A non-linear differential equation and a Fredholm determinant

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A non-linear differential equation and a Fredholm determinant

M. Mehta

To cite this version:

M. Mehta. A non-linear differential equation and a Fredholm determinant. Journal de Physique I,

EDP Sciences, 1992, 2 (9), pp.1721-1729. �10.1051/jp1:1992240�. �jpa-00246654�

Classification Physics Abstracts

02.90 05.90

A non-linear differential equation and

_a

fkedholm determinant

M-L- Mehta

(*)

C-E- Saclay, F-91191 Gif-sur-Yvette Cedex, France

(Received

4 March 1992, accepted in final form 5

June1992)

Abstract. In several branches of mathematical physics _{one comes across} the Fredholm determinant of the kernel

sin(z y)K/(z

y)K _on the finite interval

(-t, t).

Jimbo, Miwa, Mori and Sato derived _a non-linear differential equation for it. We

reinvestigate

this problem, find five equations satisfied by three functions

A(t), B(t)

and

S(t)

related to this Fredholm determinant,

and _{as a} consequence deduce the differential equation of Jimbo, Miwa, Mori and Sato.

1 Introduction.

In _a

long

and

profound

article _on

monodromy preserving deformations, Jimbo, Miwa,

Mori and Sato

[I] found,

_among other

things,

that the Fredholm determinant

F(z,ij

₌

fl

ii z>~(ijj, (i,ij

i=o

of the

integral equation

>

I(zj

=

f~ ^{Kjz, vj/(y) dv. (1.2j}

with the kernel It

satisfies a on-linear second

order differential equation.

This

same

Fredholm

eteriifinant

occurs

in

the tudy of one

dimentional ^odels, for example, the correlation functions

ansverse Ising

chain [2] or the theory of random

(*) of de

1722 JOURNAL DE PHYSIQUE I N°9

Ordering

the

eigenvalues ii

_as I >

lo

> 11 >

12

> >

0,

we define

F+(z,ij

₌

fl

ii z>~;(ijj, (1.4j

;=o

F-(z,t)

₌

ij

Ii z121+1(t)1. (15)

Thus

F+ (resp. F-)

is the Fredholm determinant of the

integral equation (1.2)

with the kernel

K+ (resp. K-),

where

1<+

(z,

_{Y) =}

lK(z, vi

+

K(-z,

_VII

(16)

The

integral equation (1.2)

is known for _a

long

^time [4], ^{its solutions} are known _as

spheroidal functions; they

_are either _{even or}

odd;

the _even

(odd)

solutions

correspond

to the l's with _an

even

(odd)

index.

Suppressing

the

dependence

on z for

simplicity,

^let us set

A(t)

₌

-) _{Jog F+(z, ii}

+

log

F_

(z, t)] (1.7)

B(t)

₌

-)~ [logf+(z,t) log F_(z,t)] (1.8)

We will be concerned with the

following

relations

(A(t)

₌

^28~(t), (1.9)

((lA)

_" Z(S(1)(~>

(l.1°)

((tB)

_{= z} lie

(S(t)~)

,

(l.ll)

and

2x(tB)

₌ z Im

(S(t)~)

,

(l.12)

with

Sill

₌

^e'~t

+

f (Ztl'~ Ii

'S=l

2

-1~~~"

^~~'~

/ _{dYl dy~}

exp ixt

(xi

_ziYi + z2Yi z2Y2 + + znyn-1 z~1Y~1 + Y~I)

= e~~~ +

f

(~)~ ^dzi

^dzn

^{dyi. dyn}

~1=1

~

~l _~t

exp ix

(tzi

_ziYi + _z2Yi Z2Y2 + + _{znyn-1 Z~IY~I} + Y~I)

(1.13)

and "Re

(Im)"

_means the "real

(imaginary)

part of ".

The function

Sit),

with

complex values,

satisfies the first order non-linear differential _equa- tion

-ij

= KS

~S* _{(S~ S*~)}

,

ii.14)

(here

I

= I and S* is the

complex conjugate

of

S). Actually equation 11.14)

^{with the initial}

condition, S(0)

₌

1,

fixes

completely

the function

Sill.

For

example, expanding

S in _powers

oft,

^the successive coefficients _can be determined

iteratively

from

equation (1.14).

^This

gives complete

information about S _near t ₌ 0 and

using equations (1,7)-(1.12)

_{we can}

get

^the same

information about

F+

and F_.

However,

it is also of interest to know their behaviour when t _{- cc.}

Equation (1.9)

_was

proved by

Gaudin [3]; we will

reproduce

his

proof

here in section 2 for

completeness.

It _was

quoted by Jimbo, Miwa,

Mori and Sato

[I]

in the form

(our notation) log

F+

ii, ii

₌

log F(I, ii

_~

/~

^dt'

(- _{~~ log} F(I, '))~~~

ii.15)

2 2

o dt

()heir Eq.(7.l18)).

If _we take

equations (1.12)

and

(1.9)

_as the definitions of

B(t)

and

A(t)

in terms of

S(t),

then

equations 11.10)

^and

(I.

II _are consequences of

equation (1.14). Moreover,

from

equations 11.9)-(1.12)

_{one can} deduce the differential

equation

of

Jimbo, Miwa,

Mori and Sato

[I], (their Eq. (7.104)),

which in _our notation reads

~ ~~

_~

~~

^~ ~ ~~~~~

~~~

^{~ ~}

~ ^~ ~

~

~

^~

~'

_~~'~~~

or the differential

equation

~ ~~

_~

~

~

~~~~~

^~

_~~~ ) ~

~

~)

^{~ ~}

^~'~~~~~

^~~'~~~

Despite

several efforts with the

help

of

prominent

_persons,

including

the four

authors,

_we

never understood the

original proof

of

equation (1.16).

Here _we deduce it from

equations

11.9)-(1.12)

in section 2. This constitutes _an

elementary,

and

perhaps

_a

different, proof

of it.

Moreover,

the four

relations, equations 11.10)-(1.12)

and

(1.14),

_are

probably

_new.

The series

expansions

of

S,

A _or B in _powers

oft,

_or in _powers of _z, does _not show that

z = I has _any

special significance. However,

_as I

lo(t)

decreases _very fast to _zero for

large

t, these series

expansions

_are

surely divergent

^when _(z( > I.

Equation (1.16)

_was used

by Mccoy

and

Tang

_[5] in their

study

of the correlation functions of the transverse

Ising chain,

and

by Basor, Tracy

and Widom [6] ^to

study

the level

spacing

functions in the

theory

of random matrices.

In another article _we will _use

equations (1.7)-(1.14)

to derive the _power series and the

asymptotic

behaviours of the

spacing probability

frunctions in the

theory

of random matrices.

In

particular,

it will be _seen that

equations (1.14)

and

(1.16)

are related _to the fifth Painlevd transcendental

functions,

while

equation (1.17)

is related _to the third Painlevd transcendent.

2. Fredhohn

deter~ninants,

resolvents and differential

equations.

From the Fredholm

theory

of

integral equations

[7], one can write

symbolically, log F(z, t)

₌

log

^det

[I zKt)

= Tr

log [I zKt)

m ~

=

j

^~ _r~y

_[K(")j

_~ n

°J n t

=

£

^~

_{dzi dzn} K(zi z2)K(z2, z3)...K(zn, zi). (2.1)

~-i n

~t

1724 JOURNAL DE PHYSIQUE I N°9

Here "det" and "Tr"

respectively

_mean "determinant" and

"trace",

and Kt is the

integral operator

^{with the kernel K} on the finite interval

j-t, ii.

For

F+(z, ii,

this _same

equation

holds

provided

_we

replace K(z,y) by K+(z, vi. Differentiating

this

equation

with respect to t, _we have

( _{log F(z, t)}

=

£ z"+~ /

_{dzi. dzn}

_(K(t,

xi

)K(zi, z2)...K(zn, t)

t

~

~~(~~i Zl)~(~l122)."~(~n ~~))

m t

= -2

£ z"+~ dzi. dzn K(t,

_xi

)K(zi, z2).. K(zn,t). (2.2)

n=o

~t

To derive the last

equality,

_we have used the fact that

Kjz, yj

₌

Kj-z, -vj (2.3j

The series in

equations (2.I)

and

(2.2)

_are

absolutely convergent

^for (z( <

I/lo,

where lo is the

largest eigenvalue

of K. The heuristic derivation

given

here _can be

perfectly justified

_{[3, 7].}

Similarly

_we have

~ ^{m t}

~ ^log F+(z,t)

= -2

£ z"+~ / _dzi

_dzn

K+(t, zi)K+(zi, z2).. I(+(zn_i, zn)K+(zn, ii

~_~ ^_t

= -2

~o £ z"+~ _dzi

_dzn

Kit,

_xi

)I<(zi, z2).. K(zn-i, zn)K+(zn, ii (2.4j

Thus

A(t)

and

Bill

defined in

equations (1.7)

^and

(1.8)

_can be written _as follows

A(t)

=

f z"+~ dzi. dzn I<(t,zi)K(zi,z2)...K(zn-i, zn)I<(zn,t) (2.5)

~1=o

~t

and

Bill

₌

~o £ z"+~

_{dzi. dzn}

I((t, zi)I((zi, z2).. K(zn-i, zn)K(zn, -t). (2.6) Differentiating Ail)

_{we get}

~~~~ ~~

~

~~

~ ~~ ~ ~~~ ~~'~~

where

Ai ₌

f _z"+~ /(

_{dzi. dzn}

_~~(j ~~~K(zi> z2i. Kizn-i> ~niK(zn>11> (2.81

n=o

m n t

A3 ₌

~j z"+~ ~j dzi. dz;-idz;+I. dzn I((t, zi)...K(z;-i,t)K(t, z;+i).. K(zn,t),

~1=1 j=I

~t

(2.10)

A4 ₌

f _z"+~ f

/~

^dzi.

dz;-idz;+I.

dzn

Kit, zi)...K(z;-i, -t)K(-t, z;+i).. K(zn,t).

n=i ;=i ^-t

(2.ll)

Now _as

K(z,y) depends only

on the difference _{z y, we can}

replace 0K(z~,t)/bt

in

A2 by -0K(zn, t)/0zn. Integration by

parts with respect to _zn

pushes

the derivative _one step ^left.

In the

integral

It

_{dzi. dzn}

_I<(t,

^~

^f~(

xi

)K(zi, z2)..

~"~~'~"

K(zn,t) (2.12)

_t

~Zn

replace 0K(zn-i, zn)/0zn by -0K(zn-i, zn)/fizn-i

and

integrate by

parts over zn-I And

so on, till the

partial

^{derivative is}

pushed

to the extreme left. These step

by

step

integrations give

A2

₌

-Ai A3

+

A4. (2.13)

Also it is easy ^to convince oneself from

equations (2.6)

and

(2.ll)

that

A4

₌

B~(t) (2.14)

With

equations (2.7), (2.13)

and

(2.14)

Gaudin's

proof

of

equation (1.9)

is

complete.

Note that _we have used

only

the property that

K(z,

_VI is _{an even} function of _{z y.}

To arrive _at

equations (1.10)-(1.12)

_we need to _use the

explicite

form

(1.3)

of

K(z, y).

With the

notation,

~~~'~~ ~~~-

~~~

t

=

j / _df

exp

ix(z vlf. (2.isl

-t

we get from

equation (2.5) by

_a

change

of variables z; -

lx;,

" 1

tA(t)

₌

£ z"+~

_{dzi. dzn}

k(I,zi)k(zi, z2)...k(zn-i, zn)k(zn,1)

~=o

~l

"

~~~~~~/_~~~~"~~" ~~~"~~"+~

eXp 1X

(yl

_{Vi Xl} +1i2Zl _Y2Z2 + + flnZn-1 YnZn + fin+lZn

yn+1) (2.16)

Sirrdlarly

" 1

tB(t)

₌

£ z"+~

_{dzi. dzn}

k(I, zi)k(zi, z2).. k(zn-i, zn)k(zn, -Ii

~1=0

~l

=

~

Ill

^"~~

ll

dzi. dzn II _{dYi. dun+i}

eXp 1X

(yl

_YlZl + _fl2Zl Y2Z2 + + flnZn-1 linZn + fin+lZn +

tin+1) (2.17)

A differentiation with respect to t

gives

((tA(t))

= Xi +

X2, (2.18)

1726 JOURNAL DE PHYSIQUE I N°9

Xi

_" ^~ +

f (~)~~~ f

/~

^dzi.

^dzn /~ ^dyi. dy;-idv;+I. dyn+1

~

n=1

~

j=i ^{~l ~t}

exp ix

(yi

_Yizi + Yj-iz;-i ⁺

tz;-i lx;

+ y;+iz; ⁺ yn+izn

yn+1),

(2.19)

~2

_" +

~

~))

^~~~

~ _/~

~~l.. dZn

/~ dYl dYj-ldYj+I dYn+1

~=i ;=i

exp ix

(yi

_{Yizi + Y;-iz;-i}

_tz;-i

₊

_lx;

_{+ y;+iz;} + yn+izn

yn+1),

(2.20)

Now it is _easy to convince oneself that

Xi ₌

)SS*

=

X2, (2.21)

with S

=

Sill given by equation 11.13),

and S* is its

complex conjugate.

Similar

manipulations

will

give

((tB(t))

= Bi +

B~, (2.22)

Bi _" ^~e~'~~ ₊

£ _(~

^~

£ _dzi.

_dzn

_{dyi dy;-idy;+i dyn+1}

2

~

2

~ ~

~)

/~ /~

n= j=

exp ix

jyi

_yizi + y;-iz;-i ⁺

iz;-i ix;

+ y;+iz; + yn+izn ⁺

yn+i),

=

)S~ ^12.231

and

exp ix

(yi

-

z ~~

= -S

quations

(2.18)

and (2,21)

_{give equation} (1.10), _while

equations

2.22)-(2.24) give

(I.ll).

To

get

_equation

(1.12) we will

alculate

the maginary part of

S~(t). For

_this

purpose let

" 1

S(t)

₌ e~~~ +

£

z"

dzi. dzn e~~~~lk(zi, z2)k(z2, z3)...k(zn-i, zn)k(zn,1). (2.25)

~=i

~l

where

k(z, y)

is

given by equation (2.15). Changing

the

sign

of each of the

integration variables,

we can write this

equation

also _as

°~ 1

S(I)

_# ^e"~ +

~

Z"

/ _{dfl.. dfn}

e

~~~~~k(fl>f2)k(f2>f3)...k(fn-I>fn)k(fn> ~~l' (~'~~)

n=1 ~~

Multiplying

the two

expressions (2.25)

and

(2.26)

_{we see} that the coefficient of z" in

S~(t)

is

Ii dfl.. dfn e"~~~ ~~~k(fli f2)...k(fn-

ii

fn)~(fin ~~)

-l I

+

~i dzi. dzn e~~~(~~~i)k(zi, z2)...k(zn-i, zn)k(zn, Ii

n-i i

+

£

_dzi

_{dz; dfi dfn-;} e~~(~i~fi )k(zi, z2)...k(z;-

i, z;

)k(z;, Ii

._~

~i

k(<i,<~j...k(<~_,-i,<~_,jk(<~_;, -i) (2.27j

Its

imaginary

_part is therefore

Idzi

_dzn

_k(I,

xi

)k(zi, z2)..

k _zn-i,

zn)k(zn, -Ii

n-I

X I _{Xl + Zn +} I +

£ _{(Z;-1 Z;)}

j=2

l

= 2x

/

_{dzi. dzn}

_{k(I, zi)k(zi, z2)k(zi, z2).. k(zn-i, zn)k(zn, Ii. (2.28)}

-1

But _on

integrating

_over all the variables y; in

equation (2.17)

_{one sees} that the coefficient of z"+~ _in

tB(t)

is

Ii _{dzi. dzn k(I,} zi)k(zi, z2)k(zi, z2)...k(zn-i, zn)k(zn, -Ii. (2.29)

-1

Thus

2xfB(t)

= zIm

S~(t). (2.30)

This is

equation (1.12).

Note that unlike

equation (1.9),

^this _one is valid

only

for the

particular

kernel

(1.3).

From

equations (1.10)-(1.12)

_we have

It ~

+

)

=

(t ~

+

B)

⁺

^{(2xtB)~ (2.31)}

~ ~

Also from

equation 11.9)

with _one differentiation _we have

I

_dt2 ^~ _dt ^~~

~

dt ~

~

~~'~~~

Eliminating

t

dB/dt

₊ B between

equations (2.31)

^and

(2.32)

^and

replacing ^28~

with

dA/dt, equation (1.9),

_we get

equation 11.16).

On the other

hand,

if _we differentiate

equation (2.31),

_use

equation (2.32),

and cancel out

the _non-zero factor t

dB/dt

+

B,

_we get

~~ dA d2_~ d ~

$

^{~ ~ "}

~w

⁺

2j

+

4x~tB j~ ~~j

1728 JOURNAL DE PHYSIQUE I N°9

Squaring

^{this and}

using equation (2.31) again

to

eliminate,

this

time,

t

dA/dt

+

A,

we get

equation il.17).

Equation 11.17)

could have been obtained also _as follows.

Multiply equation 11.14) through-

out

by

S and write it _{as a} differential

equation

for

S~. Separating

the real and

imaginary

_parts

of this

equation

and

eliminating

the real part _one gets a second order differential

equation

for the

imaginary

part of

S~,

I-e-

equation (1.17)

for

Bill.

To differentiate

Sill,

it is convenient first to write it in the

form, (Eqs. ii.13), (2.15)

and

II

_.3))>

" 1

Sill

_{= e~~~ +}

£

z"

dzi.

dzn ^e~~~~i

k(zi, z2)...k(zn-i, zn)k(zn, Ii

n=I

~l

" t

= e'~~ ₊

~j

_z"

_{dzi. dzn} e'~~iK(zi, z2).. I((zn-i, zn)K(zn, t) (2.34)

n=I

~t

so that

~

~~~~~~ ~ ~~ ~

~~

~ ~~ ~~'~~~

Si

_"

f

z"

/~ ^dzi.

^dzn

e~"iI((zi, z2)...I((zn-i, zn)~~l'~~ (2.36)

n=I ^~t

52

"

f

z"

f

/~ ^{dzi. dzn} e~"iI((zi, z2).. I((z;-i,t)K(t, z;+i).. K(zn-i, zn)K(zn,t)

~1=1 j=1 ^~~

(2.37)

53

_"

f

z"

f

^~

dzi.

dzn

e~"iI((zi, z2)...I((z;-i, -t)K(-t, z;+i).. K(zn-i, zn)I((zn, ii

n=i

=1~~

(2.38)

For

Si

_we follow the method

ofGaudin; replace 0K(zn,t)/0t by -fiK(zn,t)/fizn

and

integrate by

parts on zn, then in the

integral

~~~'' ~~" e"~~

It(zi,

_{z~)___}

°K(zn-

_{i, z~}

j

°~n I~(~"'~l

_{~~ ~}

replace 0K(zn-i, zn) /fizn by -0K(zn_

i,

zn) /0zn_1

and

integrate by

parts over zn_ i, and _so on, till the differentiation

sign

is

pushed

to the _extreme

left,

where it

finally disappears.

^Thus

Si _" -52 + 53 + ix

f

z"

dzi

dzn e~~~i

K(zi, z2).. K(zn, ii, (2.40)

n=I

~t

and from

equations (2.34), (2.35)

and

(2.40),

~

= ixs + 253

(2Al)

It is _easy to convince oneself from

equations (2.6)

and

(2.38)

that

53

_" S*.B.

(2.42)

Equations (2.41), (2.42)

and

(1.12) imply equation (1.14).

Acknowledgements.

1am thankful to Dr. C.

Tracy,

who indicated

an error in

copying equation (1.16)

in _my

book Random

Matrices,

thus

prompting

_me to

finally

^understand the

equation itself;

to my

collegues

R.

Balian,

J. des

Cloizeaux,

M.

Gaudin,

C.

Itzykson,

^and

specially

to G. Mahoux for _many discussions which made this

presentation

clearer. G. Mahoux initiated _me to the

computer

algebraic

_program MATHEMATICA which

helped

me to introduce _a factor x~ _in

the correct

place.

The

larger

audience will _{excuse me} for not

being

able to decide whether the

study presented

here is _{new or} buried somewhere in the mountain of mathematical

writings

of

our

japanese collegues.

I

hope

these few _pages will

help

others less

gifted

like _me to understand the differential

equation.

References

[1] Jimbo M., Miwa T., Mori Y. and Sato M., Density matrix of _an inpenetrable bose _gas and the fifth Painlev6 transcendent, PJ1ysica D

(1980)

80-158.

[2] See for example, Mccoy B-M-, Perk J-H-H- and Shrock R-E-, Time dependent correlation func- tions of the transverse Ising chain at the critical magnetic field, NucJ. PJ1ys. 8220

(1983)

35-47.

[3] ^{Mehta M.} L., Random Matrices, second edition

(Academic

Press, New York,

1990)

appendix

A.16.

[4] Robin L., Fonctions sph4riques de Legendre et fonctions sph6roidales, vol. 3

(Gauthier

Villars, Paris,

1959)

_p. 250, formula 3.55.

[5] Mccoy B-M- and Tang S., Connection formulae for Painlev6 functions, Physica 19D

(1986)

42-72.

[6] ^Basor E., Tracy C-A- and Widom H., Asymptotics of level spacing distributions for random matrices, Phys. Rev. Lett.

(to appear).

[7] ^{See for} example, Goursat E., Cours d'analyse Math6matiques, vol. 3,

(Gauthier

Villars, Paris,

1956)

chap.31.