POLONICI MATHEMATICI LXXIV (2000)
Integrable system of the heat kernel associated with logarithmic potentials
by Kazuhiko Aomoto (Nagoya)
Bogdan Ziemian in memoriam
Abstract. The heat kernel of a Sturm–Liouville operator with logarithmic potential can be described by using the Wiener integral associated with a real hyperplane arrange- ment. The heat kernel satisfies an infinite-dimensional analog of the Gauss–Manin con- nection (integrable system), generalizing a variational formula of Schl¨ afli for the volume of a simplex in the space of constant curvature.
1. Statement of the result. The classical variational formula for a geodesic simplex due to L. Schl¨ afli plays an important role in geometry and analysis of spaces of constant curvature. The author has extended this formula to general analytic integrals of type (2.5) (theory of Gauss–Manin connection of irregular singularity). It has an invariant expression under the group of rotations SO(n). This fact enables us to go straightforward to the analysis of infinite–dimensional function spaces in the framework of P. L´evy’s book [11], for example (see [1] about its history since V. Volterra). Several approaches have been investigated in this direction, for example, Gauss en- sembles of random matrices, white noise analysis etc. (see [5], [7], [11], [12]).
In this note, by the use of the Feynman–Kac formula ([10]), we show that the variational formula for Gauss type integrals associated with real hyper- plane arrangements gives an integrable system for a system of functionals F
0(τ
1, . . . , τ
p; I) including the heat kernel with logarithmic potentials, by taking suitable infinite-dimensional limits.
We consider the heat equation on the real line
(1.1) ∂u
∂t = 1 2
∂
2u
∂x
2− V (x)u
2000 Mathematics Subject Classification: 28C20, 34A26, 47E05.
Key words and phrases : logarithmic potentials, Wiener integral, Feynman–Kac for- mula, integrable system.
[51]
for the logarithmic potential
(1.2) V (x) = −
X
m j=1λ
jlog |x − y
j|
of a finite number of sources y
1, . . . , y
m∈ C with weights λ
1, . . . , λ
m. We assume λ
jare all positive.
The heat kernel K(t, x, 0) satisfying (1.1) with the condition (δ(x) means the Dirac delta function)
(1.3) lim
t↓0
K(t, x, 0) = δ(x) is given by the Feynman–Kac formula:
(1.4) K(t, x, 0) = lim
N →∞
1 (2π∆t)
N/2×
\
RN −1
exp
− 1 2∆
X
N ν=1(x
ν− x
ν−1)
2− ∆t
N −1
X
ν=1
V (x
ν)
dx
1∧ . . . ∧ dx
N −1for ∆t = t/N, where x
0, x
Ndenote 0, x respectively.
By the change of variables b x
ν= x
ν/ √
t, K(t, x, 0) can be written as (1.5) K(t, x, 0)
= lim
N →∞
1 (2π)
N/2√
t
\
RN
exp
−
√ N 2
X
N ν=1(b x
ν− b x
ν−1)
2− ∆t
N −1
X
ν=1
V ( √ t b x
ν)
× δ(b x
N− x/ √
t) db x
1∧ . . . ∧ db x
N −1∧ db x
Nwhere the limit on the RHS is the average defined by the Wiener integral over the set of continuous paths b x(τ ) =
√1t
x(tτ ), 0 ≤ τ ≤ 1.
Let us consider a finite-dimensional approximation to the integral (1.5).
We denote by log
±u the logarithmic functions log
+u =
log u for u > 0, 0 for u ≤ 0.
log
−u =
log(−u) for u < 0,
0 for u ≥ 0.
First we define the positive function Φ
µ(ξ), µ > −1, for ξ = (ξ
1, . . . , ξ
N)
∈ R
Nas
log Φ
µ(ξ) = − 1
2 Q(ξ) + µ log
+ξ
1+ . . . + ξ
N√ N − x
∗(1.6)
+ t N
X
m j=1λ
jX
N ν=1log
ξ
1+ . . . + ξ
ν√ N − y
∗j+
1 2 t log t ·
X
m j=1λ
jfor x
∗= x/ √
t, y
j∗= y
j/ √
t respectively, where Q(ξ) denotes the quadratic form
Q(ξ) = ξ
12+ . . . + ξ
2N.
Similarly we define the functions Φ
µ(ξ; ε
1, . . . , ε
m) by (1.7) log Φ
µ(ξ; ε
1, . . . , ε
m) = − 1
2 Q(ξ) + µ log
+ξ
1+ . . . + ξ
N√ N − x
∗+ t N
X
m j=1λ
jX
N ν=1log
εjξ
1+ . . . + ξ
ν√ N − y
j∗+ 1
2 t log t · X
m j=1λ
jwhere ε
1, . . . , ε
mdenote ±.
By the change of variables b x
ν− b x
ν−1= ξ
ν/ √
N , (1.5) can be rewritten by using the average E concerning Wiener integrals as
K(t, x, 0) = lim
µ→0
µE
exp
X
m j=1λ
jt
1
\
0
log |b x(τ ) − y
j∗| dτ (1.8)
+ (µ − 1) log
+(b x(1) − x
∗) + 1 2 t log t
X
m j=1λ
j= lim
N →∞
1 (2π)
N/2√
t lim
µ→−1
(µ + 1)
\
RN
Φ
µ(ξ) dξ
1∧ . . . ∧ dξ
Nbecause the function lim
µ→0µx
µ−1+tends to the δ function (a procedure ` a la S. Watanabe (see [8], [16])).
(1.8) can also be represented as (1.9) K(t, x, 0) = lim
N →∞
1 (2π)
N/2√
t lim
µ→−1
(µ + 1)
\
RN
Φ b
µ(ξ) dξ
1∧ . . . ∧ dξ
Nwhere
log b Φ
µ(ξ) = − 1
2 Q(ξ) + µ log
+ξ
1+ . . . + ξ
N√ N − x
∗(1.10)
+ t N
X
m j=1λ
jX
N ν=1log
ξ
1+ . . . + ξ
ν− √ N y
j∗√ ν
+ 1
2 (t log t − t) X
m j=1λ
j. In fact,
log Φ
µ(ξ) = log b Φ
µ(ξ) + X
mj=1
λ
jt
2 log t + t 2N
−N log N + X
N j=2log j
.
The last term on the RHS tends to
12(t log t − t)( P
mj=1
λ
j) as N → ∞.
Φ b
µ(ξ; ε
1, . . . , ε
m) are similarly defined from Φ
µ(ξ; ε
1, . . . , ε
m).
For given τ
1, . . . , τ
psuch that 0 < τ
1< . . . < τ
p< 1, and a continuous function ϕ(x
1, . . . , x
p) on R
p, we denote the average of ϕ(b x(τ
1), . . . , b x(τ
p)) over the set of continuous paths by
(1.11) hϕ(b x(τ
1), . . . , b x(τ
p))i
= E
ϕ(b x(τ
1), . . . , b x(τ
p))
× exp
X
mj=1
λ
jt
1
\
0
log |b x(τ ) − y
j∗| dτ +µ log
+(b x(1) − x
∗)+ 1 2 t log t
X
m j=1λ
jand its average with the restriction b x(1) = x
∗by hϕ(b x(τ
1), . . . , b x(τ
p))i
0= lim
µ↓0
µ
ϕ(b x(τ
1), . . . , b x(τ
p)) x(1) − x b
∗.
We are interested in finding a complete system of differential relations with respect to the parameters x
∗, y
1∗, . . . , y
m∗, when t is fixed.
Definition 1. For τ
1, . . . , τ
p∈ (0, 1) such that 0 < τ
1< . . . < τ
p< 1, and indices I = {i
1, . . . , i
p} ⊂ {1, . . . , m}, which may not be distinct, we define a system of functions depending not only on τ
1, . . . , τ
pbut also on x
∗, y
1∗, . . . , y
∗mby
F (τ
1, . . . , τ
p; I) =
1
(b x(τ
1) − y
∗i1) . . . (b x(τ
p) − y
∗ip)
, (1.12)
F
0(τ
1, . . . , τ
p; I) =
1
(b x(τ
1) − y
∗i1) . . . (b x(τ
p) − y
∗ip)
0
. (1.13)
In particular, for p = 0 we have F
0(φ) = h1i
0= √
t K(t, x, 0).
It is convenient to define these functions to be zero if one of τ
his negative or greater than 1.
Remark 1. More precisely, the functions F (τ
1, . . . , τ
p; I) and F
0(τ
1, . . . . . . , τ
p; I) are generalized functions with respect to τ
1, . . . , τ
p. Integration and differentiation can be done formally by the Malliavin calculus. For de- tails see [8], [16].
Theorem 1. The total differentials of F (τ
1, . . . , τ
p; I) with respect to the
parameters x
∗, y
∗1, . . . , y
m∗are given by
(1.14) δF (φ)
= X
m j=1λ
jt n
−δy
∗j 1\
0
dτ F (τ ; j) + δ(x
∗)
1
\
0
dτ τ F (τ ; j) o
−µδ(x
∗)F (φ), i.e.,
∂F (φ)
∂y
∗j= − λ
jt
1
\
0
dτ F (τ ; j), (1.15)
∂F (φ)
∂x
∗= X
m j=1λ
jt
1
\
0
dτ τ F (τ ; j) − µF (φ), (1.16)
and for p ≥ 1, we have
(1.17) δF (τ
1, . . . , τ
p; i
1, . . . , i
p) = A
I+ A
II+ A
IIIwhere A
I, A
II, A
IIIdenote the differentials with respect to δx
∗, δy
∗1, . . . , δy
m∗: A
I=
X
p h=1δ(y
i∗h−1) 1
τ
h− τ
h−1− δ(y
i∗h) τ
h+1− τ
h−1(τ
h+1− τ
h)(τ
h− τ
h−1) + δ(y
i∗h+1) 1
τ
h+1− τ
hF (τ
1, . . . , τ
h−1, τ
h+1, . . . , τ
p; ∂
hI),
A
II= X
m k=1, k6∈Iλ
kt X
p h=0δ(y
∗ih)
τh+1
\
τh
τ
h+1− τ
τ
h+1− τ
hdτ − δ(y
k∗)
τh+1
\
τh
dτ
+ δ(y
i∗h+1)
τh+1
\
τh
τ − τ
hτ
h+1− τ
hdτ
F (τ, τ
1, . . . , τ
p; k, I) + µ{−δ(x
∗) + δ(y
i∗p)}F
0(τ
1, . . . , τ
p; I),
A
III=
− X
p h=1y
i∗hδ(y
i∗h) τ
h+1− τ
h−1(τ
h− τ
h−1)(τ
h+1− τ
h) +
p−1
X
h=1
δ(y
i∗hy
i∗h+1) 1 τ
h+1− τ
hF (τ
1, . . . , τ
p; I), where we put τ
0= 0, τ
p+1= +∞, and y
i∗0= y
∗ip+1= 0.
Theorem 2. The total differentials of F
0(τ
1, . . . , τ
p; I) with respect to the parameters x
∗, y
∗1, . . . , y
m∗are given by
(1.18) δF
0(φ) = X
m j=1λ
jt n
−δy
∗j 1\
0
dτ F
0(τ ; j) + δ(x
∗)
1
\
0
dτ τ F
0(τ ; j) o
− x
∗δ(x
∗)F
0(φ),
i.e.,
∂F
0(φ)
∂y
∗j= − λ
jt
1
\
0
dτ F
0(τ ; j), (1.19)
∂F
0(φ)
∂x
∗= X
m j=1λ
jt
1
\
0
dτ τ F
0(τ ; j) − x
∗F
0(φ), (1.20)
and for p ≥ 1, we have
(1.21) δF
0(τ
1, . . . , τ
p; i
1, . . . , i
p) = B
I+ B
II+ B
IIIwhere B
I, B
II, B
IIIdenote the differentials with respect to δx
∗, δy
∗1, . . . , δy
m∗: B
I=
X
p h=1δ(y
∗ih−1
) 1
τ
h− τ
h−1− δ(y
∗ih) τ
h+1− τ
h−1(τ
h+1− τ
h)(τ
h− τ
h−1) + δ(y
∗ih+1) 1
τ
h+1− τ
hF
0(τ
1, . . . , τ
h−1, τ
h+1, . . . , τ
p; ∂
hI),
B
II= X
m k=1λ
kt X
p h=0δ(y
∗ih)
τh+1
\
τh
τ
h+1− τ
τ
h+1− τ
hdτ − δ(y
k∗)
τh+1
\
τh
dτ
+ δ(y
∗ih+1)
τh+1
\
τh
τ − τ
hτ
h+1− τ
hdτ
F
0(τ, τ
1, . . . , τ
p; k, I),
B
III=
− X
p h=1y
∗ihδ(y
∗ih) τ
h+1− τ
h−1(τ
h− τ
h−1)(τ
h+1− τ
h) − x
∗δ(x
∗) 1 1 − τ
p+
p−1
X
h=1
δ(y
∗ihy
i∗h+1) 1
τ
h+1− τ
h+ δ(y
i∗px
∗) 1 1 − τ
pF
0(τ
1, . . . , τ
p; I), where we put τ
0= 0, τ
p+1= 1 and y
∗i0= 0, y
i∗p+1= x
∗.
Theorem 3. The derivative of F
0(τ
1, . . . , τ
p; I) with respect to t is ex- pressed as
(1.22) ∂
∂t F
0(τ
1, . . . , τ
p; I)
= −V (x)F
0(τ
1, . . . , τ
p; I) − 1 2
X
m j=1λ
jy
j∗1
\
0
dτ F
0(τ, τ
1, . . . , τ
p; j, I).
Theorems 1–3 show that {F
0(τ
1, . . . , τ
p; I)}
p≥0satisfy an integrable sys-
tem with respect to the variables t, x
∗, y
∗1, . . . , y
m∗on the space of continuous
paths b x : [0, 1] → R, while {F (τ
1, . . . , τ
p; I)}
p≥0satisfy an integrable system
with respect to the variables x
∗, y
∗1, . . . , y
m∗only.
2. Arrangement of hyperplanes and a generalized Schl¨ afli for- mula. In the n-dimensional Euclidean space R
nwe consider an arrangement A of m real hyperplanes H
j(1 ≤ j ≤ m) defined by the inhomogeneous lin- ear equations
(2.1) H
j: f
j(x) = 0
for f
j(x) = P
nj=1
u
j,νx
ν+ u
j,0. The functions f
j(x) are assumed to be normalized by P
nν=1
u
2j,ν= 1.
The configuration matrix A associated with the arrangement is defined as the symmetric matrix A = (a
j,k)
mj,k=0of order m + 1, where a
j,kdenotes the inner product between the coefficients of f
j, f
k:
a
j,k= X
n ν=1u
j,νu
k,ν, m ≥ j, k ≥ 1, (2.2)
a
j,0= a
0,j= u
j,0, m ≥ j ≥ 1 and a
0,0= 1.
(2.3)
Note that a
j,j= 1.
For I = {i
1, . . . , i
p}, 0 ≤ i
1< . . . < i
p≤ m, and J = {j
1, . . . , j
p}, 0 ≤ j
1< . . . < j
p≤ m, we denote by A
JIthe subdeterminant det((a
i,j)
i∈I,j∈J), in particular we write A(I) in the case where I = J. The arrangement A is uniquely determined by the matrix A up to isomorphism of the n- dimensional orthogonal group O(n).
Let λ
1, . . . , λ
mbe real numbers and Φ(x) be the analytic function (2.4) Φ(x) = exp −
12Q(x)
f
1(x)
λ1. . . f
m(x)
λmfor Q(x) = P
nν=1
x
2ν=1. We consider the integral
(2.5) F =
\
∆
Φ(x) dx
1∧ . . . ∧ dx
nover a twisted cycle ∆ associated with the function Φ(x).
We also consider the system of integrals
(2.6) F (I) =
\
∆
Φ(x) dx
1∧ . . . ∧ dx
nf
i1. . . f
ip.
It has been proved in [3] that the functions {F (φ) and F (I), 1 ≤ p ≤ n}
form a complete system of integrals in the n-dimensional twisted de Rham cohomology which has dimension P
nj=0 m
j
.
Moreover the following variational formula holds (see Proposition 1.3 in [3], Part I).
Proposition 1 (Generalized Schl¨ afli formula).
(2.7) δF (φ) = X
m j=1λ
jδa
j,0F (j) + 1 2
X
m j,k=1, j6=kλ
jλ
kδa
j,kF (j, k)
where δ denote differentials of variation.
This formula is just a generalization of the classical Schl¨ afli formula which is a variational formula for the volume of a geodesic simplex in a space of positive constant curvature (see [2], [11], [15] etc.). In fact, to obtain Schl¨ afli’s formula from (2.7), we take as ∆ the simplicial cone defined by f
1≥ 0, . . . , f
n≥ 0 and we take the limit λ
j→ 0, for all j.
We can also get the variational formulae for the function F (I). However these are rather complicated and we do not reproduce them here.
Later on we only need the formulae in the cases where a
j,k(1 ≤ j, k ≤ m) are constants. They are described as follows (see [3], Proposition 3 in Part II, or [5], Lemma 2).
The symbols {k, I} and ∂
hI will denote the sets of indices {k, i
1, . . . , i
p} (addition of the index k to I) and {i
1, . . . , i
h−1, i
h+1, . . . , i
p} (deletion of the hth index from I) respectively.
Proposition 2. Assume that a
j,k(1 ≤ j, k ≤ m) are constants. Then
(2.8) δF (φ) =
X
m j=1λ
jδa
j,0· F (j), and for p ≥ 1, we have
A(I) · δF (I) = − X
p h=1(−1)
hδA
I
0, ∂
hI
· F (∂
hI) (2.9)
+ X
k6∈I,k≥1
λ
kδA
k, I 0, I
· F (k, I) + 1
2 δA(0, I) · F (I).
Similarly we have the recurrence relations.
Proposition 3. Let T
k±denote the shift operators corresponding to the shift λ
k7→ λ
k± 1. Then
(2.10) T
k−F (I) = F (k, I), k 6∈ I, (2.11) (λ
i1− 1)A(I) · T
i−1F (I)
= X
p h=1A
∂
1I
∂
hI
(−1)
h+1· F (∂
hI)
− X
k6∈I
λ
kA
I k, ∂
1I
· F (k, I) − A
I 0, ∂
1I
· F (I).
3. Application of the generalized Schl¨ afli formula. We denote the normalized inhomogeneous functions appearing in (1.6) as follows:
f
j,ν(ξ) = ξ
1+ . . . + ξ
ν− √ N y
∗j√ ν , ν = 1, . . . , N,
and
f
m+1,N(ξ) = ξ
1+ . . . + ξ
N− √ N x
∗√ N .
When j = m + 1, ν takes only the value N. In the sequel y
m+1∗and λ
m+1are identified with x
∗and µ respectively.
The inner product of the coefficients of f
j,νand f
k,σis given by a
j,k= (f
j,ν, f
k,σ) =
ν/ √
νσ for ν ≤ σ, σ/ √ νσ for ν ≥ σ.
By abuse of notation, we may denote it by a
j,kwithout ambiguity, since below only one function f
j,νcorresponds to each index j. We may also denote
a
j,0= a
0,j= − √
N y
j∗/ √ ν corresponding to f
j,ν.
The function b Φ
µ(ξ) can be represented as (3.1) Φ b
µ(ξ)
= exp
− 1
2 Q(ξ) + 1
2 (t log t − t) X
m j=1λ
j· n Y
mj=1
Y
N ν=1f
j,ν(ξ)
λjt/No
f
m+1,N(ξ)
µ+. Then the following lemmas can be proved by a direct computation. It is a remarkable fact that every subdeterminant is non-negative.
Lemma 1. Let p pairs of indices I = {(i
1, ν
1), . . . , (i
p, ν
p)}, {i
1, . . . , i
p} ⊂ {1, . . . , m + 1} and 0 < ν
1< . . . < ν
p≤ N + 1, be given (we assume ν
p= N + 1 if i
p= m + 1). Then
A(I) = (ν
2− ν
1) . . . (ν
p− ν
p−1) ν
2. . . ν
p, and for 1 ≤ k, h ≤ p,
A
∂
kI
∂
hI
/A(I) =
0, |h − k| > 1,
√ ν
hν
h+1ν
h+1− ν
h, k = h + 1,
√ ν
hν
h−1ν
h− ν
h−1, k = h − 1,
ν
h(ν
h+1− ν
h−1)
(ν
h+1− ν
h)(ν
h− ν
h−1) , k = h.
Lemma 2.
(3.3) A(0, I)/A(I) = 1 − X
p h=1N (y
∗ih)
2(ν
h+1− ν
h−1) (ν
h+1− ν
h)(ν
h− ν
h−1) + 2
p−1
X
h=1
N y
i∗hy
∗ih+1ν
h+1− ν
h.
Lemma 3. For 1 ≤ h ≤ p, A
I 0, ∂
hI
/A(I) = ( − 1)
h−1y
∗ih−1√ N ν
hν
h− ν
h−1(3.4)
+ (−1)
hy
∗ih√ N ν
h(ν
h+1− ν
h−1) (ν
h+1− ν
h)(ν
h− ν
h−1) + (−1)
h+1y
i∗h+1√ N ν
hν
h+1− ν
hwhere ν
0and ν
p+1are set to be 0 and +∞ respectively.
Lemma 4. Assume that the index k corresponds to the function f
k,σ. Then
A
k, I 0, I
/A(I) = y
∗ih√ N (ν
h+1− σ)
√ σ(ν
h+1− ν
h) (3.5)
− y
∗k√ N
√ σ + y
i∗h+1√ N (σ − ν
h)
√ σ(ν
h+1− ν
h) if ν
h≤ σ ≤ ν
h+1.
We can now apply the formulae (2.8), (2.9) to (1.8). We denote by ϕ(I) (I ⊂ {1, 2, . . . , m + 1}) the following integrals:
(3.6) ϕ(I) =
\
RN
Φ b
µ(ξ) 1
f
i1,ν1(ξ)f
i2,ν2(ξ) . . . f
ip,νp(ξ) dξ
1∧ . . . ∧ dξ
N. From Lemmas 1–4 and (2.8), (2.9), we deduce the following formulae.
Proposition 4.
(3.7) δϕ(I) = X
I+ X
II+ X
IIIwhere
X
I= X
p h=1δ(y
i∗h−1)
√ N ν
hν
h− ν
h−1(3.8)
− δ(y
∗ih)
√ N ν
h(ν
h+1− ν
h−1)
(ν
h+1− ν
h)(ν
h− ν
h−1) +δ(y
∗ih+1)
√ N ν
hν
h+1− ν
hϕ(∂
hI), X
II= 1
N
X
k6∈I, 1≤k≤m+1
λ
kX
p h=0X
νh<σ<νh+1
δ(y
i∗h
)
√ N (ν
h+1− σ)
√ σ(ν
h+1− ν
h) (3.9)
− δ(y
∗k) r N
σ + δ(y
∗ih+1)
√ N (σ − ν
h)
√ σ(ν
h+1− ν
h)
ϕ(k, I)
+ 1 N
X
m k=1λ
kX
p h=1, ih6=kδ log(y
∗ih− y
∗k)(ϕ(I) − ϕ(k, ∂
hI)),
(3.10) X
III= − X
p h=1y
∗ih
δ(y
∗ih
) N (ν
h+1− ν
h−1) (ν
h+1− ν
h)(ν
h− ν
h−1)
+
p−1
X
h=1
δ(y
∗ihy
∗ih+1) N
ν
h+1− ν
hϕ(I) where we put ν
0= 0, ν
p+1= +∞.
Proof of Theorems 1 and 2. Let us take the limit N →∞ of (3.7). When ν/N tends to the value τ, the function (ξ
1+ . . . + ξ
ν)/ √
N tends almost surely to the value b x(τ ) at τ of a continuous path b x.
This implies that if ν
1/N → τ
1, . . . , ν
p/N → τ
p, then ϕ(I)
(2π)N/2Np/2√ν1...νpin Proposition 4 (I ⊂ {1, . . . , m}) tends to F (τ
1, . . . , τ
p; I) defined in Defi- nition 1, i.e.,
(3.11) lim
N →∞
ϕ(I) N
p/2(2π)
N/2√ ν
1. . . ν
p= F (τ
1, . . . , τ
p; I).
In the same way, we have (3.12) lim
N →∞
lim
µ↓0
ϕ(I, m + 1) N
p/2(2π)
N/2√ ν
1. . . ν
p= F
0(τ
1, . . . , τ
p; I).
We multiply both sides of (3.7) by
(2π)N/2Np/2√ν1...νpand take the limit N →∞. Then the sum
N1P
νh<σ<νh+1
tends to the integral
Tτh+1
τh
dτ , whence (3.7) for ϕ(I) and ϕ(I, m+1) tend to the equations (1.13), (1.16) and (1.17), (1.20) respectively.
On the other hand the last term on the RHS of (3.9) tends to zero.
Theorems 1 and 2 have thus been proved.
Proof of Theorem 3. First note the following equality. Since x(tτ ) =
√ t b x(τ ), y
j= √
t y
∗jwe have log |x(t) − y
j| =
1
\
0
log |b x(τ ) − y
j∗| dτ + 1
2 (1 + log t) + 1 2 y
j∗1
\
0
dτ b x(τ ) − y
j∗. In fact as generalized Wiener functionals (Malliavin calculus), we have
log |x(t) − y
j| = d dt
t
\
0
log |x(s) − y
j| ds
= d dt
t
1
\
0
log |b x(τ ) − y
j√ t | dτ + 1 2 t log t
,
which gives the above equality by the Leibniz rule. See Remark 1.
This implies X
mj=1
λ
j 1\
0
log |b x(τ ) − y
∗j| dτ + 1
2 (log t + 1) X
m j=1λ
j= X
m j=1λ
jlog |x(t) − y
j| − 1 2 λ
jy
∗j1
\
0
dτ b x(τ ) − y
∗j. We use the identity V (x) = − P
mj=1
λ
jlog |x(t) − y
j| and get the equality (1.22).
Assume now that the indices in I are all distinct. One can prove that the formulae in Theorems 1, 2 are still valid when we consider the functions Φ b
µ(ξ; ε
1, . . . , ε
m) in place of b Φ
µ(ξ) (the latter is equal to the sum over all ε
1, . . . , ε
m.)
Let us use the same notations F (τ
1, . . . , τ
p; I), F
0(τ
1, . . . , τ
p; I) as in Theorems 1 and 2 in the case of b Φ
µ(ξ; ε
1, . . . , ε
m).
Let T
λ−kt
denote the shift operators corresponding to the shifts λ
kt 7→
λ
kt − 1. Relations between the partial differentiation ∂/∂y
k∗and the shift operators T
λ−kt
(contiguous relations) are given as follows.
When k 6∈ I,
∂
∂y
k∗F (τ
1, . . . , τ
p; I) = − λ
kt
1
\
0
dτ F (τ, τ
1, . . . , τ
p; k, I) (3.13)
= − ε
kλ
kt
3/2T
λ−kt
F (τ
1, . . . , τ
p; I), while if k ∈ I, then
(3.14) ∂
∂y
i∗hF (τ
1, . . . , τ
p; I)
= − (λ
iht − 1)
1
b x(τ
h− y
i∗h)b x(τ
1− y
∗i1) . . . b x(τ
p− y
∗ip)
− λ
iht
1\
0
F (τ, τ
1, . . . , τ
p; i
h, I)
= − (λ
iht − 1)
1
b x(τ
h− y
i∗h)b x(τ
1− y
∗i1) . . . b x(τ
p− y
∗ip)
− ε
ihλ
iht
3/2T
λ−iht
F (τ
1, . . . , τ
p; I).
The same relations are also valid for F
0(τ
1, . . . , τ
p; I).
Remark 2. The formulae of Gauss–Manin connections for (2.5) including
(2.7) which have been obtained in [3] seem to have another application.
It may be possible to extend the arguments discussed in this note to the Schr¨ odinger operators of the one-dimensional many body system
− 1 2
X
n j=1∂
2∂x
2j+ V (x
1, . . . , x
n) where V (x
1, . . . , x
n) denotes the potential
V (x
1, . . . , x
n) = − X
1≤j<k≤n
λ
j,klog |x
j− x
k| − X
1≤j≤n
X
1≤k≤m
µ
j,klog |x
j− y
k|.
More generally in the complex domain C, one can consider the operators (each z
j= x
j+ iy
j, w
j∈ C)
− 1 2
X
n j=1∂
2∂x
2j+ ∂
2∂y
2j+ V (z
1, . . . , z
n) where
V (z
1, . . . , z
n) = − X
1≤j<k≤n
λ
j,klog |z
j− z
k| − X
1≤j≤n
X
1≤k≤m
µ
j,klog |z
j− w
k|.
One may possibly obtain similar results to Theorems 1–3, although the formulae would be more complicated (see [4] for a similar argument).
Acknowledgements. The author would like to acknowledge several use- ful comments by Prof. Nobuyuki Ikeda.
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