Brewster H.D. Mathematical Physics
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52
Laplace
and
Saddle
Point Method
1 [ 2 ] 3
Sex)
=S(x
o
)+'2(x-xo)·
8
x
S(xo)
(x-xo)+O«x-xo)
)
One could also use the Morse Lemma to replace the function S
by
a
quadratic form. Then as
A
~
00
the main contribution to the integral comes
from a small neghborhood
of
xO.
In this neighborhood the terms
of
the third
order in the Taylor expansion
of
S can be neglected. Also, since the function
fis
continuous at x
o
' it can be replaced by its value at
xO.
Then the region
of
integration can be extended to the whole lin. By using the formula for the
standard Gaussian integral one gets then the leading asymptotics
ofthe
integral
F(A)
as
A~
00
(
2
)n12
112
F(A.) - exp[A.S(x
o
)]:
[
-det8;S(xo)
r
f(xo)
One can prove the general theorem.
Theorem:
Letj,
Se
C-(Q)
and let
Xo
be a nondegenerate critical point
of
the function S where it has the only maximum in
Q.
Let 0 < E <
Te/2.
Then
there
is
asymptotic
expansion
as
A
~
00
in
the
sector
Sf.
=
{A
e
ell
arg
A/
~
Te/2
-
E}
co
F(A.) -
exp[A.S(xo)]A.
-n12
L
akA.
-k
.
k=O
The coefficients a
k
are expressed in terms
of
the derivatives
of
the
functions
f and S at the point
xO.
The idea
of
the proof is the same as in the one-dimensional case and goes
as follows. First, we change the integration variables
Xi
=
xb
+ A
-1I2yi.
So, y
is
the scaled uctuation from the maximum point
xO.
The interval
of
integration should be changed accordingly, so that the maximum point is now
y =
o.
Then, we expand both functions
Sand
<p
in Taylor series at
Xo
getting
1
co
A.
-</a/-2)/2
AS(x
+
A-
I12
y)
=
AS(xo)+-y·[8;S(xo)]Y+
L
[8
a
S(xo)]ya
o 2
/al
<x!
co
A.
-la/12
m(x
+
A.1I2y)
= L
,8
a
<p(xo)ya
't'
0
/a/=O
<x.
Since the quadratic terms are
of
order
0(
1)
we leave them in the exponent
and ,expand the exponent
of
the rest in a power series. Next, we extend the
integration domain to the whole
lin and compute the standard Gaussian
integrals.
Finally, we get a power series in inverse powers
of
A.
The coefficients
Laplace
and
Saddle
Point Method
53
a
k
of
the asymptotic expansion are polynomials in the higher derivatives
aa
S(x
o
),
lal
~
3, and derivatives
aa
<p(xo),
lal
~
0, and involve inverse matrices G
=
[d2~(xO)]-l.
Remark:
If
Xo
is
a degenerate maximum point
of
the function
S,
then the
asymptotic expansion as
A
--7
00
has the form
00
N
F(A)-exp[A.S(xo)]A
-n12
L
LakiA
-rk(lnAi,
k=ot=O
where N is some positive integer and
{r
k
},
r
k
~
n12,
kEN,
is a increasing
sequence
of
nonnegative rational numbers.
The
coefficients a
k
(and a
kt
)
of
the asymptotic expansion
of
the integral
F(A) are invariants under smooth local diffieomorphisms in a neighborhood
of
Xo
and play very important role in various applications.
Boundary
Maximum
Point
Let
now
S has maximum
at
the boundary point
Xo
E
an.
We
assume, for
simplicity, that both the boundary and the function
S are smooth, i.e. S E
CO
and
an
E
CO.
Since the boundary is smooth we can smoothly parametrize it in a neigh-
borhood
of
Xo
by (n - 1) parameters
~
=
(~a),
(a =
1,
... , n - 1). Let the the
parametric equations
of
the boundary be
xi =
xi(~),
i =
1,
... , n.
Then
- (Ti)=(8i J
Ta
- a
8'f:,a
are tangent vectors to the boundary.
Let
r = rex) be the normal distance to the boundary. Then the equation
of
the boundary can be written as
rex) =
O.
a and for x E n we obtain that the vector
we
have r >
O.
Obviously,
r(x(~)
=
O.
From this equation
N=
(N
j
)=
(~)
is orthogonal to all tangent vectors and is therefore normal to the boundary. It
can be certainly normalized, since it is nowhere zero. We choose it to be the
inward normal. The normal and tangential derivatives are defined as usual
54
Laplace
and
Saddle
Point
Method
The point
Xo
is
not, in general, a critical point
of
S, since the normal
derivative
of
S at
Xo
does not have to be equal to zero.
Definition:
The.
point
Xo
is
said to be a nondegenerate boundary maximum
point
of'S
if
a,s(x
o
)
:#:
0
and the (n -
1)
X (n - I) matrix
a~
S
(x(~»
is
negative definite.
In a neighborhood
of
a non degenerate boundary maximum point the
function
S has the following Taylor expansion
Sex)
=
S(xo)+[OrS(xo)]r+.!..[o~S(xo)]r2
2
+
....
,
up to third order terms
in
r and
(~
-
~o)'
Now we replace the integral F
(A)
by an integral over a small neighborhood
of
xo' We change the variables
of
integration from xi, i =
1,
...
, n, to
(~a,
r),
a =
1,
... , n -
1,
and neglect the terms
of
third order
in
the Taylor series. We also
replace the
function/by
its value at the point
xo.
In
the remaining integral we
extend the integration to the whole space
~+
X
~n-I,
i.e. we integrate over r
from
0 to
00
and integrate over the whole tangent plane at
xo'
These integrals
are standard Gaussian integrals and we obtain the leading asymptotics as
::;
~
00
F
(A)
-
-A
-(n+I)/2(2n)(n-I)/2
exp[AS(Xo)]
x
[orS(xo)rl[
-detolS(xo)rIl2
J(xo)/(xo)
where
J(x
o
)
is
the Jacobian
of
change
of
variables.
The general form
of
the asymptotic expansion is given by the following
theorem.
Theorem: Let
j,
SE
C"(Q)
and let S have a maximum only at a non-
degenerate boundary maximum point
XOE
aQ.
Then as A
~
00,
A E
Sf:'
00
F
(A)
- A
-(n+l)/2
exp[AS(Xo)] I
akA-k
k=O
Integral Operators with Singular Kernels
Let be a bounded domain
in
~n
including the origin, 0 E
Q.
Let S be a
real valued non-positive function on
Q
of
class C
2
that has maximum equal to
zero,
S(O)
= 0, only at a nondegenerate maximum critical point
Xo
=
O.
Let
K)..:
C"(Q)
be a linear integral operator defined by
Laplace
and
Saddle
Point
Method
(
'A
)n/2
(Ktf) (x) =
21t
1 exp['AS(x -
y)]f(y)dy
Let M be a compact subset
of
n.
Then
Al~oo
(KAf)(x)
= [
-det8;S(O)
r
ll2
f(x)
uniformly for x E M. Formally
(2'A
1t
J/2
exp['AS(x -
y)]
~
[-det8;S(O)r
I/2
8(x
-
y)
STATIONARY
PHASE
METHOD
STATIONARY
PHASE
METHOD
IN
ONE
DIMENSION
Fourier
Integrals
55
Let
M=
[a,
b]
be a closed bounded interval,
S:
M
~
~
be a real valued
nonconstant
function,j
M
~
C be a complex valued nonzero function and
~
be a large positive parameter. Consider the integrals
of
the form
F
('A)
= r
f(x)
exp[i'AS(x)] dx.
The function S is called phase function and such integrals are called
Fourier Integrals.
We will study the asymptotics
of
such integrals.
As
A
~
00
the integral
F('A)
is small due to rapid oscillations
of
exp(iAS).
Lemma: (Riemann-Lebesgue)
Letfbe
an integrable function on the real
line,
i.e.fE
LI
(~).
Then
1
f(x)e
'Ax
dx
= 0(1),
('A
~
00).
Definition:
1.
A point
Xo
is
called the regular point
of
the Fourier integral
F
('A)
if
the
functionsfand
S are smooth
in
a neighborhood
of
Xo
and
S'
(x
o
)
*-
O.
2.
A point
Xo
is called the critical point
of
the integral F
('A)
if
it
is
not
a regular point.
3.
A critical point
Xo
is
called isolated critical point
if
there
is
a
neighborhood
of
Xo
that does not contain any other critical points.
4.
An interior isolated critical point
is
called stationary point.
S.
The integral over a neighborhood
of
an isolated critical point that
does not contain other critical points will be called the contribution
of
the critical point to the integral.
Clearly the main contribution comes from the critical points since close
56
Laplace
and
Saddle
Point Method
to
these points the oscillations slow down. As always, we will assume
that
functions
Sand
J are smooth, i.e.
of
class
COO(M).
Otherwise, the
singularities
of
the
functions
Sand
J
and
their
derivatives
would
contribute significantly
to
F(A).
localization Principle
Lemma:
Let SE
Co:)(lR)
be smooth function andJECQ(lR) be a smooth
function
of
compact support. Then
as
A
~
00
kJ(x)exp[iAS(X)]dx
=
O(A
-0:)
Remarks:
1.
Since the
functionJhas
compact support, the integral is, in fact,
over a finite interval.
2.
This
is
the main technical lemma for deriving the (power) asympotics
of
the Fourier integrals.
It
means that such integrals can be neglected
in a power asymptotic expansion.
3.
The Fourier integrals are in general much more subtle object than the
Laplace integrals. Instead
of
exponentially decreasing integrand one
has a rapidly oscillating one. This requires much finer estimates
and
also much stronger conditions
on
the phase function
Sand
the
inte-grand f
Theorem: Let the Fourier integral F
(A)
have finite number
of
isolated
critical points.
Then
as
A
~
00
the integral F
(A)
is
equal
to
the sum
ofthe
contributions
of
all critical points up to
O(A
---<Xl).
Thus,
the
problem
reduces
to
computing
the
asymptotics
of
the
contributions
of
critical points.In a neighborhood
of
a critical point
we
can
replace the functions
S
andJby
more simple functions and then compute some
standard integrals.
Boundary Points
If
the phase function does
not
have any stationary points,
then
by
integration by parts one can easily obtain the asymptotic expansion.
Theorem: Let
S'(x)
*-
0 \;fxEM.
Then
as A
~
00
0:)
k b
F(A) - I
(iA)-k-1
(_1_~)
(f(X)
)iAS(X)
k=O
-S'(x)
ax
S'(x)
a
The
leading asymptotics
is
F
(A)
=
(iA)-1
if(b)
exp[iAS(b)] - J (a) exp[i A Sea)]} +
O(A
-2)
The
same technique, i.e. integration by parts, applies
to
the integrals
over
an
unbounded
interval, say,
Laplace
and
Saddle
Point Method 57
F
(A)
= r f(x)exp[iAS(X)]dx
with some additional conditions
that
guarantee the converges
at
00
as well as
to
the
integrals
of
the form
F(x)
= r f(t)exp[iS(t)]dt
as x
---+
00.
Standard
Integrals
Consider the integral
<1>(A)
= r f(x)xP-IeiAxu .
Lemma: (Erdeyi) Let
ex
2
1,
f3
> °
andfis
a smooth function
on
a closed
bounded
interval [0,
a],
f E
COO([O,
aD,
that
vanish
at
x = a with all its
derivatives.
Then
as A
---+
00
where
a.
= f(k)(O) 1
r(f3+k)exP[i1tf3+k]
k k!
ex ex ex
This lemma plays he same role in the stationary phase method as Watson
lemma in
the
Laplace method.
Stationary
Point
Theorem:
Let
M =
[a,
b]
be
a closed
bounded
interval,
SECOO(M)
be
a
smooth
real valued
nonconstant
function,fE
Co
(M)
be
a complex valued
function with compact support in M. Let
S have a single isolated nondegenerate
critical
point
Xo
in
M,
i.e.
S'(x
o
)
= °
and
S"(x
o
)
*"
0.
Then
as A
---+
00
there
is
asymptotic expansion
of
the Fourier integral
F
(A)
= r f(x)exp[iAS(X)]dx
_exP[iAS(X
o
) +
[sgnS"(Xo)]i~]A
-112 I A
-k.
4
k=O
The
coefficients a
k
are
determined in
terms
of
the
derivatives
of
the
functions S
andf
at
xo'
The
leading asymptotics as A
---+
00
is
F(A) = I
21t
IA-1I2exp[iAS(Xo)+[sgns"(xo)]i~](f(Xo)+O(A-l»
S"(xo) 4
58
Laplace
and
Saddle
Point Method
To prove this theorem one does a change
of
variables in a sufficiently
small neghborhood
of
xo'
Principal Values
of
Integrals
Letfbe
a smooth function and consider the integral
r
f~X)
dx
This integral diverges, in general, at x =
O.
One can regularize it by cutting
out a symmetric neighborhood
of
the singular point
J(E)
=
[E
f~X)
dx+ r
f~X)
dx .
Definition:
If
the limit
of
J(E)
as E
~
0+
exists, then it is called the principal
value
of
the integral J
'P
f
f(x)
dx
= lim ( r
E
f(x)
dx+
f
f(x)
dx)
.Ia
x
E~O+.b
x
.1:
x
In this section we consider the asymptotics
of
the integrals
of
the form
F
(A.)
= p 1
e±iAS(x)
f(x)
~
.
as
A.
~
00.
Lemma:
Letfe
CO(JR)
be a smooth function
of
compact support. Then as
A.~oo
r +"I.x
dx
P
~e-I
f(x)-;
=±i1tf(O)+O(A-
OO
).
Theorem:
Let
fe
CQ(JR)
be a
smooth
function
of
compact
support,
Se
C-(JR) be a real valued smooth function and S'(O)
'¢
O.
Then as
A.
~
00
F(A.)
= P 1 e±iA.x
f(x)
~
= [sgnS'(O)]i1tf(O)exp[iAS(O)] + 0(1..-
00
).
Theorem:
Letfe
CQ(JR)
be a smooth function
of
compact support,
Se
C-(JR)
be a real valued smooth function. Let x = 0 be the
only
statinary point
of
the
function
S on supp
j,
and let it be nondegenerate, i.e.
S'(O)
= 0 and
S"(O)
'¢
O.
Then as
A.
~
00
there is asymptotic expansion
F
(A.)
= P 1
e±iAS(x)
f(x)
~
00
-exp[iA.S(O)]A -112 L akA
-k.
k=O
The
leading asymptotics has the form
Laplace
and
Saddle
Point
Method
59
F(A)=
eX
P
[iAS(0)+[SgnS"(0)i
1t
4
]
21t
IS"(O)I
X
A-1I2[_
S"'(O)
J(0)+J'(0)+0(A-1)]
6S"(0) .
STATIONARY
PHASE
METHOD
IN
MANY
DIMENSIONS
Let 0 be a domain in
jRn
andJ
e CQ(O) be a smooth function
of
compact
support,
Se
C'{O) be a real valued smooth function.
In
this section we study
the asympotics as
A
--?
00
of
the multi-dimensional Fourier integrals
F
(A)
= bJ(x)exp[iAS(X)]dx.
Nondegenerate Stationary Point
Localization Principle
Lemma: Let 0 be a domain in
JRn
andJe
CQ(O) be a smooth function
of
compact support,
Se
C"'(O) be a real valued smooth function without stationary
points in supp
j,
i.e. a
f'(x)
'¢
0 for
xe
supp f Then as A
--?
00
F(A)
=
O(A-<><»
This lemma is proved by integration
by
parts.
Definition: The set
S(JRn)
of
all smooth functions on
Rn
that decrease at
Ixl
--?
00
together with all derivatives faster than any power
of
Ixl
is
called the
Schwartz space.
For any integrable
functionJ
e
Ll(JRn)
the Fourier transform is defined
by
F(j)
(~)
= (21t)-nI2
~nexp(ix'~)J(x)dx
Proposition: Fourier transform is a one-to-one onto map
(bijection)F
:
S(JRn)
--?
S
(JRn),
i.e.
ifJe
S(Rn), then F(j)e
S(JRn).
The inverse Fourier transform is
F-l(j)(x) = (21t)-nI21n exp(i
x·~)
J(~)d~
Theorem:
Let
0 be a finite domain in
JRn,
J e C'" 0
(0)
be a smooth
function with compact support in and S e
C"(O)
be a real valued smooth
function.
Let
S
have
a single
stationary
point
Xo
in
and let
it
be
non-
degenerate. Then
as
A.
--?
00
there is asymptotic expansion:
00
F(A)
~
A
-n12
exp[iAS(XO)]L
ak
A
-
k
•
k=O
60
Laplace
and
Saddle
Point Method
The coefficients G
k
are determined in terms
of
derivatives
of
the functions
fand Sat
xO'
The leading asymptotics is
F (A) =
(2An
r
/2
eXP[iAS(X
O
)
+ [sgn
a;S(Xo)i~]
x Ideta;(S(Xo)I-
1I2
[f(xo)
+
0(A-
1
)]
.
Recall that sgn A = v +(A) - v _ (A) denotes the signature
of
a real symmetric
nondegenerate matrix
A, where v ±(A) are the number
of
positive and negative
eigenvalues
of
A.
Integral Operators with Singular Kernels
Let be a bounded domain
in
]R1l
including the origin, 0 E Q.
LetfE
C be
a smooth function with compact support and
SE
C"'(]RIl) be a real valued
11011-
positive smooth
functio11.
Let S have a single stationary point x =
0,
and let it to a nondegenerate,
i.e. let
S(O)
= S'(O) = 0 and
a~
(0) *
O.
Let
K;...:
ca(Q)
~
C"'(Q) be a linear
integral operator defined by
(KJ)(x)
=
(2Anr/21exP[iAS(X-
y)]f(y)dy
Then as A
~
00
(K,J)
(x) = ex
p
[
sgnD;S(O)i~
]ldeta;s(of
1l2
[f(x)
+
O(A
-I)
]
uniformly for x E
Q.
On the other hand,
if
x E
Q,
then as A
~
00
(KJ)
(x) =
0(1..--<>0)
SADDLE
POINT
METHOD
FOR
LAPLACE
INTEGRALS
Let be a contour in the complex plane and the functions f and S are
holomorphic in a neighborhood
of
this contour. In this section we will study
the asymptotics as
A
~
00
of
the Laplace integrals
F(A) =
Lf(z)exP[AS(Z)]dz
Heuristic Ideas
of
the Saddle Point Method
The idea
of
the saddle point method (or method
of
steepest descent) is
to
deform the contour in such a
way
that the main contribution
to
the integral
comes from a neighborhood
of
a single point. This is possible since the
functionsfand
S are holormorphic.
Laplace
and
Saddle
Point
Method
61
First
of
all, let us find a bound for
IF(A)I.
For a
contoury
=
Yo
finite length
/(Yo)
we have, obviously,
IF(A)I
::;1(Yo)maxj(z)exp[A
ReS(z)].
=EYo
Now, let r be the set
of
all contours obtained by smooth deformations
of
the contour
Yo
keeping the endpoints fixed. Then such an estimate
is
valid for
any contour
Y E r, hence,
IF(A)I::;
inf{l(y)maXj(Z)[AReS(Z)]}
YEr :EY
Since we are interested
in
the limit A
-t
00,
we expect that the length
of
the contour does not affect the accuracy
of
the estimate. Also, inluitively
it
is
clear that the behaviour
of
the function S
is
much more important that that
of
the function j (since S
is
in
the exponent and its variations are scaled
significantly by the large parameter
A).
Thus we expect an estimate
of
the
form
IF
(1..)1::;
C(y,f)inf{maxeXP[AReS(Z)]}
YEr
ZEY
where C
(y,j)
is
a constant that depends on the contour
yand
the
functionj
but does not depend on
A.
So, we are looking for a point on a given contour
where the maximum
of
Re S(z)
is
attained. Then, we look for a contour 1*
where the minimum
of
this maximum
is
attained, i.e. we assume that there
exists a contour
y*
where
min max
ReS(z)
YEr
ZEY
is
attained. Such a contour will be called a minimax contour.
Let
Zo
E
1..*
be the only point on the contour
y*
where the maximum
ofRe
S(z) is attained. Then, we have an estimate
IF
(1..)1
::;
C
(y*,j)
exp[A Re S(zo)]'
By deforming the contour
of
integration to
y*
we obtain
F
(A)
= t
j(z)exp[AS(z)]dz.
The asymptotics
of
this integral can be computed by Laplace method.
1.
Boundary Point. Let
Zo
be an endpoint
of
y*,
say, the initial point.
Suppose that
S'
(zo)
'*
O.
Then one can replace the integral F
(A)
by
an integral over a small arc with the initial point
zoo
Finally, integrating
by parts gives the leading asymptotics
F(l)
,1
exp[AS(Zo)]A-1[j(Zo)+O(A-
1
)].
-S
(zo)
Interior Point. Let
Zo
be an interior point
of
the contour
y*.
From