,we can write ~Y k p z, u | ν q “ πi ż Γ α p η q ˜ λ k ! ~e k ` ř λ k l “ ~b p k q l p u qp λ ´ u k q l p λ ´ u k q λ k ` ` ÿ l “ ~d p k q l p u qp λ ´ u k q l ln p λ ´ u k q ¸ e zλ dλ.
45y Cauchy formula12 πi ż Γ α p η q ˜ λ k ! ~e k ` ř λ k l “ ~b p k q l p u qp λ ´ u k q l p λ ´ u k q λ k ` ¸ e zλ dλ “ λ k ! d λ k dλ λ k »–¨˝ λ k ! ~e k ` λ k ÿ l “ ~b p k q l p u qp λ ´ u k q l ˛‚ e zλ fiflˇˇˇˇˇˇ λ “ u k “ z λ k e u k z ¨˝ ~e k ` λ k ÿ l “ ~f p k q l p u q z l ˛‚ , where ~f p k q l p u q : “ ~b p k q l p u qp λ k ´ l q ! , l “ , ..., λ k . (7.13)In order to evaluate the terms with logarithm, we observe that for any function g p λ q holomorphic along L k p η q , including λ “ u k , we have ż γ k p η q g p λ q ln p λ ´ u k q dλ “ ż L k p η q ´ g p λ q ln p λ ´ u k q ´ dλ ´ ż L k p η q ` g p λ q ln p λ ´ u k q ` dλ, where L k p η q ` and L k p η q ´ respectively are the left and right parts of L k p η q , oriented from 0 to . Sinceln p λ ´ u k q ` “ ln p λ ´ u k q ` ´ πi , we conclude that ż γ k p η q g p λ q ln p λ ´ u k q dλ “ πi ż L k p η q g p λ q dλ. (7.14)Keeping into account that the integral along Γ α can be interchanged with that along γ k , it follows that12 πi ż Γ α p η q ~ Ψ k p λ, u | ν q ln p λ ´ u k q e zλ dλ “ ż L k p η q ~ Ψ k p λ, u | ν q e zλ dλ “ ż L k p η q 8 ÿ l “ ~d p k q l p u qp λ ´ u k q l e zλ dλ. We conclude, by the standard evaluation of the remainder analogous to R N p z q considered before, andthe variation of η in the range p η ν ` , η ν q , that ż L k p η q ~ Ψ k p λ, u | ν q e zλ dλ „ e u k z ˜ ÿ l “ p´ q l ` l ! ~d p k q l p u q z ´ l ´ ¸ , z Ñ 8 in p S ν . “ z λ k e u k z ¨˝ ÿ l “ λ k ` ~f p k q l p u q z ´ l ˛‚ , where ~f p k q l p u q : “ p´ q l ´ λ k p l ´ λ k ´ q ! ~d p k q l ´ λ k ´ p u q , l ě λ k ` . (7.15)In conclusion, we have the expansion ~Y k p z, u | ν q „ z λ k e u k z ˜ ~e k ` ÿ l “ ~f p k q l p u q z ´ l ¸ , z Ñ 8 in p S ν , Notice that, by abuse of notation, if f p λ q e ´ u k λ „ ř c l z ´ l we write f p λ q „ e u k λ ř c l z ´ l . ~f p k q l p u q holomorphic in D p u c q defined in (7.13)-(7.15). Notice that, in exceptional cases, ~ Ψ k may be identically zero, so that ~f p k q l “ l ě λ k ` . (7.16) c) Case λ k P Z ´ “ t´ , ´ , ... u‚ Construction of ~Y k p z, u | ν q . We define ~Y k p z, u | ν q : “ ż L k p η q e λz ~ Ψ k p λ, u | ν q dλ ” ż L α p η q e λz ~ Ψ k p λ, u | ν q dλ. In the last equality, we have used the fact that ~ Ψ k p λ, u | ν q is analytic in D α ˆ D p u c q , where λ α “ u ck .We conclude analogously to previous cases that ~Y k p z, u | ν q is analytic in p S ν ˆ D p u c q . It is a solutionof (1.1), by (2.24), because ~ Ψ k p λ, u | ν q is analytic at λ “ u k and behaves as in (5.4)-(5.5), so that e λz p λI ´ Λ p u qq ~ Ψ k p λ, u | ν q ˇˇˇ L α “ e λz p λI ´ Λ p u qq ~ Ψ k p λ, u | ν q ˇˇˇ L k “ ´ p u k I ´ Λ p u qq ~ Ψ k p λ, u k | ν q “ . ‚ Asymptotic behaviour . We have, from (5.4)-(5.5), ~Y k p z, u | ν q “ ż L α p η q e λz ˜ p´ q λ k ~e k p´ λ k ´ q ! p λ ´ u k q ´ λ k ´ ` ÿ l ě ~b p k q l p u qp λ ´ u k q l ´ λ k ´ ¸ dλ We integrate term by term in order to obtain the asymptotic expansion (the remainder for the truncatedseries is evaluate in standard way, as R N p z q above). For the integration, we use ż L k p η q p λ ´ u k q m e λz dλ “ e u k z z m ` ż `8 e iφ x m e x dx “ e u k z z m ` m ! p´ q m ` , π ă φ ă π . We obtain, analogously to previous cases, ~Y k p z, u | ν q „ z λ k e u k z ˜ ~e k ` ÿ l “ ~f p k q l p u q z ´ l ¸ , z Ñ 8 in p S ν , where the holomorphic in D p u c q coefficients are ~f p k q l p u q : “ p´ q l ´ λ k p l ´ λ k ´ q ! ~b p k q l p u q . (7.17) Remark 7.2.
We would like to observe that ~ Ψ p sing q k p λ, u | ν q in (5.8) cannot be used to define ~Y k p z, u | ν q if u varies in the whole D p u c q . On the other hand, if u is restricted to a τ -cell, so that the eigenvalues u j are all distinct, by (7.14) we can write ~Y k p z, u | ν q “ ż L k p η q e λz ~ Ψ k p λ, u | ν q dλ “ p . q πi ż γ k p u q ~ Ψ k p λ, u | ν q ln p λ ´ u k q dλ. Then, we can use the local expansion (5.9) and the fact that ş γ k p u q reg p λ ´ u k q dλ “
0, receiving ~Y k p z, u | ν q “ πi ż γ k p u q ~ Ψ p sing q k p λ, u | ν q dλ undamental matrix solutions The vector solutions ~Y k p z, u | ν q constructed above can be arranged as columns of the matrix Y ν p z, u q : “ ” ~Y k p z, u | ν q ˇˇˇ ¨ ¨ ¨ ˇˇˇ ~Y n p z, u | ν q ı , which thus solves system (1.1). Form the general theory of differential systems, it admits analytic con-tinuation as analytic matrix valued function on R p C zt uqˆ D p u c q . Letting B “ diag A “ diag p λ , ..., λ n q ,the asymptotic expansions obtained above are summarized as Y ν p z, u | ν q z ´ B e ´ Λ p u q z „ F p z, u q “ I ` ÿ l “ F l p u q z ´ l , z Ñ 8 in p S ν ,F l p u q “ ” ~f p q l p u q | ¨ ¨ ¨ | ~f p n q l p u q ı . Therefore, the coefficients F l p u q of the formal solution Y F p z, u q “ F p z, u q z B e Λ p u q z are holomorphic in D p u c q . Moreover, the leading term is the identity I , which implies that Y ν p z, u q is a fundamental matrixsolution.Consider now another direction η , satisfying η ν ` µ ` ă η ă η ν ` µ . The above discussion can berepeated. We obtain a fundamental matrix solution Y ν ` µ p z, u q with canonical asymptotics Y F p z, u q in p S ν ` µ . Again, for η satisfying η ν ` µ ` ă η ă η ν ` µ we obtain the analogous result for Y ν ` µ p z, u q with canonical asymptotics in p S ν ` µ . This can be repeated for every ν ` hµ , h P Z , obtaining thefundamental matrix solutions Y ν ` hµ p z, u q with canonical asymptotics Y F p z, u q in p S ν ` hµ . So, Points and of Theorem 7.1 are proved.Stokes matrices are defined by (7.7). Thus, S ν ` hµ p u q “ Y ν ` hµ p z, u q ´ Y ν `p h ` q µ p z, u q is holomorphicin D p u c q . Let us consider the relations for h “ , Y ν ` µ p z, u q “ Y ν p z, u q S ν p u q , Y ν ` µ p z, u q “ Y ν ` µ p z, u q S ν ` µ p u q . (7.18)Let u be fixed in a τ -cell, so that Λ has distinct eigenvalues. From Theorem 2.3 at the fixed u we receive ` S ν p u q ˘ jk “ $’’’’&’’’’% e πiλ k α k c p ν q jk for j ă k, j “ k, j ą k, ` S ´ ν ` µ p u q ˘ jk “ $’’’’&’’’’% j ă k, j “ k, ´ e πi p λ k ´ λ j q α k c p ν q jk for j ą k. Here, for j ‰ k the ordering relation j ă k ðñ < p z p u j ´ u k qq| arg z “ τ ă u inthe τ -cell, because no Stokes rays < p z p u j ´ u k qq “ z “ τ as u varies in the τ -cell.The relation j ă k may change to j ą k when passing from one τ -cell to another only for a pair u j , u k such that u cj “ u ck . This is due to the choice of (cid:15) as in (5.1). On the other hand, c p ν q jk “ u cj “ u ck . This means that (7.9) is true at every fixed u in every τ -cell, with ordering relation j ă k precisely coinciding with that defined for j ‰ k such that u cj ‰ u ck , namely < p z p u cj ´ u ck qq ă z “ τ .Now, recall that the S ν ` hµ are holomorphic in D p u c q and the c p ν q jk are constant in D p u c q . We concludethat Stokes matrices are constant in D p u c q and hence (7.9) holds in D p u c q .The vanishing conditions (7.8) follow from the vanishing conditions (5.14) for the connection coeffi-cients, plus the fact that we can generate all the S ν ` hµ from the formula S ν ` µ “ e ´ πiB S ν e πiB .48 (Non) Uniqueness of the formal solution of (1.1) at u “ u c We prove by Laplace transform Corollary 2.1 in Background 1. Let us consider system (1.1) at the fixedpoint u “ u c , dYdz “ ˆ Λ p u c q ` A p u c q z ˙ Y (8.1)We prove that it has unique formal solution if and only if the constant diagonal entries of A p u q do notdiffer by non-zero integers . In this case, it necessarily coincides with Y F p z, u c q “ lim u Ñ u c Y F p z, u q , where Y F p z, u q “ ` I ` ř l “ F l p u q z ´ l ˘ z B e Λ p u q z is the unique formal solution defined on D p u c q in Theorems2.1 and 7.1. On the other hand, in case some diagonal entries of A p u q differ by non-zero integer λ i ´ λ j P Z zt u for some i ‰ j. (8.2)we prove that system (8.1) has a family of formal solutions with structure˚ Y F p z q “ ´ I ` ÿ l “ ˚ F l z ´ l ¯ z B e Λ p u c q z , with coefficients ˚ F l depending on a finite number of arbitrary parameters .Due to the strategy of Section 6.7, it will suffices to consider the generic case when all λ , ..., λ n R Z and A has no integer eigenvalues. Indeed, if this is not the case, the gauge transformation (6.47) relates aformal solution γ Y F to Y F at any point u , through (6.49), so that the coefficients F l of a formal expansiondo not depend on γ . We are interested in these coefficients.Consider system (1.3) under the assumptions that it is (strongly) isomonodormic in D p u c q , so that p A q ij p u c q “ u ci “ u cj . For simplicity, we order the eigenvalues as in (6.1)-(6.2). Since B p u q , ..., B n p u q are holomorphic at u c , system (1.3) at u “ u c is d Ψ dλ “ ˜ ř p j “ B j p u c q λ ´ λ ` ř p ` p j “ p ` B j p u c q λ ´ λ ` ¨ ¨ ¨ ` ř nj “ p ` ... ` p s ´ ` B j p u c q λ ´ λ s ¸ Ψ (8.3)Let G p p q be as in (6.25). The gauge transformation Ψ p λ q “ G p p q p u c q r Ψ p λ q yields d r Ψ dλ “ ˜ T p p q λ ´ λ ` s ÿ α “ D p p q α λ ´ λ α ¸ r Ψ , (8.4)where T p p q : “ T p q ` ... ` T p p q “ diag p´ λ ´ , ..., ´ λ p ´ , , ... , looomooon n ´ p q . and D p p q α : “ G p p q´ ¨ ř p ` ... ` p α j “ p ` ... ` p α ´ ` B j p u c q ¨ G p p q . The matrix coefficient in system (8.4) hasconvergent Taylor series at λ “ λ d r Ψ dλ “ λ ´ λ ˜ T p p q ` ÿ m “ D m p λ ´ λ q m ¸ r Ψ , D m “ s ÿ α “ p´ q m ` p λ ´ λ α q m D p p q α .
49e consider η ν ` ă η ă η ν and λ in the plane with branch cuts L α “ L α p η q issuing from λ , ..., λ s to infinity in direction η , as in (5.2). Close to the Fuchsian singularity λ “ λ a fundamental matrixsolution to (8.3) has Levelt form˚Ψ p p q p λ q “ G p p q ´ I ` ÿ l “ G l p λ ´ λ q l ¯ p λ ´ λ q T p p q , (8.5)where the matrix entries p G l q ij , 1 ď i ď j ď n , are recursively computed by the following formulae (seeAppendix C for an explanation of (8.5), or [22, 43]).• If T p p q ii ´ T p p q jj “ l positive integer, p G l q ij is arbitrary .• If T p p q ii ´ T p p q jj ‰ l (positive integer) p G l q ij “ T p p q jj ´ T p p q ii ` l ˜ l ´ ÿ p “ D l ´ p G l ` D l ¸ ij (sum is zero for l “ . Since we have assumed that all the λ k are not integers, the only possibility to have T p p q ii ´ T p p q jj “ l occurs for 1 ď i, j ď p , precisely the case when some diagonal entries of A differ by non-zero integers,namely T p p q ii ´ T p p q jj “ λ j ´ λ i “ l. (8.6)If this occurs, the fundamental matrix solutions (8.5) are a family depending on a finite number ofparameters due to the arbitrary p G l q ij . Thus, in the first p columns of a solution of type (8.5) ~ ˚Ψ j p λ | ν q “ ´ Γ p λ k ` q ~e k ` ÿ l “ ˚ b p j q l p λ ´ λ q ¯ p λ ´ λ q ´ λ j ´ , j “ , ..., p . the vectors ˚ b p j q l contain a finite number of parameters. By Laplace transform, we receive the first p columns of a fundamental matrix solution of (8.1) ~ ˚ Y j p z | ν q “ ż Γ p η q e zλ ~ ˚Ψ j p λ | ν q dλ, j “ , ..., p . Repeating the same computations of Section 7, we obtain, for j “ , ..., p , ~ ˚ Y j p z | ν q z ´ λ j e ´ λ z „ ~e j ` ÿ l “ ˚ b p j q l Γ p λ j ` ´ l q z l , z Ñ 8 in S ν , where S ν is given in (7.1). We repeat the same construction at all λ , ..., λ s . This yields a family offundamental matrix solutions of (8.1)˚ Y ν p z q “ ” ~ ˚ Y p z | ν q | ¨ ¨ ¨ | ~ ˚ Y n p z | ν q ı , depending on a finite number of parameters, with the behaviour for z Ñ 8 in S ν ˚ Y ν p z q „ ˚ Y F p z q “ ´ I ` ÿ l “ ˚ F l z ´ l ¯ z B e Λ p u c q z ; ˚ F l “ ” ~ ˚ f p l q | ¨ ¨ ¨ | ~ ˚ f p l q n ı , ~ ˚ f p l q j “ ~ ˚ b p l q j Γ p λ j ` ´ l q . We conclude that the formal solution is not unique whenever a condition (8.6) occurs. Only one elementin the family satisfies ˚ Y F p z q “ Y F p z, u c q . Remark 8.1.
If we choose one formal solution ˚ Y F p z q , then the corresponding ˚ Y ν p z q having ˚ Y F p z q asasymptotic expamnsion in S ν is unique. For more details on the Stokes phenomenon at u “ u c , pleaserefer to [11]. 50 Appendix A. Non-normalized Schlesinger System
Lemma 9.1.
For the Pfaffian system (3.2) defined on D p u q contained in a τ -cell, the integrabilitycondition dP “ P ^ P is the non-normalized Schlesinger system (3.3)-(3.5).Proof. For a given i P t , ..., n u , the Pfaffian system (3.2) on D p u q can be rewritten as P “ ˜ B i λ ´ u i ` ÿ j ‰ i B j λ ´ u j ¸ d p λ ´ u i q ` ÿ j ‰ i ˆ γ j ´ B j λ ´ u j ˙ d p u j ´ u i q ` n ÿ j “ γ j p u q dλ. We are interested at λ ´ u i Ñ
0, while u j ´ u i ‰ D p u q for j ‰ i . In new variables λ “ λ, y i “ λ ´ u i , y j “ u j ´ u i , j ‰ i. we receive the following expression (defining the components A j p y q below) P “ ˜ B i y i ` ÿ j ‰ i B j y i ´ y j ¸ dy i ` ÿ j ‰ i ˆ γ j ´ B j y i ´ y j ˙ dy j ` n ÿ j “ γ j p y q dλ “ : A i p y q dy i ` ÿ j ‰ i A j p y q dy j ` n ÿ j “ γ j p y q dλ. The only singular term at y i “ B i { y i in A i p y q . The components relative to dy , ..., dy n of dP “ P ^ P are B A l B y k ` A l A k “ B A k B y l ` A k A l , k ‰ l, (9.1)For k ‰ i and l “ i , from (9.1) we receive BB y k ˆ B i y i ` reg p y i q ˙ ` ˆ B i y i ` reg p y i q ˙ A k “ B A k B y i ` A k ˆ B i y i ` reg p y i q ˙ , where reg p y i q stands for an analytic term at y i “
0. We expand the above in Taylor series at y i “ y i “
0) is B B i B y k “ “ A k | y i “ , B i ‰ “ r B k , B i s u k ´ u i ` r γ k , B i s , k ‰ i. (9.2)The above gives the non-normalized Shclesinger equations (3.4)-(3.5), because B B i B y k “ B B i Bp u k ´ u i q “ B u k Bp u k ´ u i q B B i B u k “ B B i B u k , (9.3) B B i B u i “ ÿ k ‰ i Bp u k ´ u i qB u i B B i Bp u k ´ u i q “ ´ ÿ k ‰ i B B i B u k ùñ n ÿ k “ B B i B u k “ . (9.4)If we write the components of dP “ P ^ P referring to dy l ad dλ , and we substitute into them (9.3)-(9.4),we receive (3.3), namely B l γ k ´ B k γ l “ γ l γ k ´ γ k γ l . Corollary 9.1.
For every i “ , ..., n , a solution B i p u q of (3.3)-(3.5), holomorphic on a polydisc D p u q ina τ -cell, is holomorphically reducible to Jordan form on D p u q . Namely, there exists a holomorphicallyinvertible G p i q p u q such that p G p i q q ´ B i G p i q is a constant Jordan form. G p i q is a fundamental matrixsolution of the Pfaffian system (9.6) below. roof. The conditions (9.1) for k, l ‰ i can be evaluated at y i “
0, and become B A l | y i “ B y k ` A l | y i “ A k | y i “ “ B A k | y i “ B y l ` A k | y i “ A l | y i “ , k ‰ i, l ‰ i, k ‰ l. Hence, the following Pfaffian system is Frobenius integrable B G B y k “ A k | y i “ G ” ˆ B k u k ´ u i ` γ k ˙ G, k ‰ i. (9.5)Using the chain rule as in (9.3), we receive (6.7) B G B u k “ ˆ B k u k ´ u i ` γ k ˙ G, k ‰ i, B G B u i “ ´ ÿ k ‰ i ˆ B k u k ´ u i ` γ k ˙ G (9.6)Notice that for both ϕ p u q “ B i p u q and ϕ p u q “ G p u q we have n ÿ k “ B ϕ B u k “ ùñ ϕ p u q “ ϕ p u ´ u i , . . . , u n ´ u i q . (9.7)We can take a solution G p u q which holomorphically reduces B i to Jordan form. Indeedfor k ‰ i , BB y k p G ´ B i G q “ ´ G ´ B G B y k G ´ B i G ` G ´ B B i B y k G ` G ´ B i B G B y k “ p . q , p . q ´ G ´ A k | y i “ B i G ` G ´ “ A k | y i “ , B i ‰ G ` G ´ B i A k | y i “ G “ . Therefore, keeping into account (9.7), we see that B i : “ G ´ p u q B i p u q G p u qq is independent of u . Thus,there exists a constant matrix G such that G ´ B i G is a constant Jordan form, and G p i q p u q : “ G p u q G realises the holomorphic "Jordanization" . The above arguments are standard, see for example [23].If the B i p u q are holomorphic on D p u c q and the vanishing conditions (4.3) hold, the coefficients ofthe Pfaffian system (6.40) are holomorphic on D p u c q , so that G p i q p u q extends holomorphically there, andCorollary 9.1 holds on D p u c q .
10 Appendix B. Proof of Proposition 3.1
Proof.
According to Theorem 2.1, system (1.1) is strongly isomonodromic in a polydisc D p u q containedin a τ -cell of D p u c q , defined in Proposition 2.2, if and only if dA “ n ÿ j “ r ω j p u q , A s du j , ω j p u q “ r F p u q , E j s . (10.1)In this case G p q in (2.12) holomorphically reduces A p u q to constant Jordan form and satisfies dG p q “ n ÿ j “ ω j p u q du j G p q . (10.2)52uppose that (1.3) is strongly isomonodromic, so that its integrability condoitions (3.3)-(3.5) hold. Wesum (3.4) and (3.5): n ÿ k “ B i B k “ ÿ k ‰ i r B i , B k s u i ´ u k ´ ÿ k ‰ i r B i , B k s u i ´ u k ` r γ i , n ÿ k “ B k s “ r γ i , n ÿ k “ B k s . Using B k “ ´ E k p A ` I q and ř k E k “ I , the above is exactly B i A “ r γ i , A s , i “ , ..., n. (10.3)Thus, (3.4) and (3.5) imply a Pfaffian system for A of type (10.1). Notice that if γ , ..., γ n satisfy (3.3),it is immediately verified that the system (10.3) is Frobenius integrable.Let G “ G p u q be a holomorphically invertible matrix in D p u q . Then, it is straightforward to checkthat we can choose a solution of (3.3) of the form γ i “ B i G ¨ G ´ , i “ , ..., n. (10.4)Let p B k : “ G ´ B k G. (10.5)By direct computation, it is verified that (3.4)-(3.5) are equivalent to the normalized Schlesinger equa-tions for the matrices p B k , B i p B k “ r p B i , p B k s u i ´ u k , i ‰ k ; B i p B i “ ´ ÿ k ‰ i r p B i , p B k s u i ´ u k . The above equations imply that @ i “ , ..., n, B i p B “ , where p B : “ ´ ř nk “ p B k (10.6)It follows from (10.5) and (10.6) that we can choose G such a way that G ´ p u q ˜ n ÿ k “ B k p u q ¸ G p u q “ J constant Jordan form . (10.7)Now, observe that ř nk “ E k “ I , so that n ÿ k “ B k “ ´ n ÿ k “ E k p A ` I q “ ´ A ´ I. Thus, G p u q puts A in constant Jordan form, so that we can choose G p u q “ G p q p u q , where G p q is in (2.12) . In this way, (10.4) defines the γ i starting from G p q , and by the very definition we have a Frobeniusintegrable Pfaffian system for G p q p u q dG p q “ n ÿ j “ γ j p u q du j G p q . (10.8) Up to the freedom G ÞÑ GG ˚ where G ˚ commutes with the Jordan form. γ i p u q “ ω i p u q “ r F p u q , E i s . Indeed, if (10.3) holds, then a computa-tion shows that γ i “ r F , E i s satisfies (3.3). We conclude from (10.8) that (10.2) holds.Conversely, suppose that (1.1) is stronlgy isomonodromic, so that (10.1)-(10.2) hold with ω j p u q “r F , E j s . Let us define γ j p u q : “ ω j p u q ” B j G p q p u q ¨ G p q´ , so that equations (3.3) are automatically satisfied. Let A : “ ´ A ´ I , so that E k A “ B k and (10.1) arerewritten as B i A “ r ω i p u q , A s . We multiply these equations to the left by E k , with k ‰ i . We receive E k B i A “ E k r ω i p u q , A s . The l.h.s. is E k B i A “ B i B k . The r.h.s. is (recalling that γ j “ ω j ) E k r γ i , A s “ E k γ i A ´ E k A γ i “ E k γ i A ´ B k γ i “ ` E k γ i A ´ γ i B k ˘ ` r γ i , B k s . In conclusion B i B k “ ` E k γ i A ´ γ i B k ˘ ` r γ i , B k s , i ‰ k. The only terms we need to evaluate are E k γ i A ´ γ i B k “ E k r F , E i s A ´ r F , E i s B k ““ E k F E i A ` E i F B k “ E k F E i B i ` E i F E k B k . In the second line we have used E i E k “ E k E i “ E i B k “
0, for i ‰ k , and E i “ E i . Now, observe that E k F E i has zero entries, except for the entry p k, i q , which is p F q ki “ p A q ki {p u i ´ u k q . This implies that E k F E i B i ` E i F E k B k “ r B i , B k s u i ´ u k . In conclusion, we have prove that (10.1) implies (3.4). On the other hand (3.4)-(3.5) are equivalent tothe system given by (3.4) and the equations B i ÿ k B k “ r γ i , ÿ k B k s , i “ , ..., n. which are exactly (10.1) if B k “ E k A .
11 Appendix C
We prove the expression (8.5). A fundamental matrix solution in Levelt form at λ “ λ for system (8.3)is obtained from the general theory of Fuchsian systems. It is˚Ψ p λ q “ G p p q ´ I ` ÿ l “ G l p λ ´ λ q l ¯ p λ ´ λ q T p p q p λ ´ λ q R , (11.1)with R “ R ` R ` . . . R κ , κ : “ max t T p p q ii ´ T p p q jj integer u . where R is a nilpotent matrix with R ij ‰ T p p q ii ´ T p p q jj is a positive integer. We prove that R “ p G l q ij and p R l q ij are obtained recursively by substituting the seriesinto the differential system, and are as follows. 54 If T p p q ii ´ T p p q jj “ l (positive integer), p G l q ij is arbitrary, and p R l q ij “ ˜ l ´ ÿ p “ p D l ´ p G l ´ G l R l ´ p q ` D l ¸ ij , • If T p p q ii ´ T p p q jj ‰ l (positive integer) p G l q ij “ T p p q jj ´ T p p q ii ` l ˜ l ´ ÿ p “ p D l ´ p G l ´ G l R l ´ p q ` D l ¸ ij The claim that R “ u “ u c the isomonodromic funda-mental matrix solution (6.25) in the generic case (in this case the R p j q “ p p q p λ, u c q “ G p p q ¨ U p p q p λ, u c q ¨ p λ ´ λ q T p p q . (11.2)This is a fundamental matrix solution of (1.3) at u “ u c . It is a solution (11.1) with R “ R is not uniquelydetermined (see [22] and [11]; see also [17, 12] for the case of Frobenius manifolds, and [34]). Indeed,given one representative R , all the other possibilities are r R “ D ´ R D , (11.3)where D is an inverible matrix constructed below. Now, since R “ r R “
0. This proves that (8.5) is the correct form.Finally, we explain (11.3). System (1.3) at u “ u c is holomorphically equivalent to "Birkhoff-normalforms" d Ψ dλ “ ˜ T p p q λ ´ λ ` κ ÿ l “ R l p λ ´ λ q l ¸ Ψ and d r Ψ dλ “ ˜ T p p q λ ´ λ ` κ ÿ l “ r R l p λ ´ λ q l ¸ r Ψ , which are related to each other by a gauge transformations Ψ “ D p λ q r Ψ, with D p λ q “ D p I ` D p λ ´ λ q ` ¨ ¨ ¨ ` D κ p λ ´ λ q κ q , where det p D q ‰ r D , T p p q s “
0. Then, D : “ D p I ` D ` ¨ ¨ ¨ ` D κ q . Remark 11.1.
In our case, the equations R l “ l “ , , ..., κ are conditions on the entries of A p u c q .The above discussion shows that, in the isomonodromic case, such conditions turn out to be automaticallysatisfied with the only vanishing assumption p A p u c qq ab “ u ca “ u cb . These conditions are equivalentto the conditions (4.24)-(4.25) of Proposition 4.2 in [11], and probably more conveninent. We will notenter into the tedious verification of the equivalence. References [1] Balser W., Jurkat W.B., Lutz D.A., Birkhoff Invariants and Stokes’ Multipliers for Meromorphic LinearDifferential Equations, Journal Math. Analysis and Applications, 71, (1979), 48-94.[2] Balser W., Jurkat W.B., Lutz D.A., A General Theory of Invariants for Meromorphic Differential Equations;Part I, Formal Invariants, Funkcialaj Evacioj, 22, (1979), 197-221.[3] Balser W., Jurkat W.B., Lutz D.A.,A General Theory of Invariants for Meromorphic Differential Equations;Part II, Proper Invariants, Funkcialaj Evacioj, 22, (1979), 257-283.
4] Balser W., Jurkat W.B., Lutz D.A., On the Reduction of Connection Problems for Differential Equationswith Irregular Singular Points to ones with only Regular Singularities, I, SIAM J. Math Anal., Vol 12, No.5, (1981), 691-721.66-69.[5] G. D. Birkhoff:
Singular points of ordinary linear differential equations , Trans. Amer. Math. Soc. 10 (1909),436-470.[6] Bolibruch A. A., The fundamental matrix of a Pfaffian system of Fuchs type. Izv. Akad. Nauk SSSR Ser.Mat. 41 (1977), no. 5, 1084-1109, 1200.[7] Bolibruch A. A., On Isomonodromic Deformations of Fuchsian Systems, Journ. of Dynamical and ControlSystems, 3, (1997), 589-604.[8] Bolibruch A. A., On Isomonodromic Confluence of Fuchsian Singularities, Proc, Stek. Inst. Math. 221,(1998), 117-132.[9] T. Bridgeland, V. Toeldano Laredo:
Stokes factors and Multilogarithms . J. reine und angew. Math. (2013), 89-128.[10] Cotti G., Coalescence Phenomenon of Quantum Cohomology of Grassmannians and the Distribution ofPrime Numbers, arXiv:1608.06868 (2016).[11] Cotti G., Dubrovin B., Guzzetti D., Isomonodromy Deformations at an Irregular Singularity with CoalescingEigenvalues, Duke Math. J, (2019), arXiv:1706.04808 (2017).[12] Cotti G., Dubrovin B., Guzzetti D., Local Moduli of Semisimple Frobenius Coalescent Structures, SIGMA16 (2020), 040, 105 pages.(2018).[13] Cotti G., Guzzetti D., Results on the Extension of Isomonodromy Deformations to the case of a ResonantIrregular Singularity, Random Matrices Theory Appl., 7 (2018) 1840003, 27 pp.[14] Cotti G., Guzzetti D., Analytic Geometry of semisimple coalescent Frobenius Structures, Random MatricesTheory Appl. 6 (2017), 1740004, 36 pp.[15] Doetsch G. , Introduction to the Theory and Application of the Laplace Transformation, Springer (1974)[16] Dubrovin B., Geometry of 2D topological field theories, Lecture Notes in Math, 1620, (1996), 120-348.tional Congress of Mathematicians, Vol. II (Berlin, 1998), number Extra Vol. II, pages 315?326, (1998).arXiv:math/9807034[17] Dubrovin B., Painlevé trascendents in two-dimensional topological field theory, in “The Painlevé Property,One Century later” edited by R.Conte, Springer (1999).[18] Dubrovin B., On Almost Duality for Frobenius Manifolds, Geometry, topology, and mathematical physics,75-132, Amer. Math. Soc. Transl. Ser. 2, 212, (2004).[19] Galkin S., Golyshev V., Iritani H., Gamma classes and quantum cohomology of Fano manifolds: gammaconjectures, Duke Math. J. 165 (2016), no. 11, 2005-2077.[20] Guzzetti D., On Stokes matrices in terms of Connection Coefficients. Funkcial. Ekvac. 59 (2016), no. 3,383-433.[21] Guzzetti D., Notes on Non-Generic Isomonodromy Deformations, SIGMA 14 (2018), 087, 34 pages[22] Guzzetti D., Introduction to Linear ODEs in the Complex Domain and Isomonodromy Deformations –Lecture Notes for a Ph.D. Course at SISSA.[23] Haraoka Y., Linear Differential Equations in the Complex Domain, from classical theory to forefront, Su-ugaku Shobou, Tokyo (2015) (in Japanese), and Springer Lecture Notes in Mathematics (2020) (inEnglish).[24] Hurtubise J., Lambert C., Rousseau C.,
Complete system of analytic invariants for unfolded differentiallinear systems with an irregular singularity of Poincaré rank k , Moscow Math. J. 14 (2013), 309-338.[25] Hsieh P., Y. Sibuya Y., Note on Regular Perturbations of Linear Ordinary Differential Equations at IrregularSingular Points, Funkcial. Ekvac, 8, (1966), 99-108.[26] E.L. Ince:
Ordinary differential equations . Dover, (1956).
27] K. Iwasaki, H. Kimura, S. Shimomura, M. Yoshida
From Gauss to Painleve’ . Aspects of Mathematics ,(1991).[28] Jimbo M., Miwa T., Ueno K., Monodromy Preserving Deformations of Linear Ordinary Differential Equa-tions with Rational Coefficients (I), Physica, D2, (1981), 306.[29] Klimes M., Confluence of Singularities of Non-linear Differential Equations via Borel-Laplace Transforma-tions , J. Dyn. J. Dyn. Control Syst. 22 (2016)[30] Klimes M.,
Analytic Classification of Families of Linear Differential Systems Unfolding a Resonant IrregularSingularity , SIGMA 16, (2020), 006, 46 pages.[31] Klimes M.,
Confluence of Singularities in Hypergeometric Systems , Funkcialaj Ekvacioj 63, (2020), 183-197.[32] Loday-Richaud M., Remy P.,
Resurgence, Stokes phenomenon and alien derivatives for level-one lineardifferential systems . J. Differential Equations 250, (2011), 1591-1630.[33] Miwa T., Painlevé Property of Monodromy Preserving Deformation Equations and the Analyticity of τ Functions, Publ. RIMS, Kyoto Univ. (1981), 703-721.[34] Mazzocco M.. Painlevé sixth equation as isomonodromic deformations equation of an irregular system, CRMProc. Lecture Notes, , Amer. Math. Soc. 219-238, (2002).[35] P. Remy: Matrices de Stokes-Ramis et constantes de connexion pour les systèmes différentiels linéaires deniveau unique . Ann. Fac. Sci. Toulouse Math. (6) 21 (2012), no. 1, 93-150.[36] Sibuya Y., Simplification of a System of Linear Ordinary Differential Equations about a Singular Point,Funkcial. Ekvac, 4, (1962), 29-56.[37] Sibuya Y., Perturbation of Linear Ordinary Differential Equations at Irregular Singular Points, Funkcial.Ekvac, 11, (1968), 235-246.[38] R. Schäfke:
The connection problem for two neighboring regular singular points of general complex ordinarydifferential equations . SIAM J. Math. Anal. 11 (1980), 863-875.[39] R. Schäfke:
A connection problem for a regular and an irregular singular point of complex ordinary differ-ential equations . SIAM J. Math. Anal. 15 (1984), 253-271.[40] R. Schäfke:
Über das globale analytische Verhalten der Normallosungen von p s ´ B q v p s q “ p B ` t ´ A q v p s q und zweier Arten von assoziierten Funktionen . Math. Nachr. 121 (1985), 123-145[41] R. Schäfke: Confluence of several regular singular points into an irregular singular one , J. Dyn. ControlSyst. (1998), 401-424.[42] Ugaglia M., On a Poisson structure on the space of stokes matrices, International Mathematics ResearchNotices, Volume 1999, Issue 9, (1999), Pages 473-493,[43] Wasow W., Asymptotic Expansions for Ordinary Differential Equations, Dover (1965).[44] Yoshida M. , Takano K., On a linear system of Pfaffian equations with regular singular points, Funkcial.Ekvac., 19 (1976), no. 2, 175-189.(1998), 401-424.[42] Ugaglia M., On a Poisson structure on the space of stokes matrices, International Mathematics ResearchNotices, Volume 1999, Issue 9, (1999), Pages 473-493,[43] Wasow W., Asymptotic Expansions for Ordinary Differential Equations, Dover (1965).[44] Yoshida M. , Takano K., On a linear system of Pfaffian equations with regular singular points, Funkcial.Ekvac., 19 (1976), no. 2, 175-189.