atoms
Article
On the Approximate Evaluation of Some
Oscillatory Integrals
Robert Beuc *, Mladen Movre
and Berislav Horvatić
Institute of Physics, 10000 Zagreb, Croatia; movre@ifs.hr (M.M.); horvatic@ifs.hr (B.H.)
* Correspondence: beuc@ifs.hr; Tel.: +385-91-787-5082
Received: 4 March 2019; Accepted: 29 April 2019; Published: 5 May 2019
Abstract: To determine the photon emission or absorption probability for a diatomic system in the
context of the semiclassical approximation it is necessary to calculate the characteristic canonical
oscillatory integral which has one or more saddle points. Integrals like that appear in a whole range of
physical problems, e.g., the atom–atom and atom–surface scattering and various optical phenomena.
A uniform approximation of the integral, based on the stationary phase method is proposed, where
the integral with several saddle points is replaced by a sum of integrals each having only one or at
most two real saddle points and is easily soluble. In this way we formally reduce the codimension in
canonical integrals of “elementary catastrophes” with codimensions greater than 1. The validity of the
proposed method was tested on examples of integrals with three saddle points (“cusp” catastrophe)
and four saddle points (“swallow-tail” catastrophe).
Keywords: oscillatory integrals; stationary point approximation; semi-classical theory; uniform
Airy approximation
1. Introduction
Many physical processes like the atom scattering on atom, molecule, surface, or a crystal and
propagation of electromagnetic waves through a medium, are described generally by multidimensional
integrals. The value of these integrals as a function of the parameters a = {a1 , a2 , . . . an } depends
on the morphology of the phase f (a; u) of the integrand, where u = {u1 , u2 , . . . uk } are the variables
of integration.
The main contribution to the integral comes from the neighbourhood of the saddle points
∂ f (a;u )
∂ f (a;u )
∂ f (a;u )
us = {u1s , u2s , . . . uks }, where ∂u s = ∂u s = . . . = ∂u s = 0. For some parameters a it is possible
2s
1s
ks
that some higher derivatives are equal to 0 at the saddle points, which causes coalescence of these
points, increases the value of the integral that manifests itself as “rainbow and glory” [1]. To analyze
the rainbow phenomenon, it is important to know the caustic surface C(ac ), defined by the relation
∂ f (ac ;us )
∂u1s
∂2 ∂ f (a ;u )
∂∂ f (a ;u )
∂2 ∂ f (a ;u )
= ∂u c2 s = . . . = ∂u c s = ∂u c2 s = 0.
ks
1s
ks
By transforming the variables using a uniform one-to-one mapping, the phases can be transformed
into simple forms, which are classified as seven “elementary catastrophes” [2,3]. There are four
one-dimensional catastrophes: fold f (x; u) = u3 + xu, cusp f (x, y; u) = u4 + xu2 + yu, swallow-tail
f (x, y, z; u) = u5 + xu3 + yu2 + zu, butterfly f (x, y, z, w; u) = u6 + xu4 + yu3 + zu2 + wu, and three
two dimensional ones: hyperbolic
umbilic
f (x, y, z; u, v) = u3 + v3 + xuv − yu − zv, elliptic umbilic
3
2
2
2
f (x, y, z; u, v) = u − 3uv + x u + v − yu − zv, parabolic umbilic f (x, y, z, w; u, v) = u2 v + v4 + xu2 +
wv2 − yu − zv. The elementary Thom’s catastrophes in the context of the theory of atomic collisions
have been discussed in detail by Connor [4].
Atoms 2019, 7, 47; doi:10.3390/atoms7020047
www.mdpi.com/journal/atoms
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In the cuspoid case (one integration variable), the oscillatory integrals are usually written in
the form:
Z∞
I (a) =
g(u)ei f (a;u) du
(1)
−∞
where a = {a1 , a2 , . . . } is a set of parameters. As a varies, as many as K + 1 (real or complex) critical
points of the smooth, real-valued phase function f can coalesce in clusters of two or more. The function
∂n
(n) (a; u). The critical (stationary)
g has a smooth amplitude. In what follows we denote ∂u
n f (a; u) = f
points uj (a), 1 ≤ j ≤ K + 1, are defined by f (1) (a; u j ) = 0 [5].
In the case of a single real critical point the integral I (a) is in the leading order approximated
by [6]:
s
s
π
2πi
2π
i f (a;u1 )
g(u1 )ei f (a;u1 )+iσ1 4
I (a) ≈ Iq (a; u1 ) =
g ( u1 ) e
=
(2)
(
2
)
(
2
)
f (a; u1 )
f (a; u1 )
where σ1 = sgn( f (2) (a; u1 )) and the subscript q indicates a quadratic expansion of f (a; u) around u1 .
The result is easily generalized to the case of jmax (1 ≤ jmax ≤ K) isolated real critical points [7]. The
main contribution to the integral comes from the regions around the stationary points u j where the
phase function f (a; u) is slowly varying.
Since the positions of the critical points depend on a, they can move close together and coalesce
as a varies. In the uniform asymptotic evaluation of oscillatory integrals the result is expressed in
terms of certain canonical integrals [5,7] and their derivatives. Each canonical integral is characterized
by a given number of coalescing critical points. One defines a mapping u(a;t) by relating f (a;u) to the
normal form of cuspoid catastrophes ΦK (b; t) in the following way:
K
X
f (a; u(a; t)) = A(a) + ΦK (b(a); t) = A(a) + tK+2 +
bm t m
(3)
m=1
with the K + 1 functions A(a) and b(a) determined by the correspondence of K + 1 critical points of
f and ΦK .
In the simplest case of two coalescing critical points (K = 1, fold catastrophe) there is a single
point ue = u(ae ) where f (2) (a; ue ) = 0, i.e., the function f (1) (a, u) has an extremum and there are two
stationary points u1 (a) and u2 (a). In some range of the parameter a the stationary points are real and
u1 ≤ ue ≤ u2 . For a = ae the two points coalesce and u1 = ue = u2 . For other values of a the stationary
points are complex conjugate solutions of Equation (2), i.e., u1 = u∗2 .
The leading-order uniform approximation in the case of the fold catastrophe is given by [8]
√
I (a) ≈ IF (a) = π 2eiA(a)
"
g ( u1 )
√
σ1 f (2) (a;u1 )
+ √
g(u2 )
−σ1 f (2) (a;u2 ))
!
p
4
ζ(a)Ai(−ζ(a)) − iσ1 √
g(u1 )
σ1 f (2) (a;u1 )
− √
g ( u2 )
−σ1 f (2) (a;u2 ))
!
Ai0 (−ζ(a))
√
4
ζ(a)
#
(4)
where A(a) = 12 [ f (a; u2 ) + f (a; u1 )], ζ(a)3/2 = σ1 34 [ f (a; u2 ) − f (a; u1 )] and σi = sgn f (2) (a; ui ) . The
branches are chosen so that ζ(a) is real and positive if the critical points are real, or real and negative if
they are complex. (Note that σi f (2) (a; ui ) = f (2) (a; ui ) and in the case under consideration σ2 = −σ1 ).
The transitional approximation IFtr (a) reproduces the uniform approximation IF (a) on the
neighbourhood of a ≈ ae (IFtr (ae ) = IF (ae )) and enables analytical continuation from the region
of real stationary points into the region of complex ones. The transitional approximation is given by [9],
IFtr (a) = 2πeiA(a) g(ue )
where ζ(a) = σ1
2
f (3) (a;ue )
1/3
2
f (3) (a;u
e)
1/3 (
Ai(−ζ(a)) + iσ1
2
f (3) (a;ue )
1/3 f (1) (a; ue ) and A(a) = f (a; ue ) +
g ( 1 ) ( ue )
g ( ue )
(1)
1 f (a;ue )
6 f (3) (a;ue )
−
2
(4)
1 f (a;ue )
6 f (3) (a;ue )
)
Ai0 (−ζ(a))
f (4) (a; ue ).
(5)
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In order to obtain A(a) and b(a), a set of nonlinear equations has to be solved. These can be
solved in principle, but there are practical difficulties in attempting a solution [10]. On the other
hand, away from b = 0 the canonical integrals can be approximated in terms of canonical integrals Φ J
corresponding to lower-order catastrophes (i.e., J < K) [7,10–16].
The motivation to study these types of integrals originates from the investigation of optical spectra
of diatomic molecules [9]. For example, in the semiclassical approximation the matrix element of the
dipole moment D(R) for the optical transition is proportional to the integral [11]
Z∞
−∞


 Zt

i

(∆(R(t0 )) − hν)dt0 .
dtD(R(t)) exp
h

(6)
0
The radial movement of atoms is described classically, R = R(t). The phase function in the
Rt
integral (6) is f (t) = 1h (∆(R(t0 )) − hν)dt0 , where ∆(R) is the energy difference of the upper and lower
0
electronic state energies. The condition f (1) (t) = 0 gives the saddle points which satisfy the classical
Franck–Condon condition ∆(R(tc )) = hν. If there are points tt satisfying the condition f (3) (tt ) = 0, the
method suggested in Section 2 of this paper is a good choice to calculate the integral in Equation (6).
In the following sections we propose a new procedure for the approximate evaluation of oscillatory
integrals with several stationary points.
2. A New Procedure for Approximate Evaluation of Oscillatory Integrals
Let there be a point ut in the integration interval [−∞, ∞] which satisfies the condition f (3) (a; ut ) = 0.
In the neighbourhood of this point one defines a function:
F(a; u) = f (a; ut ) + f (1) (a; ut )(u − ut ) +
1 (2)
1
f (a; ut )(u − ut )2 + f (4) (a; ut )(u − ut )4
2
4!
(7)
The first derivative of this function, F(1) (a; u) = f (1) (a; ut ) + f (2) (a;ut )(u − ut ) +
has an inflection at the point ut . If sgn f (2) (a; ut ) = sgn f (4) (a; ut ) , F(1) (a; u) is
monotonic function. In the case when sgn f (2) (a; ut ) = −sgn f (4) (a; ut ) , the function F(1) (a; u) has
r
1 (4)
(a; ut )(u − ut )3 ,
3! f
two extremes at real points u1,2 = ut ∓
−
2 f (2) (a;ut )
.
f (4) (a;ut )
If there are m points up,i ∈ [−∞, ∞], i = 1, . . . , m, satisfying f (3) (a; up,i ) = 0 and σp,i =
sgn f (2) (a; up,i ) = −sgn f (4) (a; up,i ) , these points divide the interval [−∞, ∞] into m + 1 intervals
h
i
up,i−1 , up,i and the integral I (a) can be written:
Z∞
g(u)ei f (a;u) du =
I (a) =
−∞
Zup,i
m
+1
X
g(u)ei f (a;u) du
(8)
i=1 u
p,i−1
where the end points of the integration up,0 = −∞ and up,m+1 = ∞ (σp,0 = sgn limu→−∞ f (2) (a; u)
h
i
and σp,m+1 = sgn limu→∞ f (2) (a; u) have been introduced. At each interval up,i−1 , up,i the function
f (1) (a; u) has a simple property. If σp,i−1 = σp,i , the function f (1) (a; u) is monotonic on the interval
h
i
h
i
up,i−1 , up,i and has a single real saddle point. In the case σp,i−1 = −σp,i there is a point ue ∈ up,i−1 , up,i ,
f (2) (a; ue ) = 0 and the function f (1) (a; u) has an extreme at ue and two saddle points.
One defines a function fp,i (a; u) as a series expansion of the phase f (a; u) around up,i up
to the quadratic term: fp,i (a; u) = f (a; up,i ) + f (1) (a; up,i )(u − up,i ) +
1 (2)
(a; up,i )(u − up,i )2 .
2! f
Note
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(1)
(2)
(3)
that fp,i (a; up,i ) = f (a; up,i ), fp,i (a; up,i ) = f (1) (a; up,i ), fp,i (a; up,i ) = f (2) (a; up,i ), and fp,i (a; up,i ) =
f (3) (a; up,i ) = 0.
We define the integral Ip,i (a) =
R∞
g(up,i )ei fp,i (a;u) du, which has an exact solution
−∞
s
Ip,i (a) =
i( f (a;up,i )− 12
2πi
f (2) (a; up,i )
g(up,i )e
f (1) (a;up,i )2
f (2) (a;up,i )
)
(9)
Relation (8) can be written as:
I (a)
=
=
mP
+1
up,i
R
i=1 up,i−1
u
mP
+1 Rp,i
g(u)ei f (a;u) du +
m
P
i=1
g(u)ei f (a;u) du +
i=1 up,i−1
Ip,i (a) −
u
m Rp,i
P
i=1 −∞
m
P
i=1
Ip,i (a)
g(up,i )ei fp,i (a;u) du +
m R∞
P
i=1 up,i
g(up,i )ei fp,i (a;u) du −
m
P
i=1
Ip,i (a)
By combining integrals in the first three sums a simple expression is obtained:
I (a) =
m
+1
X
Ii ( a ) −
i=1
where integrals Ii (a) are
m
X
Ip,i (a)
(10)
i=1
Z∞
gi (u)ei fi (a;u) du
Ii ( a ) =
(11)
−∞
The functions gi (u) and fi (a; u) are shown in Table 1.
Table 1. Functions gi (u) and fi (a; u) for i = 1, i = m+1 and 1 < i < m + 1.
i=1
f1 (a; u) =
g1 (u) =
−∞ ≤ u < up,1
f (a; u)
g(u)
1 < i < m+1
fi (a; u) =
gi (u) =
u ≥ up,1
fp,1 (a; u)
g(up,1 )
u < up,i−1
fp,i−1 (a; u)
g(up,i−1 )
i = m+1
fm+1 (a; u) =
gm+1 (u) =
up,i−1 ≤ u < up,i
f (a; u)
g(u)
u < up,m
fp,m (a; u)
g(up,m )
up,m ≤ u ≤ ∞
f (a; u)
g(u)
u ≥ up,i
fp,i (a; u)
g(up,i )
The relation (10) can be generalized to be valid for the case m = 0 as well:

m + 1
m
X
 X


I (a) = δm,0 I (a) + (1 − δm,0 )
Ii (a) −
Ip,i (a)
i=1
(12)
i=1
So far, no approximation has been made. The relation (10) is an identity. An integral of the
function, the phase of which has several real stationary points, is divided into the sum of the integrals
Ii (a) whose phase functions fi (a; u) have either one or at most two stationary points.
(1)
If the phase function fi (a; u) has only one real saddle point and its first derivative fi (a; u) is
monotonic, the value of integral Ii (a) can be calculated using Equation (2). In the case when the
(1)
function fi (a; u) has a single extreme at the point ue and two real or complex saddle points, the
integral is easily soluble using the approximate methods described in the introduction (Equation (4)).
If the phase fi (a; u) is given by numerical points in the region where a complex pair of saddle points
contributes to the integral, the analytical continuation of (5) can be used. The numerical accuracy of
this method is determined by the accuracy of the leading-order uniform approximations (2) and (4).
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3. Results
The method outlined in Section 2 was tested on three examples that are typical for the spectra
of diatomic molecules. For simplicity we use the phase function given by the polynomial phase of
the Thom’s elementary catastrophe. The case when f (1) (t) and the difference potential ∆(R) are both
monotonic functions with a single inflection point is illustrated by the analysis of the Pearcey integral
P(x ≥ 0, y) in Section 3.1.1. In Section 3.1.2 with the Pearcey integral P(x < 0, y) we analyze the case
when the function f (1) (t) and the difference potential ∆(R) have two extremes and one inflection point.
Finally, in Section 3.2 we illustrate the case when the difference potential has an extreme near the
turning point by the analyses of the swallow-tail catastrophe integral S(x < 0, 0, z). For simplicity, we
take g(u) = 1 in all the examples. The dependence of the integral (6) on the variable transition dipole
moment was discussed by Beuc et al. in [9].
3.1. Cusp Catastrophe (K = 2)
Let us consider the Pearcey integral, the canonical integral for the cusp catastrophe (x, y ∈ R):
Z∞
ei(u
P(x, y) =
4 +xu2 + yu)
du
(13)
−∞
(Other notations also appear in the literature). The Pearcey integral is symmetrical with respect to
variable y: P(x, y) = P(x, −y). For the numerical integration of the Pearcey integral we used the form:
∞
i 3π 2
5π
π R
4
P(x, y) = 2ei 8 e−t +e 4 xt cos(ei 8 yt)dt [13]. The numerical evaluation of the integral and all
0
other calculations in this paper were done using the Wolfram Mathematica 11.3 computing system.
The phase function in (6) is f (x, y; u) = u4 + xu2 + yu. There are three saddle points defined by
the condition f (1) (x, y; u) = 4u3 + 2xu + y = 0 (Figure 1 and Figure 3a):
u1 (x, y) =
u2 (x, y) =
u3 (x, y) =
√
√
1+i 3
1−i 3
x
2√ U (x, y)
12√ U (x,y) −
1+i 3
1−i 3
x
2 U (x, y)
12 U (x,y) −
1
x
− 6 U(x,y) + U (x, y)
(14)
qp
3
8 3
where U (x, y) = 12
δ(x, Y) − y and δ = 27
x + y2 . If δ < 0, all saddle points are real and if δ > 0, one
saddle point is real and the other two are complex conjugates of each other (Figure 1 and Figure 3a).
q
There are two bifurcation points ue1,e2 = ∓ − 61 x defined by the relation f (2) (x, y; u) = 12u2 + 2x = 0.
(2)
f (2) (x,y;up )
f (x,y;up )
x
There is a single point up = 0 where f (3) (x, y; up ) = 0 and (4)
, i.e., sgn (4)
= sgn(x).
= 12
f
(x,y;up )
f
In the special case x = 0 one has ue1 = ue2 = up = 0.
Atoms 2019, 7, 47
(x,y;up )
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(1)
(f1) (1, y; u )
u1 1, ,uu2 2, ,and
Figure1.1.Saddle
Saddlepoints
points( (u
and uu33 )) of
Figure
of the
the function
function f (1, y; u). .
3.1.1. Case x > 0
If x ≥ 0 , then δ is always positive, the saddle point u3 ( x, y ) is real and the points u1 ( x, y ) and
u2 ( x, y ) are complex conjugates (Figure 1).
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u u
u
f (1) (1, y; u ) .
Figure 1. Saddle points ( 1 , 2 , and 3 ) of the function
3.1.1.3.1.1.
CaseCase
x > 0x > 0
If x ≥If 0,
δ isδ always
saddlepoint
pointu u(3x(, x,
) is
real
points
y) and
is alwayspositive,
positive, the
the saddle
y ) yis
real
andand
the the
points
u1 ( x,uy1)(x,
and
x ≥then
0 , then
3
u2 (x, yu )( are
complex
conjugates
(Figure
1).
2 x, y ) are complex conjugates (Figure 1).
The function
f (1) (f1,(1) y;
accordingtotoEquation
Equation
value
of Pearcey
the Pearcey
monotonic and,
and, according
(2),(2),
thethe
value
of the
The function
(1, u
y;)uis
) ismonotonic
integral
can
be
approximated
as
integral can be approximated as
s
P( x ≥ 0, y) ≈ Pq ( x ≥ 0, y) =
P(x ≥ 0, y) ≈ Pq (x ≥ 0, y) =
πi
i ( u34 + xu32 + yu3 )
πie i(u4 +xu2 + yu3 )
e 3 3
2
6u3 + x
(15)
6u32 + x
(15)
From Figure 2 and Table 2, we can freely estimate that the difference of the approximation
From Figure 2 and Table 2, we can freely estimate that the difference of the approximation
Pq (x, y)
p
x 2 + y 2 > 5 is
P ( x, y ) and the exact values of P( x, y ) is smaller than few percent if the condition
and theq exact values of P(x, y) is smaller than few percent if the condition x2 + y2 > 5 is satisfied.
satisfied.
(b)
(a)
(c)
Figure 2. Function P(x, y) (a), function Pq (x, y) (b) and the relative difference
Pq (x,y)−P(x,y)
P(x,y)
(c) for
x ∈ [0, 10] and y ∈ [0, 10].
Table 2. Comparison values of P(x, y) and approximate function Pq (x, y) for y = 0 and x = 0.
x
P(x,0)
Pq (x,0)
y
P(0,y)
Pq (0,y)
1
2
3
4
5
6
7
8
9
10
1.20838 + 0.779288i
0.924029 + 0.729007i
0.754294 + 0.657362i
0.646979 + 0.593695i
0.573931 + 0.541858i
0.520848 + 0.500054i
0.480234 + 0.465936i
0.447916 + 0.437618i
0.421413 + 0.413718i
0.399166 + 0.393242i
1.25331 + 1.25331i
0.886227 + 0.886227i
0.723601 + 0.723601i
0.626657 + 0.626657i
0.560499 + 0.560499i
0.511663 + 0.511663i
0.473708 + 0.473708i
0.443113 + 0.443113i
0.417771 + 0.417771i
0.396333 + 0.396333i
1
2
3
4
5
6
7
8
9
10
1.55093 + 0.427893i
1.12475 − 0.17608i
0.384485 − 0.642953i
−0.385925 − 0.545144i
−0.670195 + 0.0710806i
−0.235367 + 0.592027i
0.430078 + 0.415615i
0.510179 − 0.260969i
−0.1039 − 0.542069i
−0.532222 − 0.0242518i
1.09286 + 0.353605i
0.837873 − 0.359347i
0.244423 − 0.757992i
−0.434337 − 0.578749i
−0.66748 + 0.0754601i
−0.214718 + 0.594539i
0.442629 + 0.405754i
0.506489 − 0.270769i
−0.112749 − 0.540578i
−0.532896 − 0.0165838i
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3.1.2.
Case7,x47< 0
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According to Section 2, when x < 0 using the relation (10) the Pearcy integral can be written as:
P( x, y) (a), function Pq ( x , y ) (b) and the relative difference
Figure 2. Function
Pq ( x, y ) − P ( x, y )P(x, y) = I1 (x, y) + I2 (x, y) − Ip,1 (x, y)
R∞ ] .
R∞
(c) for x ∈R[∞0,10] and y ∈ [ 0,10
(16)
P ( x, y )
=
ei f1 (x,y;u) du +
ei f2 (x,y;u) du −
ei fp,1 (x,y;u) du
−∞
−∞
−∞
 and x = 0 .
Table 2. Comparison values of P( x, y ) and approximate function Pq ( x, y ) for y = 0


 f (x, y; u) u < 0
Here the phase functions have the form fp,1 (x, y; u) = xu2 + yu, f1 (x, y; u) = 
,

 fp (x,
x
P(x,0)
Pq(x,0)
y
P(0,y)
Pqy;
(0,y)
u) u ≥ 0

1
1.20838
+1.25331i
1
1.55093 +0.427893i
1.09286 +0.353605i
+0.779288i
fp (x, y; u) u <1.25331
0

f22 (x, 0.924029
y; u) = 
.
It
is
easy
to
show
that
f
(
x,
y,
u
)
=
f
(
x,
−y,
−u
)
,
I2 (x, y) –0.359347i
= I1 (x, −y),
+0.729007i
0.886227
+0.886227i
2
1.12475
–0.17608i
0.837873
2
1


f (x, y; u) u ≥ 0
3 0.754294 +0.657362i 0.723601 +0.723601i 3
0.384485 –0.642953i
0.244423 –0.757992i
and the Pearcey integral can be decomposed exactly as
4 0.646979 +0.593695i 0.626657 +0.626657i 4
–0.385925–0.545144i
–0.434337 –0.578749i
r +0.0710806i
5 0.573931 +0.541858i 0.560499 +0.560499i 5 –0.670195
–0.66748
+0.0754601i
2
π −i y +i π
4x
4
P
(
x,
y
)
=
I
(
x,
y
)
+
I
(
x,
−y
)
−
e
(17)
1
1
6 0.520848 +0.500054i 0.511663 +0.511663i 6
–0.235367
–0.214718 +0.594539i
|x| +0.592027i
7 0.480234 +0.465936i 0.473708 +0.473708i 7
0.430078 +0.415615i
0.442629 +0.405754i
q
8 0.447916 +0.437618i 0.443113 +0.443113i 8
0.510179 –0.260969i
0.506489 –0.270769i
The function f1 (x, y; u) has on the interval u ∈ [−∞, ∞] only one bifurcation point ue1 = − |x|
6,
9 0.421413 +0.413718i 0.417771 +0.417771i 9
–0.1039 –0.542069i
–0.112749
–0.540578i



y) y ≤ 0
u2 (x,
(2)
10 0.399166
+0.393242i 0.396333 +0.396333i 10 –0.532222 –0.0242518i
–0.532896
–0.0165838i
where f1 (x, y; ue1 ) = 0, and two saddle points e
u1 (x, y) = u1 (x, y), e
u2 (x, y) = 
y


y>0
2|x|
x
<
0
(Figure
3b).
3.1.2. Case
(a)
(b)
(1)
(1)
(1) 1, y ; u ) are shown in
Figure 3.
3. Saddle
Saddle points
points (u
( u1,,uu2,,and
andu u)3 of
) ofthe
thefunctions
functionsf (f1) (−1,
(1, yy;; uu)) and
Figure
and ff1 ((−−1,
y; u) are shown in
3
1 2
(a)
and
(b),
respectively.
(a,b), respectively.
Using
relation
(4), the2,integral
can be
as: Pearcy integral can be written as:
According
to Section
when Ix1 (<x,0 y)using
theapproximated
relation (10) the
 P ( x, y ) = I ( x, y ) + I ( x, y ) − I ( x, y )

1
p ,1


2p
4
1

 ζ(∞x, y)Ai(−ζ(x, y))∞ − i q
+ q∞



πeiA(x,y)  q 1
6u21 +x
1


 Ai0 (−ζ(x,y)) 


− q
 √

4
ζ(x,y) 
−6u2 Θ(−y)−x
1
(18)
2 x
−6u2 Θ(−y)−x
(16)
if ( x6u
, y ; u+
)
2
=  e2if1 ( x , y ; u ) du +  eif2 ( x , y ; u ) du −  e p ,1 1 du
−∞
−∞
−∞
 2 + u2 + 3y (u + u ) y ≤ 0


x
u
2
1

2
2
1
1
where,
A(x, y)
=
,
ζ(x,f y( x),3/2
y; u ) u < 0=

4
3y
y2
2
2

+ 2f pu,11( x−, y2x; u ) = xu + yuy,> f01 ( x, y; u ) = 
,
Here the phase functions have thexuform
1
3y
  f p ( x, y; u ) u ≥ 0
2
2


x u1 − u2 + 2 (u1 − u2 ) y ≤ 0
σ1 38 
.
 f p ( x, yy;2u ) u < 0
,xu
. Ity is
easy
to show that f 2 ( x, y, u ) = f1 ( x, − y, −u ) , I 2 ( x, y ) = I1 ( x, − y ) , and
f 2 ( x
y; 2u )+= 3y

u
+
>
0
1
2
2x
1
 f ( x, y; u ) u ≥ 0
We define the Airy approximation of the Pearcey integral as:
the Pearcey integral can be decomposed exactly as
r
π− i y2−i+i πy2 +i π
4
PA (x,Py()x,=y )IF=(Ix,( yx,) y+
I
(
x,
−y
)
−
(19)
FI ( x, − y ) − π e e4 x 44x
(17)
)
+
1
1
x |x|
I1 (x, y) ≈ IF (x, y) =
Atoms 2019, 7, 47
8 of 13
Paris obtained the asymptotic form of P(x,y) by considering its analytic continuation to arbitrary
complex variables x and y [15]. In Table 3. we compare some values of P(x,y) for large negative values
of x when y = 2 and 4 to the asymptotic values [15] and the present work.
Table 3. Values of P(x,y) obtained for large negative values of x when y = 2 and 4 compared to the
asymptotic values [15] and the present work.
x
y
P(x,y)
Asymptotic [11]
PA (x,y)
−4
−6
−8
−4
−6
−8
2
2
2
4
4
4
1.96341 − 0.73419i
0.96527 + 0.46413i
1.00422 − 0.11480i
0.14360 + 0.90244i
0.29478 − 0.84373i
0.75372 − 0.23933i
1.97363 − 0.72605i
0.96537 + 0.46415i
1.00422 − 0.11480i
−
0.29399 − 0.84356i
0.75371 − 0.23933i
1.96482 − 0.72731i
0.96366 + 0.46538i
1.004077 − 0.11392i
0.14063 + 0.90013i
0.29629 − 0.84406i
0.75379 − 0.23889i
Kaminski [16] rewrites (8) as a sum of two contour integrals, one of which has exactly two relevant
coalescing saddle points. This allows him to apply a cubic transformation introduced by Chester,
Friedman, and Ursell [17] and to construct a uniform asymptotic expansion of (7) as x → −∞ with δ
varying in an interval containing 0. The leading-order approximation was already given by Connor [12]
and Connor and Farrelly [13]. In Table 4 the values of P(x, y) are compared to Kaminski’s results [16]
3
and the approximation PA (x, y) at some points on the caustic y = − 32 x 2 .
Table 4. Comparison of values of P(x, y) on the caustic with Kaminski [16] and PA (x, y).
x
y
P(x,y)
Kaminski [10]
PA (x,y)
−1.0
−2.0
−3.0
−4.0
−5.0
−6.0
−7.0
−8.0
−9.0
−10.0
0.544331
1.5396
2.82843
4.35465
6.08581
8.
10.0812
12.3168
14.6969
17.2133
2.14158 + 0.0990191i
0.962205 − 0.450083i
1.13215 + 1.19182i
−0.142478 + 1.20972i
−0.888104 + 0.979074i
−1.10157 + 0.582286i
−0.249906 − 0.91133i
0.321769 − 0.468203i
0.495502 + 0.309572i
−0.704129 + 0.779039i
2.34415 + 0.00118008i
0.926925 − 0.428207i
1.14743 + 1.19594i
−0.143582 + 1.2217i
−0.890885 + 0.983784i
−1.09951 + 0.581515i
−0.249866 − 0.914663i
0.324275 − 0.466919i
0.495034 + 0.311661i
−0.704954 + 0.779772i
2.1003 + 0.156994i
0.965935 − 0.448303i
1.12358 + 1.19408i
−0.146125 + 1.20649i
−0.889185 + 0.975844i
−1.1015 + 0.58047i
−0.248282 − 0.910954i
0.321939 − 0.467325i
0.494746 + 0.309898i
−0.70467 + 0.778148i
Atoms 2019, 7, 47
9 of 13
From Tables 3 and 4, and Figure 4, we estimate
estimate that
that the
the difference
difference of
of the
thepapproximation
approximation PPAA ((xx,, yy))
2
and the
the exact
exact values
values of
of PP((x,x,yy)) is
few percent
percent if
if the
the condition
condition xx22 +
is smaller
smaller than
than aa few
is satisfied.
satisfied.
and
+ yy2 >>44 is
(a)
Figure 4. Cont.
(b)
Atoms 2019, 7, 47
9 of 13
(a)
(b)
(c)
) (a),
, y )and(b)theand
Figure 4.4. Function
FunctionP(P
function
the absolute
of the functions’
Figure
function
absolute
value of value
the functions’
difference
x,(yx), y(a),
PA (x,PyA)( x(b)
[0, 10
]
∆P
= PA (x,ΔyP) −
for
0] and
.
= PP(x,( xy,)y )(c)
−P
( xx, y∈) [−10,
x ∈y[ −∈10,0
y
∈
0,10
difference
(c) for
and
.
]
[ ]
A
3.2. Swallow-Tail Catastrophe (K = 3)
3.2. Swallow-Tail Catastrophe (K = 3)
The swallow-tail canonical integral is defined by:
The swallow-tail canonical integral is defined by:
Z∞ ∞
5
2 2
+xu
+ zu+
xu 33++
yuyu
) zu) du
i(ui(5u+
S(x, Sy,( xz,)y=
, z ) = e e
du
−∞
(20)
(19)
−∞
As a further example, we consider a special case of the swallow-tail integral, i.e., S ( x,0, z ) —the
As a further example, we consider a special case of the swallow-tail integral, i.e., S(x, 0, z)—the
oddoid integral
is also
also real,
real, and
and for
for
oddoid
integral of
of the
the order
order two
two [18].
[18]. For
Forthe
thereal
realxxand
andyythe
thefunction
function SS((xx,,0,
0,zz)) is
∞
∞
R
3 3
numericalevaluation
evaluationthe
theequation
equation
) =22  cos
++
zuzu
the numerical
S(Sx,( x0,,0,
z)z =
used.This
Thisintegral
integral is
is of
cos( uu55++xuxu
) duduis isused.
00
interest
interest in
in the
the study
study of
of bound-continuum
bound-continuum [19]
[19] and
and bound-bound
bound-bound [20]
[20] Franck–Condon
Franck–Condon factors.
factors. The
The
analysis
is
applied
to
the
domain
x
<
0.
analysis is applied to the domain x < 0 .
In that case the phase function is f (x, 0, z; u) = u5 + xu3 + zu and it is antisymmetric with respect
to the variable u: f (x, 0, z; −u) = − f (x, 0, z; u). There are four saddle points ui defined by the condition
f (1) (x, 0, z; u) = 5u4 + 3xu2 + z = 0:
q
3
10
rq
u1 (x, z) = −u4 (x, z) = −
x2 − 20
9 z−x
r
q
q
3
u2 (x, z) = −u3 (x, z) = − 10
−x − x2 − 20
9 z
(21)
The condition f (2) (x, 0, z; ue ) = 20u3e + 6xue = 0 defines three real bifurcation points uei of the
q
q
q
|x|
|x|
“fold” type: ue1 = −ue3 = − 3|x|
,
u
=
0.
As
there
are
two
real
points
(u
=
−
,
u
=
e2
p2
p1
10
10
10 )
satisfying the conditions f (3) (x, z; upi ) = 0 and sgn f (2) (x, z; upi ) = −sgn f (4) (x, z; upi ) , according to
Section 2, the integral S(x, 0, z) can be written as:
S(x, 0, z)
=
=
3
P
Ii (x, z) −
i=1
3 R∞
P
i=1 −∞
where,
2
P
i=1
Ipi (x, z)
ei fi (x,0,z;u) du −
2 R∞
P
ei fpi (x,0,z;u) du
i=1 −∞
√
√
5
3
10
3
10
2
2
(−x) 2 + u z + x + u
(−x) 2 ,
fp1 (x, 0, z; u) = − fp2 (x, 0, z; −u) =
250
20
5
(22)
Atoms 2019, 7, 47
10 of 13



fp,1 (x, 0, z; u)



u ≤ up,1

, f2 (x, 0, z; u) = 
f (x, 0, z; u)


u > up,1


 fp,2 (x, 0, z; u)



 fp,2 (x, 0, z; u) u < up,2
f3 (x, 0, z; u) = 
.

 f (x, 0, z; u)
u ≥ up,2
u < up,1



 f (x, 0, z; u)
f1 (x, 0, z; u) = 

 fp1 (x, 0, z; u)
up,1 ≤ u ≤ up,2 , and
u > up,2
Since fp1 (x, 0, z; u) = − fp2 (x, 0, z; −u), it follows that Ip2 (x, z) = Ip1 (x, z)∗ and
∗
2
P
i=1
Ipi (x, z) =
2ReIp1 (x, z). Also, f1 (x, 0, z; u) = − f3 (x, 0, z; −u), I3 (x, z) = I1 (x, z) , and I1 (x, z) + I3 (x, z) = 2ReI1 (x, z).
Using Equation (9) to calculate Ip1 (x, z), one can write Equation (22) in the form:
s
S(x, 0, z) = 2ReI1 (x, z) + I2 (x, z) −
4
40π2
|x|3
19x4 − 600x2 z − 2000z2
π
cos
+
√
3/2
4
1600 10|x|
!
(23)
This expression exactly represents the function S(x, 0, z). To find an approximate solution of
the integral S(x, 0, z) one needs to calculate integrals I1 (x, z) and I2 (x, z), using the approximation
described by Equation (4).
u1 (x, y) = u1 (x, y), e
u2 (x, z) =
The function f1 (x, 0, z; u) has two saddle points (see Figure 5b): e

x2


z≥ 4

u2 (x, y) 2 . Applying Equation (4) one gets,

5
3 2

z < x4

3 z + 20 x
− √
2
40|x|
I1 (x, z) ≈ IF1 (x, z) =




πeiA1 (x,z)  q
1
10u31 +3xu1
+
1
q
−10e
u32 −3xe
u2


 p

 4 ζ (x, z)Ai(−ζ (x, z)) − i q 1
−
 1

1
10u3 +3xu
1
1
1
q
−10e
u32 −3xe
u2


 Ai0 (−ζ (x,z)) 
 √ 1

 4 ζ (x,z) 
1
whereAtoms
A1 (2019,
x, z)7,=47 21 [ f1 (x, z; e
u2 ) − f1 (x, z; u1 )].
u2 ) + f1 (x, z; u1 )] and ζ1 (x, z)3/2 = 34 [ f1 (x, z; e
10 of 13
(a)
(b)
(c)
(1)
(1)
(11)
u u u3 , and u4 ) of the functions (f1) ( −1, 0, z ; u ) (a), f ( −1, 0, z ; u ) (b),
Figure
5. Saddle
points
Figure
5. Saddle
points
(u1 , (u21 ,, u23 ,, and
u4 ) of the functions f (−1, 0, z; u) (a), f1 (−1, 0, z; u) (b), and
(1)
(1)
f ( −1, 0, z ; u ) (c).
and0, z;2 u) (c).
f2 (−1,
In that case the phase function is f ( x,0, z; u ) = u 5 + xu 3 + zu and it is antisymmetric with respect
to the variable u: f ( x,0, z; − u ) = − f ( x,0, z; u ) . There are four saddle points ui defined by the
condition f (1) ( x,0, z; u ) = 5u 4 + 3 xu 2 + z = 0 :
u1 ( x, z ) = −u4 ( x, z ) = −
3
10
x 2 − 209 z − x
(20)
(24)
Atoms 2019, 7, 47
11 of 13




u2 (√x, z)
The function f2 (x, 0, z; u) has two symmetrical saddle points: e
u3 (x, z) = 
10


3 z+
−
40|x| 2
(2)
2π
−10e
u33 − 3xe
u3
3 2
20 x
(2)
e
u4 (x, z) = −e
u3 (x, z) (Figure 5c). Since f2 (x, z; e
u3 ) = − f2 (x, z; e
u4 ) and f2 (x, z; e
u3 ) = − f2
approximation of the integral I2 (x, z) has a simple form,
I2 (x, z) ∝ IF2 (x, z) = q
z≤
z>
u4 ) ,
2 (x, z; e
q
4
ζ2 (x, z)Ai(−ζ2 (x, z))
x2
4
x2
4
,
the
(25)
where ζ2 (x, z)3/2 = − 23 f2 (x, z; e
u3 ) .
Finally, we write the approximation of the integral S(x, 0, z) as:
s
SA (x, 0, z) = 2ReIF1 (x, z) + IF2 (x, z) −
4
40π2
|x|3
19x4 − 600x2 z − 2000z2
π
cos
+
√
3/2
4
1600 10|x|
!
(26)
In Table 5 we compare the values of the functions S(x, 0, z) and SA (x, 0, z) at the caustics i.e., at
the points where the function f (1) (x, 0, z; u) has an extreme. These comparisons together with the
comparison of functions in Figure 6 clearly
p show that the function SA (x, 0, z) is a good approximation
of the function S(x, 0, z) if the condition x2 + y2 > 3 is satisfied.
Table 5. Comparison of the values of S(x, 0, z) and SA (x, 0, z) on the caustics z = 0 and z =
x
−1
−2
−3
−4
−5
−6
Atoms 2019, 7, 47
−7
−8
−9
−10
z
S(x,0,0)
SA (x,0,0)
0.
0.
0.
0.
0.
0.
0.
0.
–10
0.
0.
2.269084
2.409949
0.596406
1.292366
0.215598
0.247304
0.670358
0.834808
0.767679
1.155092
0.767679
1.913266
2.394455
0.604199
1.281152
0.212031
0.245451
0.667798
0.83448
0.767563
1.154554
0.767563
0.
z = (9x2 )/20
45.
0.45
1.8
4.05
7.2
11.25
16.2
22.05
28.8
–0.891812
36.45
45.
S(x,0,z)
2.043860
1.663543
−0.519597
−0.915491
0.821846
0.660728
−0.394882
0.204247
–0.890310
1.055443
−0.891812
(a)
9 2
20 x .
SA (x,0,z)
1.661334
1.612423
−0.508896
−0.908131
0.816979
0.660188
12 of 13
−0.392698
0.205210
1.053662
−0.890310
(b)
Figure 6. Cont.
(c)
Atoms 2019, 7, 47
12 of 13
(a)
(b)
(c)
,0,0,zz)) (a),
S A ((x,
x,0,
(b) and
and the
the difference
difference of
of the
the functions
functions
Figure
Figure 6.
6. Function
FunctionSS( x(x,
(a), function
function S
0, zz)) (b)
A
]. ] .
x,0,
− SS((x,x,0,
∈ [ −10
10,10
SSAA((x,
0, zz)) −
0, zz)) (c)(c)for
10,
0] and
z ∈ z[−10,
forx ∈x [∈
[10,0
] and
4. Discussion and Conclusions
4. Discussion and Conclusions.
We have shown that the original oscillatory integral can be expressed exactly as a sum of integrals,
We have shown that the original oscillatory integral can be expressed exactly as a sum of
each having either one or at most two real stationary points. The construction of these integrals
integrals, each having either one or at most two real stationary points. The construction of these
introduces new phase functions that are smooth (infinitely differentiable), but only a few first derivatives
integrals introduces new phase functions that are smooth (infinitely differentiable), but only a few
are continuous in a characteristic point. This means that the proposed method is limited to the leading
first derivatives are continuous in a characteristic point. This means that the proposed method is
term only (i.e., the integral is the same as the leading term plus the residue that cannot be further
limited to the leading term only (i.e., the integral is the same as the leading term plus the residue that
treated as in the standard application of asymptotic analysis and iterated to obtain an asymptotic
cannot be further treated as in the standard application of asymptotic analysis and iterated to obtain
expansion [6]). However, the method has practical applications, especially in cases where the phase
an asymptotic expansion [6]). However, the method has practical applications, especially in cases
function (or its first derivative!) is tabulated.
where the phase function (or its first derivative!) is tabulated.
The validity of the proposed method was tested on examples of integrals with three saddle points
The validity of the proposed method was tested on examples of integrals with three saddle
(“cusp” catastrophe) and four saddle points (“swallow-tail” catastrophe). The examples chosen are
points (“cusp” catastrophe) and four saddle points (“swallow-tail” catastrophe). The examples
typical of the spectra of diatomic molecules, but the method described in this paper can be used
chosen are typical of the spectra of diatomic molecules, but the method described in this paper can
for numerical computation of canonical integrals occurring in other physical fields as well, e.g., the
be used for numerical computation of canonical integrals occurring in other physical fields as well,
propagation of electromagnetic, sound or fluid waves, and particularly within the semiclassical theory
e.g., the propagation of electromagnetic, sound or fluid waves, and particularly within the
of atom–atom and atom–surface scattering, chemical reactions etc.
semiclassical theory of atom–atom and atom–surface scattering, chemical reactions etc.
Author
R.B. made
made the
the calculation,
calculation, and
and prepared
prepared the
the figures.
figures. All
All the
the authors
authors R.B.,
R.B., M.M.
M.M. and
and B.H.
B.H.
Author Contributions:
Contributions: R.B.
discussed the problem, wrote the paper, and approved the final version of the manuscript.
discussed the problem, wrote the paper, and approved the final version of the manuscript.
Funding: This research received no additional external funding.
Funding: This research received no additional external funding.
Conflicts of Interest: The authors declare no conflict of interest.
Conflicts of Interest: The authors declare no conflict of interest.
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