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