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Connection formulae for Painlevé V functions
Physica 20D (1986) 187-216 North-Holland, Amsterdam
CONNECTION FORMULAE FOR PAINLEVk V FUNCTIONS II. THE 8 FUNCTION BOSE GAS PROBLEM Barry M. MCCOY and Shuang TANG Instilule for
Theoretical Physics, State University of New York ut Stony Brook, Stony Brook, New York 11794,
W.4
Received 15 April 1985
We continue our study of those Painlevt V equations which admit a one-parameter family of solutions analytic at the origin and solve the global problem of connecting the 2-parameter expansion about I = 0 with the 2-parameter expansion valid for I -) f iw. A special case of our results is applied to the study of the one-particle reduced density matrix of the one-dimensional gas of impenetrable bosons. We also use our results to obtain a 2-parameter connection formula for a class of Painleve III functions.
1. Introduction The Painleve V equation (y-1)* t*
(“y+&-i)+yyt-‘+6y(y+l)(y-l)-‘,
(1.1)
with (y=
y=4(1-28)
-p+*,
6= -8,
(1.2)
plays an important role in the study of the correlation functions of the transverse Ising model [l, 21 and the impenetrable Bose gas [3]. The first paper of this series [2] studied the asymptotic behavior for t + m, which is the physical region for the transverse Ising model. In this present paper we study the asymptotic behavior for t=
-ix
and
x+
+cc,
(1.3)
which is the physical region for the bose gas problem. For the physical applications it proves useful to consider the Painleve equation in the u form of Jimbo and Miwa [4],
(1.4) where y f 1 = (a;)* f 2ixu,” - 80, - 4( n* - r?‘) + 8x0,’ (a; * 2i8)* + 4n2 0167-2789/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(1.5)
188
B. M. McCoy and S. Tang/ Connection formulae for PainleG V functions Xl
For the impenetrable Bose gas the off-diagonal one-particle reduced density matrix was shown by Jimbo, Miwa, Mori and Sato [3] to be obtainable from (1.4) with n = 1 and 8 = 0 as
p(x) = poexp ~xdx’o,(x’)x’-‘,
0.6)
where we specify the boundary condition for (1.4) as
q(x) = - 4.x’+ $X3 + O(x4),
(1.7a)
h=2/lr.
(1.7b)
with
The problem of physical interest is the determination of the behavior of al(x) [equivalently p(x)] as x-, +co. Somewhat more generally Creamer, Thacker and Wilkinson [5] showed that if one generalizes the impenetrable Bose gas to the Bose gas with pairwise interaction cS(xi - xi) then the first c-l correction to the c + cc [impenetrable] limit of p(x) involves the derivative of al(x) with respect to A. These authors initiated the study of the X dependence of the x + 00 asymptotic behavior of al(x) by both numerical and analytic means and one motivation for this present study is to provide a derivation of their several conjectures. Even though the special case n = 1 is perhaps of the most direct physical interest it is most useful to consider all u,, with integer n on the same footing. The reason is that if we define X,(x)
(1.8)
= X~(0)~xdx’u,(x’)x?
then X,(x) satisfies the Toda equation
2
ln X,(x) =Xn_lXn+l/X,‘,
(I .9)
which in terms of a, reads x
‘u;
xu; - a,
- 2 + 2U”= u”+t + a,_,.
(1.10)
Thus, since from (1.4) we may require a,, = u_ “, as long as the boundary conditions on (1.4) are compatible with (1.10) we may generate u, for any n from a knowledge of u,. The PainlevC V equation (1.1) (or 1.4) with the restrictions on (Y,p, y, and 6 given by (1.2) with n an integer has the important property that there exists a one-parameter family of solutions analytic at x = 0. If the dependence of the one constant of integration is chosen to be compatible with (1.10) this analytic family has the form
u,(x) =
2 u,(n)x2’+x2n+1 fJb,(n)x2’, I-1
I-0
(1.11)
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189
where for n 2 1
al(n)
= -
4(n2 - 82) 4n*_ 1
(1.12)
and
b,(n) =
2(P - e*) (-1)“2XfJ 1(2/ - 1)2(21 + 1) *
(1.13)
1=1
For n=O q(x)
= -2xx-4(P+B*)x*
+ 0(x3).
(1.14)
More generally as x + 0 we have the 2-parameter family of solutions which we follow Jimbo [6] and parametrize as
gc1t(28
q)(x) ==g+
+
u)
-
0)(4e-wi/2x)1+a
,(fI’,,(28 +u)(4e-wi/2x)1-o+
0(,x*(‘-“)).
(1.15)
The object of this present paper is to calculate the behavior of u,(x) as x + 00 for either the boundary condition (1.11) or (1.15). We will carry this out in several steps. In section 2 we study the case n = 0. For the case of the solutions analytic at x = 0, aa has an asymptotic expansion as x --) 00 valid except when X is real and greater than Bcot rr13 q)(x)
=Ax
+ B + x-V_,(s)
+ x-*F_*(s)
+ x-3F_,(s)
+ 0(x-“),
(1.16)
where s=x+x,+klnx,
(1.17a)
k = A/4,
(1.17b)
~=+[8*+A*/4],
(1.17c)
~_,=~[~*+,4*/4]sin4s-~A(8*+A2/4)~
(1.17d)
F_,=
(1.17e)
-~A(~*+A2/4)sin4s+C_,~s4~+K-2~
with c_,=fiB(4B-4+A*),
(1.17f)
K-,=$(8*+%)
0 .lW
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B. M. McCoy
andS.
Tang/ Connection formulae for Painle& Vfunctions II
and 1
4(282+3A2-6)-(B2+~)]~i~4s
F-3=512 + C_,cos4s
+ K_,,
(1.17h)
where C_, and K_, are still to be determined. A principle result of section 2 is to show that the two constants of integration k and x,, are given in terms of X by cosn@-isin&
(1.18)
and e4ixo
= 2iskI’( -ik + 8/2)r( -ik - 19/2) I’(ik + 8/2)I’(ik - e/2) ’
(1.19)
These one parameter connection formulae are derived as a special case from more general 2-parameter connection formulae for the initial condition (1.15). The results (1.16), (1.18), and (1.19) break down when
x > 8 cot se.
(1.20)
because then k has an imaginary part of f i and hence the term x -‘F_,(s) in section 3 where we show that when (1.20) holds. cos 2s
is of order x. We treat this case
(1.21)
where S> =x+x,>
+k,
lnx,
(1.22)
k, =$A,
and
G%>)=
f12+ 3(4k2, +l) 4k2, - 1+ 8’ + 2k> cot2s> + 2 4[sin2s,12
(1.23) ’
The connection formulae are now hsin7rB--cosnB k>=&In (6
(1.24)
1
and e4ix,’
=
_28ik,
I’(-ik, r(ik,
+++8/2)r(-ik, +)+e/2)r(ik,
+f-e/2) +f-e/2)
’
(1.25)
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In section 4 we study the case n > 0. The x + 00 expansions are now derived from (1.16) and (1.21) by use of the Toda equation (1.10). We denote the asymptotic expansion which has no poles (the generahzation of (1.16)) by 4(x) and the asymptotic expansion which does have poles (generalization of (1.21)) by a,( x ). Then we have
6”(X) =/lx
B(n) + x-lP_l(Z”, n) + X-2E_2(?“,n) + o(x-3),
+
(1.26)
with S,=x+4,+Alnx,
(1.27a)
4
B(n) =f[P+A2/4-nq,
(1.27b)
~_1(~,“)=6[(82+A2/4-n2)2+n2A2]1’2sin4s-~A[82+A2/4+n2],
(1.27~)
F_2(~,n)=
(1.27d)
-$4[(B2+A2/4-n2)2+n2A2]sin4s+~_2,
where k_2 is a constant to be determined and
i&(x) = XFl(s,,n) + F,(s,, PI)+ x-lF_l(S,, n) + 0(x-2),
(1.28)
where
5,=x+x,+~hlx, 4
(1.29a)
cos 2s F,(s, n) = 2=+-4,
(1.29b)
F&n)=; [ 82+&2-l
1
1 cos2s ++m+
f32+n2+3(A2/4+1) 4(sin2s)2
(1.29~) ’
and
(1.29d) We then show that if the phase angles are chosen to satisfy n
-
(1.30a)
X2n=X;+2Cff2,-1, I-1 n X2n+l
(1.30b)
=xo+2&,,, I-1 ”
XZn+l
=x0’
+2
c ffzi-1+ I-1
a2n+1,
(1.3oc)
n-l X2n = x0+2
c l-l
~ZI+~Zn,
(1.30d)
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B. M. McCoy and S. Tang/ Connection formulae for PainlevP V functions II
with e4ia,
8*
=
+
A*/4 - n* + inA
(1.31)
([~2+A2/4-n2]2+n2A2)1’2 that for x --) 00 the general case with X not real above Scot m!I satisfies ~2”W
(1.32a)
= ~2”b)
and fJ*n+A-d
(1.32b)
= 52;n+lb)9
whereas if X > Scot Ire Q*“(X)
(1.33a)
= ~2nb)
and O*n+dX)
(1.33b)
= ~2;n+1W.
In (1.32) A = 4k and x0 are given by (1.18) and (1.19) whereas in (1.33) A = 4k, (1.24) and (1.25). Therefore, for the particular case of the Bose gas where 8 = 0 and A > l/m q(x)
-f+x-‘{+(4kt
=4k>x+2k;
-1)sin4s’+Sk,
~0~4s’ -ak,(4k;
and xc are given by
+l)}
+0(x-*),
with S’=x+xc+k,lnx,
(1.34)
k,=&ln(h?r-l),
(1.35)
and e4ix
0’ = _28ik.
T*( -ik, r*(ik,
+ i) +$)
.
(1.36)
The result (1.35) was conjectured in ref. 5. When h + 2/s k, -0
(1.37a)
and e 4ixo<
=
_28ik,
e-4ik,(r+21n2)
where y is Euler’s constant.
7
(1.37b)
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These results were also first numerically conjectured in ref. 5. Finally we conclude in section 5 with an extension of our connection formulae to a class of Painleve III functions.
2. Thecasen-0 We begin our calculation by considering the asymptotic expansion of u,(x) given by (1.16) and (1.17). It is easily verified by direct substitution that (1.16) does indeed satisfy (1.4). Thus, the purpose of the section is to calculate x0 and k in terms of the initial conditions (1.14) and (1.15). As in the previous paper [2] our method will be to use the deformation theory of Jimbo and Miwa [4]. However, as there are major differences between ref. 2 and the present case it is not out of place to begin our considerations with a brief recapitulation of the basic procedures. We characterize our PainlevC V function by its relation with the 2 x 2 linear matrix differential equation ~=_4(z)Y(z),
(2.1)
where
A(z) = -2i[ i
_“X] + i&(x)
(2.2)
+&A,(x),
with
A,(x)
1. duo - z1 dx l.du,, 8
=
--aldx-2
l.du,, KG-Z l.du, a%-
8 (2.3a) ’ I
and 1. do,
a1dx
Al(X) = Y
1
I
l.du,
-a'dx
* I
(2.3b)
We uniquely define solutions to this linear equation by the requirement that as 1z ) + 00 Y(z) =
(1+
Q(z-‘))[ “6’“’e*!z].
(2.4)
The linear equation (2.2) has regular singularities at z = 0 and z = 1 near which Y(z) behaves as
Y(z) =
C,(x)[ ‘“0”z_#”
(2Sa)
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194
and e/2
0
Cl,
(2Sb)
(z- 1)-e’2 1 where
G’b)~ob)Go(x) = [Y2
$21
G;‘(+,b)G,b)
$21.
(2.6a)
and
= [Y2
(2.6b)
The relation this linear equation has with the PainlevC V function is the theorem [4] that when u,,(x) obeys (1.4) then the connection matrices Co and C’ are independent of x. More precisely, taking into account the fact that the above definition defines Co and C’ only up to pre and post multiplication by diagonal matrices we will focus on the following two quantities which deformation theory guarantees are independent of x:
IO=-
coca 11
22
coca 12
21
(2.7a)
’
and
(2.7b)
We obtain our connection formulae by using the small x expansion (1.15) and the large x expansion (1.16) in the equation (2.1). In each case the invariants I0 and I’ are calculated in terms of the boundary conditions. Then by equating these two independent forms of the invariants we obtain the desired results. We thus must approximately solve (2.1) for x = 0 and x + cc. The case x = 0 was previously studied by Jimbo [6]. The result for the invariants is
10~
_e-nia
Al=
_e-nia
[sina(B+f~)-Ssinn(8-~a)][l-F~e~~~] sin rr( 8 + fo) + Seeni sinn(B - &)] [l + S-l]
(2.8a)
and [sins( 8 + )a) + Selrio sin?r(8-+a)][1 [sins(B+ta)-SsinIr(B-ta)][I-s’--’e-~i~]’
+ S-i]
(2.8b)
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195
where s is related to the i in the boundary condition (1.15) by r(e+:e+l)P(?)
_ (2.8~)
s=z-401(B_:~+l)T2(~)~. In the special case where u = 0 and the boundary condition (1.14) is to be used
(S+s’-‘)d = -2iX
(2.9)
and the invariants reduce to (2.10a) (2.10b) Here we make only one additional remark. In order to equate the invariants which we will calculate for x --* co with (2.8) we must use the same conventions on branch cuts and Stokes lines as those of ref. 6. When t = e -vi/z x is real Jimbo completes the definition of Y(z) by cutting the z plane from 0 to 1 and on z = i + i y from 0 < y < 00. The criteria for this latter cut is (2.11a)
argt(z-+)=a/2. Now if t = e’*x and $I goes continuously from 0 to -n/2 arg(z-f)+
then (2.11a) is preserved if
+a.
(2.11b)
Thus to be consistent with ref. 6 we must cut the plane from + 1 to - cc as shown in fig. 1. Furthermore, because the phase of (2.11b) is + s the cut going from + 1 to - cc must be thought of as passing above the point z = 0 and hence the connection matrix Co is obtained by letting z + 0 from the lower half plane. We now turn to the calculation of I0 and I’ for x --) cc. In subsection 2.1 we approximately calculate Y(z) for z # 0, 1 by the WKB technique. In subsection 2.2 we approximately solve (2.1) for z = 0 and z = 1. Then in subsection 2.3 we connect these 3 solutions together to calculate the invariants. The major difference with the previous paper [2] is that in the present case the turning points of the WKB solution coincide with the monodromy points z = 0, and 1. Hence the same functions will serve both as turning point and as monodromy connection functions. Finally in subsection 2.4 we equate the two different expressions for the invariants to obtain the desired connection formulae.
2.1. WKB solution for z # 0,l When x x=-1 we use the expansion (1.16) to find that to order 1 yz
( s)e-4is
(2.12)
B. M. McCoy and S. Tang/ Connection formulae for Painlevk V functionsII
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Fig. 1. The branch cut conventions in the z plane used in the case x + + co. To be consistent with ref. 6 the point z = 0 must be thought of as lying below the cut.
and hence (2.1) becomes
Y(z).
(2.13)
We seek a solution of the form Y,(z)
=e
**ixr{&f(z)
+x-l&‘(z)
+ 0(x-2)},
(2.14)
Then to order x the equation (2.13) is simply *2irf,*(z)
= [ -tx
(2.15)
and hence G(z)
= [ ;]
(2.16a)
and f&(z) =
01-
fi I
(2.16b)
B.M. McCoy and S. Tang/ Connection formulae for PainlevP Vfunctions II
197
To obtain fO* we must proceed to the order 1 term in (2.13). This gives
*2if,*(z)
+
A&*(z) =[-$ i]
f,+(z)
Then, using (2.16) we find that for fz the lower equation is
(2.18) so that
fo+(z) =
c+($y4.
(2.19a)
Similarly, f,-(z)_C-
(zz-l 1-iA’4*
(2.19b)
Therefore, using the boundary condition (2.4) to determine C * we find that the leading WKB solution to (2.1) valid for z # 0,l is Y(r) = IL
y+l,
(2.20)
where y+=
e2ixz(&.)iA’4( y)
y_=
e-2ixr
(2.21a)
and
(*)-iA’4( ;).
(2.21b)
2.2. Approximate solutions near z = 0 and z = 1 The WKB solution (2.20) breaks down near z = 0 and 1 and the approximate equation (2.13) must be studied anew in these two cases. Consider first z = 0 and define a new scaled variable .E=xz,
(2.22)
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B. M. McCoy and S. Tang/ Connection formulae for PainlevP Vfunctions II
where Z is of order 1. Then calling Y(r) = F(T)
(2.23)
and letting x + cc we find from (2.13) that F satisfies
(2.24)
We reduce this to a second order equation by first writing the two component equations separately as
(2.25a) and s=(Zi+g)Y,++(-y--t)Y,.
(2.25b)
Thus if we solve (2.25a) for r, as (2.26) and substitute into (2.25b), we obtain
(2.27) which may be solved in terms of the + confluent hypergeometric
F)l) = (4iZ) e/2e-2i~$[!++~,
function as [7]
l+fl;4ir]
(2.28a)
and 1-t 8; -4iZ
1.
(2.28b)
We then may find the component Y, from (2.26) using the relation (71 o,c;x)=~[(a-c+1)~(a+1,c;x)-~(a,c;x)] =$[(a-
c+x)~(u,c;X)-I/(u-1,c,x)].
(2.29)
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Thus we obtain two linearly independent solutions of the approximate equation valid near z = 0
J;(l)
=
C(l)(4iz)8/2
e-2iZ
I
#($+;,
1+8;4iE)
(~+~)$(I+Q!+~,
l+f3;4i?)
and (-~+~)#(l-~+~, y(2) = c7*)( -4iz)
1
(2.30a)
l+B,-4iZ)
e/2 e*il
(2.30b) #(-y+;,
1+8;
[
-4iZ)
’ I
We also require solutions valid near z = 1. Here we define s=x(z-1)
(2.31)
F(a) = Y(z).
(2.32)
and
Then letting x + oo in (2.13) we find that y satisfies
(2.33)
Thus !$_ dz -
(2.34a)
~=(2i_~)E,+fe4i$(~+:)~~
(2.34b)
and
and hence p2;= -~e4i~(~-:)1(~+(2i_~)~~)
(2.35)
and so d*& ~++~+(4+f(2i-A)-$}~I=0.
(2.36)
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Therefore we find that near z = 1 the two linearly independent solutions are
p(1) =
#(-y+i, c(l)(4iz)‘/*
l+B;4ii)
e-2it
(2.37a) -e4i’(~+~)+(l-~+~,
1+8;4ii)
e-4is($-~)#(l+~+~,
1+8;
i
and
v(2) = c(*)(
-
4iz)‘/* e*ii
+(Y+i,
[
1+
e; -4iI
-4ii) (2.37b)
1
i
TO complete the definition of the solutions (2.30) and (2.37) we must define the branch cut implicit in il/. This will be done in the next subsection. 2.3. Calculation of the invariants We now calculate the invariants I’ and I0 by using the approximate solution (2.21), (2.30) and (2.37) to connect the boundary condition (2.4) with the definition of the connection matrices (2.5). From the boundary condition (2.4) we define the solution Y(l) such that in the lower half-plane
y(1) t: y is exponentially
(2.38a) small. Similarly we define the solution Y0 such that in the upper half-plane
y(2) z y +
(2.38b)
is exponentially small. Here Y, are the WKB solutions (2.21). Furthermore, from our definition of branch cuts and Stokes line (2.38a) also holds on the real axis approached from below and (2.38b) holds on the real axis approached from above where Y, are no longer exponentially small but oscillate. However, when the solution Y(*) is continued around the branch point at z = 1 to the lower side of the cut from 1 to - 00 an additional contribution from Y_ is picked up and hence on the segment of the real axis 0 < z < 1 approached from below (2.39)
y(*) = Y, + c_ Y_.
The situation is summarized in fig. 2. We consider first matching solutions near z = 1. Consider first Y (*). From fig. 2, (2.38b) and (2.21a) as z + l* above the cut y(2)
~
e2ixe2iiXiA/4~-iA/4
0 0
1 ’
(2.40)
where .Zhas phase e Oinfor Z > 0 and has phase eni for z’< 0. This must be matched to a linear combination of p(‘)(Z) and 8(*)(Z) in the limit Z + f cc. For this we need the asymptotic expansion of #(a, b, z).
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Fig. 2. The relation between the exact solutions Y(l) and Y(*) and the WKB approximate solution Y,.
Remembering -;a
that # is a multivalued function we note [8] that as z + cc for
+$lr,
$(a, b, z) = z-,
(2.41a)
while for z = e * 3”i~zx with x + + cc
(2.41b) Thus, if in F(2) of (2.37b) we define branches so that arg( -4i&) = -a/2, arg(-4iZ)
= + $,
arg(-CZ)
= -g,
for for
Z > 0, ,Z< 0
(2.42a) on top of the cut
(2.42b)
and for
Z< 0
on bottom of the cut,
(2.42~)
we see that for Z + f 00 on top of the cut p(2)
+
c(2)(
_ 4iZ)-‘A14
e2ii
.
y
(
1
(2.43)
This matches (2.40) if e(2)
= e2ixXiA/44iA/4
ewA/8e
(2.44) Next consider Y cl). As z + 1* from below the cut y(l)
z
e -2ixe-2iiX-iA/4ZiA/4
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B. M. McCoy and S. Tang/ Connectionformulae for PainlevC V junctions II
where ? has phase e Onifor Z > 0 and has phase eeni for Z < 0. This solution will match to f(l) of (2.37a) if the phase of iZ in # is chosen as arg(4ii)=T,
for
arg(4iZ) = - ;
Z>O,
for
(2.45a)
.Z< 0
below cut.
(2.45b)
and if c;(l)
= e-2ixx-iA/44-iA/4e-rAA/8
(2.46)
We now may determine the connection matrix C’. For this we need the matrix G,(x) defined by (2.6b) where for A,(x) we use the appropriate matrix in (2.13). Thus we find
(2.47)
and hence from
C’=lim
2-l
(z-
l)-e’2
0
0
(z - l)e’2
1G,‘(x)Y(z),
(2.48)
we have the 4 explicit equations
Y21) ,
(2.49a)
(2.49b)
Ci2 = ,LF~ (2 - 1) e’2{ Yj’) - e-4isYJ1)},
(2.49~)
(z - 1) “‘{ YJ2)- e-4iSY$2)}.
(2.49d)
C:, = !,I
We now may expand Y(l) and Yc2) using r(1~(“‘c;E);rr(a_c+l)+il-
c)
.r(c-- 1) r(a) *
(2.50)
B. M. McCoy and S. Tang/ Connection formulae for PainlevP V functions II
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Thus, noting that the phases are defined from (2.42) and (2.46) we find as Z + 0
r(
iqA
ij +(4iz)P0~j_$~:)
Y(l) = c;‘1)(4ii)8/2
(2Sla) _
G-0)
AL+;
ei4s
)i r +;
(
(
L
+
1
r(e) .
(4it’)-’ r
(
1-y+;
11
and
yC2)
e
and consequently (-il
=
‘I2
c(2)( _ 4iz)
-6;(l)
(2Slb)
we obtain 4012e”i@/48r( _ 8) rl_!.+-;
'
(2.52a)
(
) ci2= ct2,4e/:e-"ie'4er(.-e)e-4i",
(2.52b)
($+$r(+-$) (2.52~)
and ci2
=
-
$2)
4-8/2e~iO/4r(~)ee-4i~
ri+' A$+; !
(2.52d) 1
Thus we compute the invariant
(2.53)
We now turn to the calculation of the invariant I0 obtained from the connection matrix at z = 0. Here, in contrast to the point z = 1 the solutions Y(l) and Yt2) do not play symmetric roles in the calculation.
204
B. M. McCoy and S. Tang/ Connectionformulae for PuinlevP Vfunctions II
Consider first Y(l). Since we are calculating the connection matrix for z + 0 from below the cut we see from fig. 2 that the procedure for matching Y(l) to r (the solution valid near z = 0) is identical with the matching procedure used near z = 1. Accordingly if we define the phase of 4it in the function Y(l) of (2.30a) by arg4iI = a/2,
for
E> 0
(254a)
and arg4iZ = -n/2
for
Z
(2.54b)
then we may use the asymptotic expansion (2.41a) and find that near z = 1 y(1) 3 qqw
(2.55)
where c;l)
= XiA/44iA/4
e-nA/8
(2.56)
The behavior of Y@) near z = 0 is, however, quite different because as indicated in fig. 2 the behavior in 0 < z < 1 is given by Y, + C- Y_ instead of by Y, alone. The constant C_ is calculated by matching the form Y, + C- Y_ with the asymptotic expansion of y(2) = @)jw)(
9
(2.57)
valid for Z + - 00 when -i.Z has phase -3/2a (2.42~). For this region we must use the asymptotic expansion (2.41b). Thus as a matching condition we have
e2i*e2iiXiA/4z
--iA/
[ 01
1
+ c_
e-
2ixe-2iiX-iA/4ZiA/4
[0 1
1
2nie”i/2(l+iA/4+38/2) e-4ii(
1
I
_4z)iA/4-
~(,,!5,!!)
277ie r(++j)r(y_;)
e-4ii(-4~)iA’4-e’2-1
(2.58)
B. M. McCoy and S. Tang/ Connection formulae for Painlev6 Vfunctions II
205
and hence we find -4isqiA/4
_ 2aic:(2)e2iXxiA/4
e-3nA/8
c_=
(2.59)
=
_ 2rie-4i%
where in the last line the definition of s (1.17a) has been used. We now may calculate the behavior of Y(*)(z) as z + 0 along the positive real axis by matching Yt2) = y++ c_ Y_,
(2.60)
with z + O+ to the linear combination cl”‘jw’(
2) +
@)jm(
3,
(2.61)
with I + + cc on the real axis. In (2.61) we define the phase of 4it in y(l) as +n/2 in Y(*) as -n/2. Hence in both cases expansion (2.41a) may be used. Thus we find eni(iA/4)X
-iA/4ziA/4
e Zil
ni(-iA/4)XiA/4,--iA/4e-2ii
1 (
=
@)(4iz)-iA/4,-2ii
i (
+ cjZ)( )
_4iZ)iA/4e2if
f (
and the phase of -4iZ
)
0) (2.62)
and hence @) =
e9rA/8
c_XiA/44iA/4
(2.63a) and @?) =
e-3nA/8X-iA/44-iA/4
(2.63b)
We now may calculate the connection matrix at z = 0 from the definition (2.5a) if we note that as x --, cc 1 G,‘(+
1
.1 -1
_-
+;
!A+; I
*
(2.64)
206
B. M. McCoy and S. Tang/ Connection formulae for Painlevh V functions II
Thus
cp, = feoz-
e/2 yp _ yp } 3 1
(2.65a)
Cf2 = lim z -‘I2 { Y/‘) - YJ2)} ,
(2.65b)
2-0
(2.65~)
(2.65d)
Here we may use (2.50) to first write that as Z --f 0
(2.66a)
and y(2) 5 qqW)
+
CqF(2) 2
cj’)(4ij)8/2
( 1 r(!++i 11 rLee)B
r 4-T I
[
+
(4it)-’
r(e)
=z
(2.66b) cjO”‘(4iF)
e/2 iA ( 4
+! “j+7$,
+Q2)(-4iz)e/2 r( i
r(-0) $4
+(4i~)-@+q)+;)
:)
+ (-4i’)-er
_$i$ (
)I
B. M. McCoy and S. Tang/Connection
207
formulae for Painlevi V/unctions II
and thus from (2.65) obtain
(2.67a)
(2.67b)
co
=
_
cpsr(
l2
per(
-e)e"'~/4(4xp2
(!g_;)r(~_;)
-e)e-"'~/4(4X)e'2
(2.67~)
- (!4+!)+!.$_;)
and
c0
= q%q
22
e) tYi@/4(4,)
-e’2
+ per(e)
(!.$+;)q!+!)
e”‘@/4!4x)
-#‘2
(2.67d)
(!.A+;)+!++;).
Therefore we find the required invariant 11 22 zo=- coca coca 21
12
C_
=
+ 4-iA/4e-nA/2enifl/2
=
9
(2.68)
c_ _ 4-iA/4e-nA/2e-ai@/2
where from (2.63)
e-4ix04+3iA/4e-rrA/4 C_=
-2ai
(2.69)
r(i+y+;)r($:).
208
B. M. McCoy and S. Tang/Connection
formulae for PainleuP Vfunctions II
2.4. The connection formulae
We now may equate the expressions for the invariants calculated for x = 0 with those calculated for X--,00.
We consider first the one parameter case where the invariants are given by (2.11b). For the invariant I’ we equate (2.10b) with (2.53). Thus
-lrA/4+inO/2
iA&‘- 1 lri0 iX8-’ + 1 = e
_
e?rA/4-id/2
_
enA/4+in8/2
eail?e e-vA/4-inO/2
(2.70)
which may be solved for A to give e ~A/2=c0Sd?-AXg-1sind
(2.71)
or, in terms of k (1.17b) (2.72) Similarly for the invariant I0 we equate (i.lOa) with (2.68). Thus
c_diA/4
M-l
enA/
+
enib’/2
+ 1
(2.73) c-4
iA/
enA/
_
e-eiO/2
from which if we solve for c_ and use (2.74) we obtain (2.75)
Then eliminating A&’ on the right using (2.71) and using (2.69) for c_ yields e-4ix
O=
-_&A-iAr
x {2c0sTe-
erAI
- eCTA'A/2}
(2.76)
B.M. McCoy and S. Tang/Connection
209
formulae for PainlevO Vfunctions II
from which, using (2.74) again we obtain the desired result
++(!L+!) e-4ixo
=
4-‘A
(2.77) +!q)+~+;)
Thus we have derived the one-parameter connection formulae (1.18) and (1.19). We now may repeat the same procedure and derive the general 2-parameter connection formulae by using the 2-parameter invariants (2.8). Again we consider first I’. Then
sin7T i. ++i
[sina(8+fa)+Se”i0sina(B-$r)][1+S-1]
1
-e-*‘u[sinn(e+t~)-ssinn(8-io)l[l-S-1e-~i~]=e-lesin~~“_“) 4
(2.78)
2
which can be solved for A to give
- i sin
~0
[
$evio/2 sins
(
8-i
1
+S=1eP+“/2sinn(B+~)]) (2.79)
Similarly for I0 we have -Ssins(B-fO)][l-S-le”io] [sin++te) sin 8( 0 + $u) + Seeni sinn(B-$a)][1
_e-aio
C_4iA/4
enA/
+
+S-l]
enib’/2
=
(2.80) c-4
iA/4enA/2
_
e-niO/2
This can be solved for e-4i*o to give
1
e-4ix0 =
s4
_,
r1+q+; ( . IA
)
r1+‘A-8 ‘4 (
2)
r g+ ( 4
cos~[sinn(t9+~)+sin7r(B-~)]
!+(+:)
210
B. M. McCoy and S. Tang/ Connection formulae for PainlevO V funcrions II
x { - t(sinlr(28 + e--A/2
cos~[sinrr(fl+q)+sina(8-s)] ria/2gsin 7f( 8 - fa) + e ‘“i’/2~-1sina(B+ta)]},
-isinae[e-
1
e -4ix,, = _
+ u) + sina(28 - u))
(2.81)
.(l+~+;)r(l+$;)r(~+;)$$;)
__
~4
-4
4a2
cos?J-[sin,(e+;)+sinr(B-q)]
x sinrr(28+e)+sin7r(2e-a) ( - (e-
~A/2+e-A/2)~~~~
+2sinysin7rB
S-
r
SILT
[. sinrr
l .
(
(e+? “1+smn. (e-z 71
13+Z~
l
1
-Ssma -.
(
e-Zu
l
)1l
(2.82)
In this form it is clear that as u + 0 we regain (2.76) in which the s dependence disappears. However, for u # 0 it is, of course, not possible to express x,, in terms of A alone.
3. The case
x>ecotse
The question now arises of where in the complex X plane the expansion for uo(x) of (1.16) and the connection formulae (1.18) and (1.19) are valid. If cos rre - A&’ sin n/3 is not real and negative then
and so for real X, (3.2) and consequently A>ecOtse
the term x-‘F_,
is in fact smaller than the leading term. However, when X is real and (3.3)
then x-‘F_, is of the same order of magnitude as the leading term Ax and the expansion (1.16) breaks down. This breakdown is analogous to the one which occurred in the previous paper [2] for t + co when I Im h I > 0. However, in that previous paper there was no compelling physical reason to pursue the special case further. In contrast in this present problem the physical case of the Bose gas f3= 0 and X = 2/77 lies in the region where (3.3) holds and thus we are forced to do a further analysis.
211
B. hi. McCoy and S. Tang/ Connection formulae for PainlevP Vfunctions II
We begin our study by considering anew the differential equation for aa expansion for large x of the form
(1.4) and seek an asymptotic
q)(x) = XFl’(S>) + F,‘(S>) + 0(x-‘),
(3.4)
where S>
=x+xc+k,lnx
(3.5)
We find from (1.4) that F,’ satisfies (&“‘)2
= 16(F1”)2(1
+ a&“)
(3.6)
the solution of which is
F,‘(s,)=2~+A,.
(3.7)
z
This solution may be systematically extended to higher order terms and is the result given in (1.21)-(1.23) of the introduction. The question now arises of the relation between the two apparently different expansions (1.16) and (1.21). The answer is that they are basically the same expansion and that (1.21) can be obtained from (1.16) by the substitution k=k,+;
(3.8)
and e4ixdk,
+i/2)
=
-2-4
[({r+(t)‘]e4idCk>j.
(3.9)
One way to see the relationship between the two expansions is to consider the higher order terms in the expansion (1.16). It is not difhcuh to show that in order x- “+’ the term with the highest oscillation is (3.10) When (3.8) holds with k, _
i4x-2”+
12-4n
[(
!!)I
real this may be rewritten as +
( :)l]n[X-inei4n(r+roi*,
hx)
_
X2ne-i4n(x+x,+k,
Inx)]_
(3.11)
The second term in this expression is order x for ah n. Therefore the term of order x obtained from (1.16) when (3.8) holds is 4k,
+2i+4i
n_l f 2-4”
(3.12)
212
B. M. McCoy and S. Tang/ Connection formulae for Painle& V functions II
Moreover, from (1.19) r -ik,+++g e4ixo(k,
+i/2)
(
= 28ik,2-4
r
-ik,++-g
I(
rik,-i+i
(
rik,-i-i )(
1
= -2si&,2_4[~k,+~~+i:l;j’!i::::~~~
=2-4[(~)2+(:)2]e4ir(~,~,
where xc
(k ,)
x 4k,
(3.13)
is given by (1.25). Thus (3.12) becomes
+2i + 4i f
e-4ins> 1
n=l
=x{A,
+2cot2s,}.
(3.14)
This is precisely the leading term xF, derived above with the additional pieces of information that k , and x0’ are given by (1.24) and (1.25). Further evidence for the relation between the two expansions comes from noting that if in (1.21) we let k,=k-i/2 then
F,‘(s>) +2k2+
;
=B,
(3.15)
which is indeed the order 1 term of (1.16). More generally the terms in (1.16) of order x-2n with the highest oscillation sum up as before to give the cotangent term in the order 1 term of (1.21). The above discussion is not meant to be an exhaustive discussion of the relation between the series (1.16) and (1.21). A full discussion recognizes that q,(x) can only have simple poles of residue xP where xp is the location of the pole. The term F,’ in (1.21) has a double pole and hence appears to violate the single pole property. However, since +J-
>
cot2s,
(3.16)
= 2(sin2s,)-2
we may interpret the double pole as indicating that the poles of u,(x) do not occur precisely at s, = fm(21+ 1) but rather at slightly shifted positions. This shift could be accommodated by writing cc
s,
=x+x,)
+k,
lnx+
c a,x-I.
(3.17)
I=1
However, we forbear doing this because a single line of poles in u,(x) is far from the whole story. In fact q,(x) has a doubly periodic array of poles whose period parallelogram slowly changes we move about in
B. M. McCoy and S. Tang/ Connection formulae for Painleve V functions II
213
the complex x plane. This is the same phenomenon for PainlevC V functions which Boutroux [9] found for Painleve I and II. It requires a much fuller discussion in the context of the expansion as x + cc along an arbitrary ray in the complex plane.
4. Thecasen>O As discussed in the introduction the physical problem of the impenetrable Bose gas requires the asymptotic expansion of u,,(x) for n = 1. Thus, in this section we make contact with this physical problem by generalizing our considerations form n = 0 to arbitrary positive integer n. There is a feature of this general result which should be mentioned before we begin the calculations. When n = 0 the x + cc expansion in the general case (1.16) does not have poles while the special case A > Bcot at9 does have poles (1.21). When n > 0 this generalizes to the statement that in the general case u,(x) has no poles for n even but does have poles for n odd. Conversely, for the special case X > 8 cot 7~8u,(x) has poles for n even and has no poles for n odd. We begin our analysis by generalizing the local expansion (1.16) and (1.21) to the case n > 0. We denote the expansion with no poles as IS,,and the expansion with poles as a,. The results are given in (1.26) and (1.28). (We could let the constant A be different in the two expansion but this will prove unnecessary.) The object of this section is to express u”(x) in terms of c”(x) and G,,(x) and to express X, and Z.n in terms of x0. Then the connection formulae of A and x0 in terms of A (or s and a) calculated in section 2 will also solve the connection problem for n > 0. Our tool for relating the n > 0 case to n = 0 is the Toda equation (1.9) written in terms of u,(x) (1.10). We begin by using 5” for a,, in (1.10). Then to order x we find
?In
X2grl n X6’-IT
I
PI,,
-2+2Gn=x
-++A. F’,_B [
(4-l)
Thus, defining (Y, by (1.31) we find that using (1.27~) we have . _ 2A-4cos451n4~;s4a n
X2gu
n
x6; - 3”
n
1+
0(l).
(4.2)
The Toda eq. (1.10) says that this should equal the order x term in a,,, + a,,_,. Clearly the right-hand side of (4.2) is not equal to &,+1 + I?,,_r. However, if we consider 0, k 1 from (1.28) we see that to order x a,,,
+ a,_, = 2A + 2 =2A-4
coszs,_t sin2s,_, cos2(s,_,
+2
cos2s,+1 sin2I,+,
sin2[s,_, + S,+i] + s,+J - cos2(s,+1 -s,_,)
.
(4.3)
Thus comparing with (4.2) we see that if 21,=x,+l+x,_l
(4.4a)
and (4.4b)
214
B.M. McCoy and S. Tang/ ~o~~ecti~~o~~uiae~ar Painlevh Vfunctions II
then (at least to leading order)
Now we may run the calculation in the opposite order. Thus using (1.28) we may show that
xt n xii,’ - ii;,
-2+2ii;,=2Ax+02+
$
1$i
(p+
4-x-l
-~‘-1 g
_*2
- 1 sin4ZR+ $Xcos4Zn + 0(x-2). i 1
(4.6)
On the other hand from (4.26) ==2Ax+P+ a”-* + %‘n+l
$
- ~(n-1)2-
-$+1)2
e2+ A2 4 - (n - 1)2)2+ (n - I)2A2]r’* sin4$,_, i-g
1 I(
e2+g-(n + 1)‘)’ + (n + 1)2A2]1’2 siu4Sn+r $
-i-n*+ 1
+ 0(x-2).
fi4.7)
Therefore, if we write SS+r = S, + 2, *r - Z, we may equate (4.6) to (4.7) and obtain (4.8a)
KII+1-~,=(Y,+r, zT,_, - x, = -a,_r*
(4.8b)
Thus combining (4.4) and (4.8) we find n
x2n = ZO +
(4.9a)
2 C a2i_1, 1-l R
XZn+l
n
I-l L X2*+1
_ x2,
(4.9b)
=X,+2~a,,=~,+2~a,,, f-1
n = %
=Z,+2
f
2 C a21-1 I-1
(4.9c)
+~z~+I~
n-1
n-l
x az,faz,=~,+2 I-1
I-1
C azf+az,.
(4.9d)
We thus conclude that for the general case where h is not real greater than Beat n8
UZnc4 = ~2nW
(4.10a)
B. M. McCoy and S. Tang/
Connection jormulae for PainlevP V junctions II
215
and
%+1(X)
whileif
(4.10b)
= b2n+dXh
X>Bcot&
(J*“(x) = c*,(x)
(4.11a)
%+1(x)
(4.11b)
and =&+1(x)*
5. Connection formula for Painlevk III The results in the preceding section can be used to study a class of PainIeve III functions. As we discussed previously [2], if we define wm by
I
1 -
yv=
1
WI11
*
1
(5.1)
’
where y, satisfies the PainIeve V equation (1.1) with a = fi = 0, then wm satisfies the PainlevC III equation
with y = -6 = 1, and a = - j? = (1 - 28). Let t = -ix For X -C@cot 8~. _ l-(~)1’2e-2is
w
“‘-
1+
(i
ediS> +@I
=
1 _
%I
=
+ cc. Then we may use (2.12).
+otL)
‘:i:)“2e-*i~
where s=x+x,+klnx, For X > cot Blr yv
and let x
(5.3)
x ’
A=4k.
(1 i
,
(5.4)
e*iS>
1+e2i~,
+’
(
i
1
(5.5)
’
where s,=x+x,S +k, lnx On the other hand, at t + 0, we have [2] y,=l-41+0s
* 28-a (
28+a
1
t”=1-4i+“S
a 2e-o
(
-28+a
(-ix)” 1
(5.6)
B. M. McCoy and S. Tang/
216
Connection formulae for PainlevC Vfunctions II
and from (5.1) n 28-o
WI11 =
S
28
(
1
(-4ix)“.
(5.7)
For the analytic case, as t + 0
Yv=
ihe-’ + 1 iX8-‘-1
(5.8)
and [iXB_’ - l]l” WI11 =
- [iX8-’ + 1]1’2
[iXB-l - 1]112+ [iX8-l+
The relations between x0, (1.25), respectively.
k, xl,
and
1]112 * k,
(5.9)
and u, i or X shown by (2.81), (2.78), (1.18), (1.19) (1.24) and
Acknowledgements
We are most pleased to thank Prof. M. Ablowitz, Prof. M. Jimbo, Prof. M. Kruskal, Prof. T. Miwa, Prof. J.H.H. Perk, and Prof. H. Segur for many useful discussions about connection formulae. We are most indebted to Prof. Creamer, Dr. H. Thacker, and Dr. D. Wilkenson for making unpublished notes available to us. One of us (B.M.) gratefully acknowledges the hospitality of Prof. R. Schrieffer and the Institute for Theoretical Physics at the Univ. of California at Santa Barbara where this work was started under NSF grant PHY77-27084 and PHY82-17853 supplemented by funds from the National Aeronautics and Space Administration. This work is partially supported by the National Science Foundation under DMR8206390-A-01.
References PI B.M. McCoy, J.H.H. Perk and R.E. Shrock, Nuclear Physics B220 [FS8] (1983) 35, 269. I21B.M. McCoy and S. Tang, Physica 19D (1986) 42 . [31 M. Jimbo, T. Miwa, Y. Mori and M. Sato, Physica 1D (1980) 80. [41 M. Jimbo and T. Miwa, Physica 2D (1981) 407.
[51D.B. Creamer, H.B. Thacker and D. Wilkinson, Phys. Rev. D23 (1981) 3081. I61M. Jimbo, Publ. RIMS, Kyoto Univ. 18 (1982) 1137. [71 “Higher Transcendental
Functions,” Erdelyi et al., vol. 1, p. 258.
181Ref. 7. pp. 278 and 264. [91 P. Boutroux, Ann. Ec. Norm. 30 (1913) 255; 31 (1914) 99. For a summary see E. Hille, Ordinary Differential Equations in the
Complex Domain (Wiley, New York, 1976) pp. 444-448.
CONNECTION FORMULAE FOR PAINLEVk V FUNCTIONS II. THE 8 FUNCTION BOSE GAS PROBLEM Barry M. MCCOY and Shuang TANG Instilule for
Theoretical Physics, State University of New York ut Stony Brook, Stony Brook, New York 11794,
W.4
Received 15 April 1985
We continue our study of those Painlevt V equations which admit a one-parameter family of solutions analytic at the origin and solve the global problem of connecting the 2-parameter expansion about I = 0 with the 2-parameter expansion valid for I -) f iw. A special case of our results is applied to the study of the one-particle reduced density matrix of the one-dimensional gas of impenetrable bosons. We also use our results to obtain a 2-parameter connection formula for a class of Painleve III functions.
1. Introduction The Painleve V equation (y-1)* t*
(“y+&-i)+yyt-‘+6y(y+l)(y-l)-‘,
(1.1)
with (y=
y=4(1-28)
-p+*,
6= -8,
(1.2)
plays an important role in the study of the correlation functions of the transverse Ising model [l, 21 and the impenetrable Bose gas [3]. The first paper of this series [2] studied the asymptotic behavior for t + m, which is the physical region for the transverse Ising model. In this present paper we study the asymptotic behavior for t=
-ix
and
x+
+cc,
(1.3)
which is the physical region for the bose gas problem. For the physical applications it proves useful to consider the Painleve equation in the u form of Jimbo and Miwa [4],
(1.4) where y f 1 = (a;)* f 2ixu,” - 80, - 4( n* - r?‘) + 8x0,’ (a; * 2i8)* + 4n2 0167-2789/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(1.5)
188
B. M. McCoy and S. Tang/ Connection formulae for PainleG V functions Xl
For the impenetrable Bose gas the off-diagonal one-particle reduced density matrix was shown by Jimbo, Miwa, Mori and Sato [3] to be obtainable from (1.4) with n = 1 and 8 = 0 as
p(x) = poexp ~xdx’o,(x’)x’-‘,
0.6)
where we specify the boundary condition for (1.4) as
q(x) = - 4.x’+ $X3 + O(x4),
(1.7a)
h=2/lr.
(1.7b)
with
The problem of physical interest is the determination of the behavior of al(x) [equivalently p(x)] as x-, +co. Somewhat more generally Creamer, Thacker and Wilkinson [5] showed that if one generalizes the impenetrable Bose gas to the Bose gas with pairwise interaction cS(xi - xi) then the first c-l correction to the c + cc [impenetrable] limit of p(x) involves the derivative of al(x) with respect to A. These authors initiated the study of the X dependence of the x + 00 asymptotic behavior of al(x) by both numerical and analytic means and one motivation for this present study is to provide a derivation of their several conjectures. Even though the special case n = 1 is perhaps of the most direct physical interest it is most useful to consider all u,, with integer n on the same footing. The reason is that if we define X,(x)
(1.8)
= X~(0)~xdx’u,(x’)x?
then X,(x) satisfies the Toda equation
2
ln X,(x) =Xn_lXn+l/X,‘,
(I .9)
which in terms of a, reads x
‘u;
xu; - a,
- 2 + 2U”= u”+t + a,_,.
(1.10)
Thus, since from (1.4) we may require a,, = u_ “, as long as the boundary conditions on (1.4) are compatible with (1.10) we may generate u, for any n from a knowledge of u,. The PainlevC V equation (1.1) (or 1.4) with the restrictions on (Y,p, y, and 6 given by (1.2) with n an integer has the important property that there exists a one-parameter family of solutions analytic at x = 0. If the dependence of the one constant of integration is chosen to be compatible with (1.10) this analytic family has the form
u,(x) =
2 u,(n)x2’+x2n+1 fJb,(n)x2’, I-1
I-0
(1.11)
8. M. McCoy and S.
Tang/Connectionformulae for PainlevP Vfunctions II
189
where for n 2 1
al(n)
= -
4(n2 - 82) 4n*_ 1
(1.12)
and
b,(n) =
2(P - e*) (-1)“2XfJ 1(2/ - 1)2(21 + 1) *
(1.13)
1=1
For n=O q(x)
= -2xx-4(P+B*)x*
+ 0(x3).
(1.14)
More generally as x + 0 we have the 2-parameter family of solutions which we follow Jimbo [6] and parametrize as
gc1t(28
q)(x) ==g+
+
u)
-
0)(4e-wi/2x)1+a
,(fI’,,(28 +u)(4e-wi/2x)1-o+
0(,x*(‘-“)).
(1.15)
The object of this present paper is to calculate the behavior of u,(x) as x + 00 for either the boundary condition (1.11) or (1.15). We will carry this out in several steps. In section 2 we study the case n = 0. For the case of the solutions analytic at x = 0, aa has an asymptotic expansion as x --) 00 valid except when X is real and greater than Bcot rr13 q)(x)
=Ax
+ B + x-V_,(s)
+ x-*F_*(s)
+ x-3F_,(s)
+ 0(x-“),
(1.16)
where s=x+x,+klnx,
(1.17a)
k = A/4,
(1.17b)
~=+[8*+A*/4],
(1.17c)
~_,=~[~*+,4*/4]sin4s-~A(8*+A2/4)~
(1.17d)
F_,=
(1.17e)
-~A(~*+A2/4)sin4s+C_,~s4~+K-2~
with c_,=fiB(4B-4+A*),
(1.17f)
K-,=$(8*+%)
0 .lW
190
B. M. McCoy
andS.
Tang/ Connection formulae for Painle& Vfunctions II
and 1
4(282+3A2-6)-(B2+~)]~i~4s
F-3=512 + C_,cos4s
+ K_,,
(1.17h)
where C_, and K_, are still to be determined. A principle result of section 2 is to show that the two constants of integration k and x,, are given in terms of X by cosn@-isin&
(1.18)
and e4ixo
= 2iskI’( -ik + 8/2)r( -ik - 19/2) I’(ik + 8/2)I’(ik - e/2) ’
(1.19)
These one parameter connection formulae are derived as a special case from more general 2-parameter connection formulae for the initial condition (1.15). The results (1.16), (1.18), and (1.19) break down when
x > 8 cot se.
(1.20)
because then k has an imaginary part of f i and hence the term x -‘F_,(s) in section 3 where we show that when (1.20) holds. cos 2s
is of order x. We treat this case
(1.21)
where S> =x+x,>
+k,
lnx,
(1.22)
k, =$A,
and
G%>)=
f12+ 3(4k2, +l) 4k2, - 1+ 8’ + 2k> cot2s> + 2 4[sin2s,12
(1.23) ’
The connection formulae are now hsin7rB--cosnB k>=&In (6
(1.24)
1
and e4ix,’
=
_28ik,
I’(-ik, r(ik,
+++8/2)r(-ik, +)+e/2)r(ik,
+f-e/2) +f-e/2)
’
(1.25)
B. M. McCoy and S. Tang/ Connection formulae for Painlevk V functionsII
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In section 4 we study the case n > 0. The x + 00 expansions are now derived from (1.16) and (1.21) by use of the Toda equation (1.10). We denote the asymptotic expansion which has no poles (the generahzation of (1.16)) by 4(x) and the asymptotic expansion which does have poles (generalization of (1.21)) by a,( x ). Then we have
6”(X) =/lx
B(n) + x-lP_l(Z”, n) + X-2E_2(?“,n) + o(x-3),
+
(1.26)
with S,=x+4,+Alnx,
(1.27a)
4
B(n) =f[P+A2/4-nq,
(1.27b)
~_1(~,“)=6[(82+A2/4-n2)2+n2A2]1’2sin4s-~A[82+A2/4+n2],
(1.27~)
F_2(~,n)=
(1.27d)
-$4[(B2+A2/4-n2)2+n2A2]sin4s+~_2,
where k_2 is a constant to be determined and
i&(x) = XFl(s,,n) + F,(s,, PI)+ x-lF_l(S,, n) + 0(x-2),
(1.28)
where
5,=x+x,+~hlx, 4
(1.29a)
cos 2s F,(s, n) = 2=+-4,
(1.29b)
F&n)=; [ 82+&2-l
1
1 cos2s ++m+
f32+n2+3(A2/4+1) 4(sin2s)2
(1.29~) ’
and
(1.29d) We then show that if the phase angles are chosen to satisfy n
-
(1.30a)
X2n=X;+2Cff2,-1, I-1 n X2n+l
(1.30b)
=xo+2&,,, I-1 ”
XZn+l
=x0’
+2
c ffzi-1+ I-1
a2n+1,
(1.3oc)
n-l X2n = x0+2
c l-l
~ZI+~Zn,
(1.30d)
192
B. M. McCoy and S. Tang/ Connection formulae for PainlevP V functions II
with e4ia,
8*
=
+
A*/4 - n* + inA
(1.31)
([~2+A2/4-n2]2+n2A2)1’2 that for x --) 00 the general case with X not real above Scot m!I satisfies ~2”W
(1.32a)
= ~2”b)
and fJ*n+A-d
(1.32b)
= 52;n+lb)9
whereas if X > Scot Ire Q*“(X)
(1.33a)
= ~2nb)
and O*n+dX)
(1.33b)
= ~2;n+1W.
In (1.32) A = 4k and x0 are given by (1.18) and (1.19) whereas in (1.33) A = 4k, (1.24) and (1.25). Therefore, for the particular case of the Bose gas where 8 = 0 and A > l/m q(x)
-f+x-‘{+(4kt
=4k>x+2k;
-1)sin4s’+Sk,
~0~4s’ -ak,(4k;
and xc are given by
+l)}
+0(x-*),
with S’=x+xc+k,lnx,
(1.34)
k,=&ln(h?r-l),
(1.35)
and e4ix
0’ = _28ik.
T*( -ik, r*(ik,
+ i) +$)
.
(1.36)
The result (1.35) was conjectured in ref. 5. When h + 2/s k, -0
(1.37a)
and e 4ixo<
=
_28ik,
e-4ik,(r+21n2)
where y is Euler’s constant.
7
(1.37b)
B.M. McCoyand S. Tang/Connection formulaefor PainlevPVfunctionsII
193
These results were also first numerically conjectured in ref. 5. Finally we conclude in section 5 with an extension of our connection formulae to a class of Painleve III functions.
2. Thecasen-0 We begin our calculation by considering the asymptotic expansion of u,(x) given by (1.16) and (1.17). It is easily verified by direct substitution that (1.16) does indeed satisfy (1.4). Thus, the purpose of the section is to calculate x0 and k in terms of the initial conditions (1.14) and (1.15). As in the previous paper [2] our method will be to use the deformation theory of Jimbo and Miwa [4]. However, as there are major differences between ref. 2 and the present case it is not out of place to begin our considerations with a brief recapitulation of the basic procedures. We characterize our PainlevC V function by its relation with the 2 x 2 linear matrix differential equation ~=_4(z)Y(z),
(2.1)
where
A(z) = -2i[ i
_“X] + i&(x)
(2.2)
+&A,(x),
with
A,(x)
1. duo - z1 dx l.du,, 8
=
--aldx-2
l.du,, KG-Z l.du, a%-
8 (2.3a) ’ I
and 1. do,
a1dx
Al(X) = Y
1
I
l.du,
-a'dx
* I
(2.3b)
We uniquely define solutions to this linear equation by the requirement that as 1z ) + 00 Y(z) =
(1+
Q(z-‘))[ “6’“’e*!z].
(2.4)
The linear equation (2.2) has regular singularities at z = 0 and z = 1 near which Y(z) behaves as
Y(z) =
C,(x)[ ‘“0”z_#”
(2Sa)
B. M. McCoy and S. Tang/ Connection formulae for Painlevd V functionsII
194
and e/2
0
Cl,
(2Sb)
(z- 1)-e’2 1 where
G’b)~ob)Go(x) = [Y2
$21
G;‘(+,b)G,b)
$21.
(2.6a)
and
= [Y2
(2.6b)
The relation this linear equation has with the PainlevC V function is the theorem [4] that when u,,(x) obeys (1.4) then the connection matrices Co and C’ are independent of x. More precisely, taking into account the fact that the above definition defines Co and C’ only up to pre and post multiplication by diagonal matrices we will focus on the following two quantities which deformation theory guarantees are independent of x:
IO=-
coca 11
22
coca 12
21
(2.7a)
’
and
(2.7b)
We obtain our connection formulae by using the small x expansion (1.15) and the large x expansion (1.16) in the equation (2.1). In each case the invariants I0 and I’ are calculated in terms of the boundary conditions. Then by equating these two independent forms of the invariants we obtain the desired results. We thus must approximately solve (2.1) for x = 0 and x + cc. The case x = 0 was previously studied by Jimbo [6]. The result for the invariants is
10~
_e-nia
Al=
_e-nia
[sina(B+f~)-Ssinn(8-~a)][l-F~e~~~] sin rr( 8 + fo) + Seeni sinn(B - &)] [l + S-l]
(2.8a)
and [sins( 8 + )a) + Selrio sin?r(8-+a)][1 [sins(B+ta)-SsinIr(B-ta)][I-s’--’e-~i~]’
+ S-i]
(2.8b)
B.M. McCoy and S. Tang/Connection
formulae for PainlevP Vfunctions I1
195
where s is related to the i in the boundary condition (1.15) by r(e+:e+l)P(?)
_ (2.8~)
s=z-401(B_:~+l)T2(~)~. In the special case where u = 0 and the boundary condition (1.14) is to be used
(S+s’-‘)d = -2iX
(2.9)
and the invariants reduce to (2.10a) (2.10b) Here we make only one additional remark. In order to equate the invariants which we will calculate for x --* co with (2.8) we must use the same conventions on branch cuts and Stokes lines as those of ref. 6. When t = e -vi/z x is real Jimbo completes the definition of Y(z) by cutting the z plane from 0 to 1 and on z = i + i y from 0 < y < 00. The criteria for this latter cut is (2.11a)
argt(z-+)=a/2. Now if t = e’*x and $I goes continuously from 0 to -n/2 arg(z-f)+
then (2.11a) is preserved if
+a.
(2.11b)
Thus to be consistent with ref. 6 we must cut the plane from + 1 to - cc as shown in fig. 1. Furthermore, because the phase of (2.11b) is + s the cut going from + 1 to - cc must be thought of as passing above the point z = 0 and hence the connection matrix Co is obtained by letting z + 0 from the lower half plane. We now turn to the calculation of I0 and I’ for x --) cc. In subsection 2.1 we approximately calculate Y(z) for z # 0, 1 by the WKB technique. In subsection 2.2 we approximately solve (2.1) for z = 0 and z = 1. Then in subsection 2.3 we connect these 3 solutions together to calculate the invariants. The major difference with the previous paper [2] is that in the present case the turning points of the WKB solution coincide with the monodromy points z = 0, and 1. Hence the same functions will serve both as turning point and as monodromy connection functions. Finally in subsection 2.4 we equate the two different expressions for the invariants to obtain the desired connection formulae.
2.1. WKB solution for z # 0,l When x x=-1 we use the expansion (1.16) to find that to order 1 yz
( s)e-4is
(2.12)
B. M. McCoy and S. Tang/ Connection formulae for Painlevk V functionsII
196
Fig. 1. The branch cut conventions in the z plane used in the case x + + co. To be consistent with ref. 6 the point z = 0 must be thought of as lying below the cut.
and hence (2.1) becomes
Y(z).
(2.13)
We seek a solution of the form Y,(z)
=e
**ixr{&f(z)
+x-l&‘(z)
+ 0(x-2)},
(2.14)
Then to order x the equation (2.13) is simply *2irf,*(z)
= [ -tx
(2.15)
and hence G(z)
= [ ;]
(2.16a)
and f&(z) =
01-
fi I
(2.16b)
B.M. McCoy and S. Tang/ Connection formulae for PainlevP Vfunctions II
197
To obtain fO* we must proceed to the order 1 term in (2.13). This gives
*2if,*(z)
+
A&*(z) =[-$ i]
f,+(z)
Then, using (2.16) we find that for fz the lower equation is
(2.18) so that
fo+(z) =
c+($y4.
(2.19a)
Similarly, f,-(z)_C-
(zz-l 1-iA’4*
(2.19b)
Therefore, using the boundary condition (2.4) to determine C * we find that the leading WKB solution to (2.1) valid for z # 0,l is Y(r) = IL
y+l,
(2.20)
where y+=
e2ixz(&.)iA’4( y)
y_=
e-2ixr
(2.21a)
and
(*)-iA’4( ;).
(2.21b)
2.2. Approximate solutions near z = 0 and z = 1 The WKB solution (2.20) breaks down near z = 0 and 1 and the approximate equation (2.13) must be studied anew in these two cases. Consider first z = 0 and define a new scaled variable .E=xz,
(2.22)
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B. M. McCoy and S. Tang/ Connection formulae for PainlevP Vfunctions II
where Z is of order 1. Then calling Y(r) = F(T)
(2.23)
and letting x + cc we find from (2.13) that F satisfies
(2.24)
We reduce this to a second order equation by first writing the two component equations separately as
(2.25a) and s=(Zi+g)Y,++(-y--t)Y,.
(2.25b)
Thus if we solve (2.25a) for r, as (2.26) and substitute into (2.25b), we obtain
(2.27) which may be solved in terms of the + confluent hypergeometric
F)l) = (4iZ) e/2e-2i~$[!++~,
function as [7]
l+fl;4ir]
(2.28a)
and 1-t 8; -4iZ
1.
(2.28b)
We then may find the component Y, from (2.26) using the relation (71 o,c;x)=~[(a-c+1)~(a+1,c;x)-~(a,c;x)] =$[(a-
c+x)~(u,c;X)-I/(u-1,c,x)].
(2.29)
B. M. McCoy and S. Tang/ Connection formulae for PainlevP Vfunctions II
199
Thus we obtain two linearly independent solutions of the approximate equation valid near z = 0
J;(l)
=
C(l)(4iz)8/2
e-2iZ
I
#($+;,
1+8;4iE)
(~+~)$(I+Q!+~,
l+f3;4i?)
and (-~+~)#(l-~+~, y(2) = c7*)( -4iz)
1
(2.30a)
l+B,-4iZ)
e/2 e*il
(2.30b) #(-y+;,
1+8;
[
-4iZ)
’ I
We also require solutions valid near z = 1. Here we define s=x(z-1)
(2.31)
F(a) = Y(z).
(2.32)
and
Then letting x + oo in (2.13) we find that y satisfies
(2.33)
Thus !$_ dz -
(2.34a)
~=(2i_~)E,+fe4i$(~+:)~~
(2.34b)
and
and hence p2;= -~e4i~(~-:)1(~+(2i_~)~~)
(2.35)
and so d*& ~++~+(4+f(2i-A)-$}~I=0.
(2.36)
B. M. McCoy and S. Tang/ Connection formulae for Pain/e& V functionsII
200
Therefore we find that near z = 1 the two linearly independent solutions are
p(1) =
#(-y+i, c(l)(4iz)‘/*
l+B;4ii)
e-2it
(2.37a) -e4i’(~+~)+(l-~+~,
1+8;4ii)
e-4is($-~)#(l+~+~,
1+8;
i
and
v(2) = c(*)(
-
4iz)‘/* e*ii
+(Y+i,
[
1+
e; -4iI
-4ii) (2.37b)
1
i
TO complete the definition of the solutions (2.30) and (2.37) we must define the branch cut implicit in il/. This will be done in the next subsection. 2.3. Calculation of the invariants We now calculate the invariants I’ and I0 by using the approximate solution (2.21), (2.30) and (2.37) to connect the boundary condition (2.4) with the definition of the connection matrices (2.5). From the boundary condition (2.4) we define the solution Y(l) such that in the lower half-plane
y(1) t: y is exponentially
(2.38a) small. Similarly we define the solution Y0 such that in the upper half-plane
y(2) z y +
(2.38b)
is exponentially small. Here Y, are the WKB solutions (2.21). Furthermore, from our definition of branch cuts and Stokes line (2.38a) also holds on the real axis approached from below and (2.38b) holds on the real axis approached from above where Y, are no longer exponentially small but oscillate. However, when the solution Y(*) is continued around the branch point at z = 1 to the lower side of the cut from 1 to - 00 an additional contribution from Y_ is picked up and hence on the segment of the real axis 0 < z < 1 approached from below (2.39)
y(*) = Y, + c_ Y_.
The situation is summarized in fig. 2. We consider first matching solutions near z = 1. Consider first Y (*). From fig. 2, (2.38b) and (2.21a) as z + l* above the cut y(2)
~
e2ixe2iiXiA/4~-iA/4
0 0
1 ’
(2.40)
where .Zhas phase e Oinfor Z > 0 and has phase eni for z’< 0. This must be matched to a linear combination of p(‘)(Z) and 8(*)(Z) in the limit Z + f cc. For this we need the asymptotic expansion of #(a, b, z).
B.M. McCoy and S. Tang/ Connectionformulae
forPainlevk Vfunctions
II
201
Fig. 2. The relation between the exact solutions Y(l) and Y(*) and the WKB approximate solution Y,.
Remembering -;a
that # is a multivalued function we note [8] that as z + cc for
+$lr,
$(a, b, z) = z-,
(2.41a)
while for z = e * 3”i~zx with x + + cc
(2.41b) Thus, if in F(2) of (2.37b) we define branches so that arg( -4i&) = -a/2, arg(-4iZ)
= + $,
arg(-CZ)
= -g,
for for
Z > 0, ,Z< 0
(2.42a) on top of the cut
(2.42b)
and for
Z< 0
on bottom of the cut,
(2.42~)
we see that for Z + f 00 on top of the cut p(2)
+
c(2)(
_ 4iZ)-‘A14
e2ii
.
y
(
1
(2.43)
This matches (2.40) if e(2)
= e2ixXiA/44iA/4
ewA/8e
(2.44) Next consider Y cl). As z + 1* from below the cut y(l)
z
e -2ixe-2iiX-iA/4ZiA/4
202
B. M. McCoy and S. Tang/ Connectionformulae for PainlevC V junctions II
where ? has phase e Onifor Z > 0 and has phase eeni for Z < 0. This solution will match to f(l) of (2.37a) if the phase of iZ in # is chosen as arg(4ii)=T,
for
arg(4iZ) = - ;
Z>O,
for
(2.45a)
.Z< 0
below cut.
(2.45b)
and if c;(l)
= e-2ixx-iA/44-iA/4e-rAA/8
(2.46)
We now may determine the connection matrix C’. For this we need the matrix G,(x) defined by (2.6b) where for A,(x) we use the appropriate matrix in (2.13). Thus we find
(2.47)
and hence from
C’=lim
2-l
(z-
l)-e’2
0
0
(z - l)e’2
1G,‘(x)Y(z),
(2.48)
we have the 4 explicit equations
Y21) ,
(2.49a)
(2.49b)
Ci2 = ,LF~ (2 - 1) e’2{ Yj’) - e-4isYJ1)},
(2.49~)
(z - 1) “‘{ YJ2)- e-4iSY$2)}.
(2.49d)
C:, = !,I
We now may expand Y(l) and Yc2) using r(1~(“‘c;E);rr(a_c+l)+il-
c)
.r(c-- 1) r(a) *
(2.50)
B. M. McCoy and S. Tang/ Connection formulae for PainlevP V functions II
203
Thus, noting that the phases are defined from (2.42) and (2.46) we find as Z + 0
r(
iqA
ij +(4iz)P0~j_$~:)
Y(l) = c;‘1)(4ii)8/2
(2Sla) _
G-0)
AL+;
ei4s
)i r +;
(
(
L
+
1
r(e) .
(4it’)-’ r
(
1-y+;
11
and
yC2)
e
and consequently (-il
=
‘I2
c(2)( _ 4iz)
-6;(l)
(2Slb)
we obtain 4012e”i@/48r( _ 8) rl_!.+-;
'
(2.52a)
(
) ci2= ct2,4e/:e-"ie'4er(.-e)e-4i",
(2.52b)
($+$r(+-$) (2.52~)
and ci2
=
-
$2)
4-8/2e~iO/4r(~)ee-4i~
ri+' A$+; !
(2.52d) 1
Thus we compute the invariant
(2.53)
We now turn to the calculation of the invariant I0 obtained from the connection matrix at z = 0. Here, in contrast to the point z = 1 the solutions Y(l) and Yt2) do not play symmetric roles in the calculation.
204
B. M. McCoy and S. Tang/ Connectionformulae for PuinlevP Vfunctions II
Consider first Y(l). Since we are calculating the connection matrix for z + 0 from below the cut we see from fig. 2 that the procedure for matching Y(l) to r (the solution valid near z = 0) is identical with the matching procedure used near z = 1. Accordingly if we define the phase of 4it in the function Y(l) of (2.30a) by arg4iI = a/2,
for
E> 0
(254a)
and arg4iZ = -n/2
for
Z
(2.54b)
then we may use the asymptotic expansion (2.41a) and find that near z = 1 y(1) 3 qqw
(2.55)
where c;l)
= XiA/44iA/4
e-nA/8
(2.56)
The behavior of Y@) near z = 0 is, however, quite different because as indicated in fig. 2 the behavior in 0 < z < 1 is given by Y, + C- Y_ instead of by Y, alone. The constant C_ is calculated by matching the form Y, + C- Y_ with the asymptotic expansion of y(2) = @)jw)(
9
(2.57)
valid for Z + - 00 when -i.Z has phase -3/2a (2.42~). For this region we must use the asymptotic expansion (2.41b). Thus as a matching condition we have
e2i*e2iiXiA/4z
--iA/
[ 01
1
+ c_
e-
2ixe-2iiX-iA/4ZiA/4
[0 1
1
2nie”i/2(l+iA/4+38/2) e-4ii(
1
I
_4z)iA/4-
~(,,!5,!!)
277ie r(++j)r(y_;)
e-4ii(-4~)iA’4-e’2-1
(2.58)
B. M. McCoy and S. Tang/ Connection formulae for Painlev6 Vfunctions II
205
and hence we find -4isqiA/4
_ 2aic:(2)e2iXxiA/4
e-3nA/8
c_=
(2.59)
=
_ 2rie-4i%
where in the last line the definition of s (1.17a) has been used. We now may calculate the behavior of Y(*)(z) as z + 0 along the positive real axis by matching Yt2) = y++ c_ Y_,
(2.60)
with z + O+ to the linear combination cl”‘jw’(
2) +
@)jm(
3,
(2.61)
with I + + cc on the real axis. In (2.61) we define the phase of 4it in y(l) as +n/2 in Y(*) as -n/2. Hence in both cases expansion (2.41a) may be used. Thus we find eni(iA/4)X
-iA/4ziA/4
e Zil
ni(-iA/4)XiA/4,--iA/4e-2ii
1 (
=
@)(4iz)-iA/4,-2ii
i (
+ cjZ)( )
_4iZ)iA/4e2if
f (
and the phase of -4iZ
)
0) (2.62)
and hence @) =
e9rA/8
c_XiA/44iA/4
(2.63a) and @?) =
e-3nA/8X-iA/44-iA/4
(2.63b)
We now may calculate the connection matrix at z = 0 from the definition (2.5a) if we note that as x --, cc 1 G,‘(+
1
.1 -1
_-
+;
!A+; I
*
(2.64)
206
B. M. McCoy and S. Tang/ Connection formulae for Painlevh V functions II
Thus
cp, = feoz-
e/2 yp _ yp } 3 1
(2.65a)
Cf2 = lim z -‘I2 { Y/‘) - YJ2)} ,
(2.65b)
2-0
(2.65~)
(2.65d)
Here we may use (2.50) to first write that as Z --f 0
(2.66a)
and y(2) 5 qqW)
+
CqF(2) 2
cj’)(4ij)8/2
( 1 r(!++i 11 rLee)B
r 4-T I
[
+
(4it)-’
r(e)
=z
(2.66b) cjO”‘(4iF)
e/2 iA ( 4
+! “j+7$,
+Q2)(-4iz)e/2 r( i
r(-0) $4
+(4i~)-@+q)+;)
:)
+ (-4i’)-er
_$i$ (
)I
B. M. McCoy and S. Tang/Connection
207
formulae for Painlevi V/unctions II
and thus from (2.65) obtain
(2.67a)
(2.67b)
co
=
_
cpsr(
l2
per(
-e)e"'~/4(4xp2
(!g_;)r(~_;)
-e)e-"'~/4(4X)e'2
(2.67~)
- (!4+!)+!.$_;)
and
c0
= q%q
22
e) tYi@/4(4,)
-e’2
+ per(e)
(!.$+;)q!+!)
e”‘@/4!4x)
-#‘2
(2.67d)
(!.A+;)+!++;).
Therefore we find the required invariant 11 22 zo=- coca coca 21
12
C_
=
+ 4-iA/4e-nA/2enifl/2
=
9
(2.68)
c_ _ 4-iA/4e-nA/2e-ai@/2
where from (2.63)
e-4ix04+3iA/4e-rrA/4 C_=
-2ai
(2.69)
r(i+y+;)r($:).
208
B. M. McCoy and S. Tang/Connection
formulae for PainleuP Vfunctions II
2.4. The connection formulae
We now may equate the expressions for the invariants calculated for x = 0 with those calculated for X--,00.
We consider first the one parameter case where the invariants are given by (2.11b). For the invariant I’ we equate (2.10b) with (2.53). Thus
-lrA/4+inO/2
iA&‘- 1 lri0 iX8-’ + 1 = e
_
e?rA/4-id/2
_
enA/4+in8/2
eail?e e-vA/4-inO/2
(2.70)
which may be solved for A to give e ~A/2=c0Sd?-AXg-1sind
(2.71)
or, in terms of k (1.17b) (2.72) Similarly for the invariant I0 we equate (i.lOa) with (2.68). Thus
c_diA/4
M-l
enA/
+
enib’/2
+ 1
(2.73) c-4
iA/
enA/
_
e-eiO/2
from which if we solve for c_ and use (2.74) we obtain (2.75)
Then eliminating A&’ on the right using (2.71) and using (2.69) for c_ yields e-4ix
O=
-_&A-iAr
x {2c0sTe-
erAI
- eCTA'A/2}
(2.76)
B.M. McCoy and S. Tang/Connection
209
formulae for PainlevO Vfunctions II
from which, using (2.74) again we obtain the desired result
++(!L+!) e-4ixo
=
4-‘A
(2.77) +!q)+~+;)
Thus we have derived the one-parameter connection formulae (1.18) and (1.19). We now may repeat the same procedure and derive the general 2-parameter connection formulae by using the 2-parameter invariants (2.8). Again we consider first I’. Then
sin7T i. ++i
[sina(8+fa)+Se”i0sina(B-$r)][1+S-1]
1
-e-*‘u[sinn(e+t~)-ssinn(8-io)l[l-S-1e-~i~]=e-lesin~~“_“) 4
(2.78)
2
which can be solved for A to give
- i sin
~0
[
$evio/2 sins
(
8-i
1
+S=1eP+“/2sinn(B+~)]) (2.79)
Similarly for I0 we have -Ssins(B-fO)][l-S-le”io] [sin++te) sin 8( 0 + $u) + Seeni sinn(B-$a)][1
_e-aio
C_4iA/4
enA/
+
+S-l]
enib’/2
=
(2.80) c-4
iA/4enA/2
_
e-niO/2
This can be solved for e-4i*o to give
1
e-4ix0 =
s4
_,
r1+q+; ( . IA
)
r1+‘A-8 ‘4 (
2)
r g+ ( 4
cos~[sinn(t9+~)+sin7r(B-~)]
!+(+:)
210
B. M. McCoy and S. Tang/ Connection formulae for PainlevO V funcrions II
x { - t(sinlr(28 + e--A/2
cos~[sinrr(fl+q)+sina(8-s)] ria/2gsin 7f( 8 - fa) + e ‘“i’/2~-1sina(B+ta)]},
-isinae[e-
1
e -4ix,, = _
+ u) + sina(28 - u))
(2.81)
.(l+~+;)r(l+$;)r(~+;)$$;)
__
~4
-4
4a2
cos?J-[sin,(e+;)+sinr(B-q)]
x sinrr(28+e)+sin7r(2e-a) ( - (e-
~A/2+e-A/2)~~~~
+2sinysin7rB
S-
r
SILT
[. sinrr
l .
(
(e+? “1+smn. (e-z 71
13+Z~
l
1
-Ssma -.
(
e-Zu
l
)1l
(2.82)
In this form it is clear that as u + 0 we regain (2.76) in which the s dependence disappears. However, for u # 0 it is, of course, not possible to express x,, in terms of A alone.
3. The case
x>ecotse
The question now arises of where in the complex X plane the expansion for uo(x) of (1.16) and the connection formulae (1.18) and (1.19) are valid. If cos rre - A&’ sin n/3 is not real and negative then
and so for real X, (3.2) and consequently A>ecOtse
the term x-‘F_,
is in fact smaller than the leading term. However, when X is real and (3.3)
then x-‘F_, is of the same order of magnitude as the leading term Ax and the expansion (1.16) breaks down. This breakdown is analogous to the one which occurred in the previous paper [2] for t + co when I Im h I > 0. However, in that previous paper there was no compelling physical reason to pursue the special case further. In contrast in this present problem the physical case of the Bose gas f3= 0 and X = 2/77 lies in the region where (3.3) holds and thus we are forced to do a further analysis.
211
B. hi. McCoy and S. Tang/ Connection formulae for PainlevP Vfunctions II
We begin our study by considering anew the differential equation for aa expansion for large x of the form
(1.4) and seek an asymptotic
q)(x) = XFl’(S>) + F,‘(S>) + 0(x-‘),
(3.4)
where S>
=x+xc+k,lnx
(3.5)
We find from (1.4) that F,’ satisfies (&“‘)2
= 16(F1”)2(1
+ a&“)
(3.6)
the solution of which is
F,‘(s,)=2~+A,.
(3.7)
z
This solution may be systematically extended to higher order terms and is the result given in (1.21)-(1.23) of the introduction. The question now arises of the relation between the two apparently different expansions (1.16) and (1.21). The answer is that they are basically the same expansion and that (1.21) can be obtained from (1.16) by the substitution k=k,+;
(3.8)
and e4ixdk,
+i/2)
=
-2-4
[({r+(t)‘]e4idCk>j.
(3.9)
One way to see the relationship between the two expansions is to consider the higher order terms in the expansion (1.16). It is not difhcuh to show that in order x- “+’ the term with the highest oscillation is (3.10) When (3.8) holds with k, _
i4x-2”+
12-4n
[(
!!)I
real this may be rewritten as +
( :)l]n[X-inei4n(r+roi*,
hx)
_
X2ne-i4n(x+x,+k,
Inx)]_
(3.11)
The second term in this expression is order x for ah n. Therefore the term of order x obtained from (1.16) when (3.8) holds is 4k,
+2i+4i
n_l f 2-4”
(3.12)
212
B. M. McCoy and S. Tang/ Connection formulae for Painle& V functions II
Moreover, from (1.19) r -ik,+++g e4ixo(k,
+i/2)
(
= 28ik,2-4
r
-ik,++-g
I(
rik,-i+i
(
rik,-i-i )(
1
= -2si&,2_4[~k,+~~+i:l;j’!i::::~~~
=2-4[(~)2+(:)2]e4ir(~,~,
where xc
(k ,)
x 4k,
(3.13)
is given by (1.25). Thus (3.12) becomes
+2i + 4i f
e-4ins> 1
n=l
=x{A,
+2cot2s,}.
(3.14)
This is precisely the leading term xF, derived above with the additional pieces of information that k , and x0’ are given by (1.24) and (1.25). Further evidence for the relation between the two expansions comes from noting that if in (1.21) we let k,=k-i/2 then
F,‘(s>) +2k2+
;
=B,
(3.15)
which is indeed the order 1 term of (1.16). More generally the terms in (1.16) of order x-2n with the highest oscillation sum up as before to give the cotangent term in the order 1 term of (1.21). The above discussion is not meant to be an exhaustive discussion of the relation between the series (1.16) and (1.21). A full discussion recognizes that q,(x) can only have simple poles of residue xP where xp is the location of the pole. The term F,’ in (1.21) has a double pole and hence appears to violate the single pole property. However, since +J-
>
cot2s,
(3.16)
= 2(sin2s,)-2
we may interpret the double pole as indicating that the poles of u,(x) do not occur precisely at s, = fm(21+ 1) but rather at slightly shifted positions. This shift could be accommodated by writing cc
s,
=x+x,)
+k,
lnx+
c a,x-I.
(3.17)
I=1
However, we forbear doing this because a single line of poles in u,(x) is far from the whole story. In fact q,(x) has a doubly periodic array of poles whose period parallelogram slowly changes we move about in
B. M. McCoy and S. Tang/ Connection formulae for Painleve V functions II
213
the complex x plane. This is the same phenomenon for PainlevC V functions which Boutroux [9] found for Painleve I and II. It requires a much fuller discussion in the context of the expansion as x + cc along an arbitrary ray in the complex plane.
4. Thecasen>O As discussed in the introduction the physical problem of the impenetrable Bose gas requires the asymptotic expansion of u,,(x) for n = 1. Thus, in this section we make contact with this physical problem by generalizing our considerations form n = 0 to arbitrary positive integer n. There is a feature of this general result which should be mentioned before we begin the calculations. When n = 0 the x + cc expansion in the general case (1.16) does not have poles while the special case A > Bcot at9 does have poles (1.21). When n > 0 this generalizes to the statement that in the general case u,(x) has no poles for n even but does have poles for n odd. Conversely, for the special case X > 8 cot 7~8u,(x) has poles for n even and has no poles for n odd. We begin our analysis by generalizing the local expansion (1.16) and (1.21) to the case n > 0. We denote the expansion with no poles as IS,,and the expansion with poles as a,. The results are given in (1.26) and (1.28). (We could let the constant A be different in the two expansion but this will prove unnecessary.) The object of this section is to express u”(x) in terms of c”(x) and G,,(x) and to express X, and Z.n in terms of x0. Then the connection formulae of A and x0 in terms of A (or s and a) calculated in section 2 will also solve the connection problem for n > 0. Our tool for relating the n > 0 case to n = 0 is the Toda equation (1.9) written in terms of u,(x) (1.10). We begin by using 5” for a,, in (1.10). Then to order x we find
?In
X2grl n X6’-IT
I
PI,,
-2+2Gn=x
-++A. F’,_B [
(4-l)
Thus, defining (Y, by (1.31) we find that using (1.27~) we have . _ 2A-4cos451n4~;s4a n
X2gu
n
x6; - 3”
n
1+
0(l).
(4.2)
The Toda eq. (1.10) says that this should equal the order x term in a,,, + a,,_,. Clearly the right-hand side of (4.2) is not equal to &,+1 + I?,,_r. However, if we consider 0, k 1 from (1.28) we see that to order x a,,,
+ a,_, = 2A + 2 =2A-4
coszs,_t sin2s,_, cos2(s,_,
+2
cos2s,+1 sin2I,+,
sin2[s,_, + S,+i] + s,+J - cos2(s,+1 -s,_,)
.
(4.3)
Thus comparing with (4.2) we see that if 21,=x,+l+x,_l
(4.4a)
and (4.4b)
214
B.M. McCoy and S. Tang/ ~o~~ecti~~o~~uiae~ar Painlevh Vfunctions II
then (at least to leading order)
Now we may run the calculation in the opposite order. Thus using (1.28) we may show that
xt n xii,’ - ii;,
-2+2ii;,=2Ax+02+
$
1$i
(p+
4-x-l
-~‘-1 g
_*2
- 1 sin4ZR+ $Xcos4Zn + 0(x-2). i 1
(4.6)
On the other hand from (4.26) ==2Ax+P+ a”-* + %‘n+l
$
- ~(n-1)2-
-$+1)2
e2+ A2 4 - (n - 1)2)2+ (n - I)2A2]r’* sin4$,_, i-g
1 I(
e2+g-(n + 1)‘)’ + (n + 1)2A2]1’2 siu4Sn+r $
-i-n*+ 1
+ 0(x-2).
fi4.7)
Therefore, if we write SS+r = S, + 2, *r - Z, we may equate (4.6) to (4.7) and obtain (4.8a)
KII+1-~,=(Y,+r, zT,_, - x, = -a,_r*
(4.8b)
Thus combining (4.4) and (4.8) we find n
x2n = ZO +
(4.9a)
2 C a2i_1, 1-l R
XZn+l
n
I-l L X2*+1
_ x2,
(4.9b)
=X,+2~a,,=~,+2~a,,, f-1
n = %
=Z,+2
f
2 C a21-1 I-1
(4.9c)
+~z~+I~
n-1
n-l
x az,faz,=~,+2 I-1
I-1
C azf+az,.
(4.9d)
We thus conclude that for the general case where h is not real greater than Beat n8
UZnc4 = ~2nW
(4.10a)
B. M. McCoy and S. Tang/
Connection jormulae for PainlevP V junctions II
215
and
%+1(X)
whileif
(4.10b)
= b2n+dXh
X>Bcot&
(J*“(x) = c*,(x)
(4.11a)
%+1(x)
(4.11b)
and =&+1(x)*
5. Connection formula for Painlevk III The results in the preceding section can be used to study a class of PainIeve III functions. As we discussed previously [2], if we define wm by
I
1 -
yv=
1
WI11
*
1
(5.1)
’
where y, satisfies the PainIeve V equation (1.1) with a = fi = 0, then wm satisfies the PainlevC III equation
with y = -6 = 1, and a = - j? = (1 - 28). Let t = -ix For X -C@cot 8~. _ l-(~)1’2e-2is
w
“‘-
1+
(i
ediS> +@I
=
1 _
%I
=
+ cc. Then we may use (2.12).
+otL)
‘:i:)“2e-*i~
where s=x+x,+klnx, For X > cot Blr yv
and let x
(5.3)
x ’
A=4k.
(1 i
,
(5.4)
e*iS>
1+e2i~,
+’
(
i
1
(5.5)
’
where s,=x+x,S +k, lnx On the other hand, at t + 0, we have [2] y,=l-41+0s
* 28-a (
28+a
1
t”=1-4i+“S
a 2e-o
(
-28+a
(-ix)” 1
(5.6)
B. M. McCoy and S. Tang/
216
Connection formulae for PainlevC Vfunctions II
and from (5.1) n 28-o
WI11 =
S
28
(
1
(-4ix)“.
(5.7)
For the analytic case, as t + 0
Yv=
ihe-’ + 1 iX8-‘-1
(5.8)
and [iXB_’ - l]l” WI11 =
- [iX8-’ + 1]1’2
[iXB-l - 1]112+ [iX8-l+
The relations between x0, (1.25), respectively.
k, xl,
and
1]112 * k,
(5.9)
and u, i or X shown by (2.81), (2.78), (1.18), (1.19) (1.24) and
Acknowledgements
We are most pleased to thank Prof. M. Ablowitz, Prof. M. Jimbo, Prof. M. Kruskal, Prof. T. Miwa, Prof. J.H.H. Perk, and Prof. H. Segur for many useful discussions about connection formulae. We are most indebted to Prof. Creamer, Dr. H. Thacker, and Dr. D. Wilkenson for making unpublished notes available to us. One of us (B.M.) gratefully acknowledges the hospitality of Prof. R. Schrieffer and the Institute for Theoretical Physics at the Univ. of California at Santa Barbara where this work was started under NSF grant PHY77-27084 and PHY82-17853 supplemented by funds from the National Aeronautics and Space Administration. This work is partially supported by the National Science Foundation under DMR8206390-A-01.
References PI B.M. McCoy, J.H.H. Perk and R.E. Shrock, Nuclear Physics B220 [FS8] (1983) 35, 269. I21B.M. McCoy and S. Tang, Physica 19D (1986) 42 . [31 M. Jimbo, T. Miwa, Y. Mori and M. Sato, Physica 1D (1980) 80. [41 M. Jimbo and T. Miwa, Physica 2D (1981) 407.
[51D.B. Creamer, H.B. Thacker and D. Wilkinson, Phys. Rev. D23 (1981) 3081. I61M. Jimbo, Publ. RIMS, Kyoto Univ. 18 (1982) 1137. [71 “Higher Transcendental
Functions,” Erdelyi et al., vol. 1, p. 258.
181Ref. 7. pp. 278 and 264. [91 P. Boutroux, Ann. Ec. Norm. 30 (1913) 255; 31 (1914) 99. For a summary see E. Hille, Ordinary Differential Equations in the
Complex Domain (Wiley, New York, 1976) pp. 444-448.