Masujima M. Applied Mathematical Methods in Theoretical Physics

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7.4 Inhomogeneous Wiener–Hopf Integral Equation of the Second Kind 267

where

H (x, y) ≡ y

−

(x − y) +y

(x − y) +



+∞

−

(x − z)y

(z − y)dz. (7.4.42)

It is noted that the existence of the resolvent kernel H(x, y) given above is solely due

to the analyticity of 1/Y

−

(k) in the lower half plane and that of Y

(k) in the upper

half plane. Thus, when index ν = 0, we have a unique solution, solely consisting of a

particular solution alone to Eq. (7.4.11).

Case 2:Indexν>0.

When index ν is positive,wehaveν independent homogeneous solutions given by

Eq. (7.3.40). We observed in Section 7.3 that

(k) → k

as k →∞, (7.4.43)

where Y

(k) are given by Eqs. (7.3.34) and (7.3.35). On the other hand, in solving

for a particular solution, we want Y

(k)tobesuchthat(1)1/Y

−

(k)(Y

(k)) is analytic

in the lower half plane (the upper half plane),

(k) → 1as



→∞. (7.4.44)

We construct W

(k)as

(k) = Y

(k)/

j=1

(k − p

(j)), Im p

(j) ≤ A,1≤ j ≤ ν, (7.4.45)

where the locations of p

(j)’s are quite arbitrary as long as Im p

(j) ≤ A.Wenotice

that W

(k) satisfy requirements (1) and (2):

(1)

1/W

−

(k) =

j=1

(k − p

(j))/Y

−

(k)

is analytic in the lower half plane, while

(k) = Y

(k)/

j=1

(k − p

(j))

is analytic in the upper half plane;

(2)

(k) → 1as



→∞. (7.4.46)

Thus we use W

(k), Eq. (7.4.45), instead of Y

(k), in the construction of the

resolvent H(x, y).

268 7 Wiener–Hopf Method and Wiener–Hopf Integral Equation

Case 3:Indexν<0.

When index ν is negative, we have no nontrivial homogeneous solution. From

Eqs. (7.3.46) and (7.3.47), we have

(k) → 1/k



→∞. (7.4.47)

Then we have

1/Y

−

(k) → k



→∞, (7.4.48)

while

(k)

−

(k) → 0as



→∞. (7.4.49)

By Liouville’s theorem,

−

(k) can grow at most as fast as k

−1



→∞,

−

(k) = (Y

(k)

−

(k))

−

(k) ∼ k

−1



→∞. (7.4.50)

In general, we have

−

(k)  0as



→∞,

so that a particular solution to the inhomogeneous problem may not exist. There

are some exceptions to this. We analyze (Y

(k)

−

(k))

−

more carefully. We know

(k)

−

(k))

−

=−

2πi



−C

(ζ )

−

(ζ )

ζ − k

dζ. (7.4.51)

Expanding 1/(ζ − k) in power series of ζ/k,

1/(ζ − k ) =−(1/k)



1 +

+···+

−1

+···





< 1,

we write

(k)

−

(k))

−

2πi



−C

∞



j=0





(ζ )

−

(ζ )dζ

∞



j=0

2πi



−C

(ζ )

−

(ζ )dζ. (7.4.52)

In view of Eqs. (7.4.22) and (7.4.52), we realize that

−

(k) = (Y

(k)

−

(k))

−

(k) → 0as



→∞, (7.4.53)

7.4 Inhomogeneous Wiener–Hopf Integral Equation of the Second Kind 269

if and only if

2πi



−C

(ζ )

−

(ζ )dζ = 0, j = 0, ...,

− 1. (7.4.54)

If this condition is satisﬁed, we get from the j =

term onward,

−

(k) → C/k



→∞, (7.4.55)

so that

−

(k)canbeinvertedforφ(x), which is the unique solution to the inhomo-

geneous problem. To understand this solvability condition (7.4.54), we ﬁrst recall

the Parseval identity,



+∞

−∞

h(k)

g(−k) = 2π



+∞

−∞

h(y)g(y)dy, (7.4.56)

where

h(k)and

g(k) are the Fourier transforms of h(y)andg(y), respectively. Then

we consider the homogeneous adjoint problem. Recall that for a real kernel,

adj

(x, y) = K(y, x).

Thus, corresponding to the original homogeneous problem,

φ(x) = λ



+∞

K(x − y)φ(y)dy,

there exists the homogeneous adjoint problem,

adj

(x) = λ



+∞

K(y − x)φ

adj

(y)dy, (7.4.57)

whose translation kernel is related to the original one by

adj

(ξ) = K(−ξ). (7.4.58)

Now, when we take the Fourier transform of the homogeneous adjoint problem,

we ﬁnd



1 −λ

K(−k)



adj

−

(k) =−

adj

(k), (7.4.59)

where the only difference from the original equation is the sign of k inside

K(−k).

However, since 1 − λ

K(−k)isjustthereﬂectionof1−λ

K(k) through the origin, a

zero of 1 −λ

K(k) in the upper half plane corresponds to a zero of 1 −λ

K(−k)in

the lower half plane, etc. Thus, when the original 1 −λ

K(k) has a negative index

ν<0 with respect to a line, Im k = k

= A, the homogeneous adjoint problem

1 − λ

K(−k) has a positive index

relative to the line Im k = k

=−A.So,in

270 7 Wiener–Hopf Method and Wiener–Hopf Integral Equation

that case, although the original problem may have no solutions, the homogeneous

adjoint problem has

independent solutions. Now presently, 1 −λ

K(k)isfound

to have the decomposition

1 −λ

K(k) = Y

−

(k)/Y

(k),

with

(k) → 1/k

as k →∞.

We conclude that

1 −λ

K(−k) = Y

−

(−k)/Y

(−k) = Y

adj

−

(k)/Y

adj

(k), (7.4.60)

from which we recognize



adj

−

(k) = 1/Y

(−k) analytic in the lower half plane, → k

adj

(k) = 1/Y

−

(−k) analytic in the upper half plane, → k

Thus the homogeneous adjoint problem reads

adj

−

(k)

adj

−

(k) =−Y

adj

(k)ψ

adj

(k) ≡ G(k), (7.4.61)

with

G(k) = C

+ C

k +···+C

−1

. (7.4.62)

We then know that the

independent solutions to the homogeneous adjoint

problem are of the form

adj

−

(k) ={1/Y

adj

−

(k), k/Y

adj

−

(k), ..., k

−1

adj

−

(k)},

which is equivalent to

adj

−

(k) ={Y

(−k), kY

(−k), ..., k

−1

(−k)}. (7.4.63)

As such, to within a constant factor, which is irrelevant, we can write the solvability

condition (7.4.54) as



−C

(ζ )

−

(ζ )dζ = 0, j = 0, 1, ...,

− 1, (7.4.64)

which is equivalent to



−C

adj

−(j)

(−ζ )

−

(ζ )dζ = 0, j = 0, 1, ...,

−1, (7.4.65)

7.4 Inhomogeneous Wiener–Hopf Integral Equation of the Second Kind 271

where

adj

−(j)

(ζ ) ≡ ζ

(−ζ ), j = 0, 1, ...,

− 1. (7.4.66)

By the Parseval identity (7.4.56), the solvability condition (7.4.65) can now be written



+∞

adj

(j)

(x)f (x)dx = 0, j = 0, 1, ...,

− 1, (7.4.67)

where

adj

(j)

(x) =

2π



+∞−iA

−∞−iA

iζ x

adj

−(j)

(ζ )dζ , j = 0, 1, ...,

− 1, x ≥ 0. (7.4.68)

Namely, if and only if the inhomogeneous term f (x) is orthogonal to all of the

homogeneous solutions φ

adj

(x) of the homogeneous adjoint problem, the inhomogeneous

equation (7.4.1) has a unique solution, when index ν is negative.

Summary of the Wiener–Hopf integral equation:

φ(x) = λ



∞

K(x − y)φ(y)dy, x ≥ 0,

adj

(x) = λ



∞

K(y − x)φ

adj

(y)dy, x ≥ 0,

φ(x) = f (x) +λ



∞

K(x − y)φ(y)dy, x ≥ 0.

(1) Index ν = 0:

Homogeneous problem and its homogeneous adjoint problem have no

solutions.

Inhomogeneous problem has a unique solution.

(2) Index ν>0:

Homogeneous problem has ν independent solutions and its homogeneous

adjoint problem has no solutions.

Inhomogeneous problem has a nonunique solutions (but there are no

solvability conditions).

(3) Index ν<0:

Homogeneous problem has no solutions, and its homogeneous adjoint

problem has

independent solutions.

Inhomogeneous problem has a unique solution, if and only if the

inhomogeneous term is orthogonal to all

independent solutions to the

homogeneous adjoint problem.

272 7 Wiener–Hopf Method and Wiener–Hopf Integral Equation

7.5

Toeplitz Matrix and Wiener–Hopf Sum Equation

In this section, we consider the application of the Wiener–Hopf method to the

inﬁnite system of the inhomogeneous linear algebraic equation,



X =



f ,or



= f

, (7.5.1,2)

where the coefﬁcient matrix M is real and has the Toeplitz structure,

= M

n−m

. (7.5.3)

Case A: Inﬁnite Toeplitz matrix.

The system of the inﬁnite inhomogeneous linear algebraic equations

(7.5.1) becomes

∞



m=−∞

n−m

= f

, −∞ < n < ∞. (7.5.4)

We look for the solution {X

}

+∞

m=−∞

under the assumption of the uniform convergence

of {X

}

+∞

m=−∞

and {M

}

+∞

m=−∞

∞



m=−∞

< ∞ and

∞



m=−∞

< ∞. (7.5.5)

Multiplying Eq. (7.5.4) by ξ

and summing over n,wehave



n,m

n−m



. (7.5.6)

The left-hand side of Eq. (7.5.6) can be expressed as



n,m

n−m



n−m

= M(ξ)X(ξ),

with the interchange of the order of the summations, where X(ξ )andM(ξ )are

deﬁned by

X(ξ ) ≡

∞



n=−∞

, M(ξ ) ≡

∞



n=−∞

. (7.5.7,8)

7.5 Toeplitz Matrix and Wiener–Hopf Sum Equation 273

We also deﬁne

f (ξ ) ≡

∞



n=−∞

. (7.5.9)

Thus Eq. (7.5.6) takes the following form:

M(ξ)X(ξ) = f (ξ ). (7.5.10)

We assume the following bounds on M

and f



O(a

−

)asn →−∞,

O(b

−n

)asn →+∞,

a, b > 0, (7.5.11a)





O(c

−

)asn →−∞,

O(d

−n

)asn →+∞,

c, d > 0. (7.5.11b)

Then M(ξ ) is analytic in the annulus in the complex ξ plane,

1/a <

< b,1< ab, (7.5.12a)

and f (ξ ) is analytic in the annuls in the complex ξ plane,

1/c <

< d,1< cd. (7.5.12b)

Hence we obtain

X(ξ ) =

f (ξ )

M(ξ)

∞



n=−∞

provided M(ξ) = 0. (7.5.13)

X(ξ ) is analytic in the annulus

max(1/a,1/c) <

< min(b, d). (7.5.12c)

By the Fourier series inversion formula on the unit circle,weobtain

2π



2π

dθ exp[−inθ ]X(exp[iθ ]) =

2πi

dξξ

−n−1

X(ξ ), (7.5.14)

with

M(ξ) = 0for

= 1. (7.5.15)

274 7 Wiener–Hopf Method and Wiener–Hopf Integral Equation

Next, consider the eigenvalue problem of the following form:

∞



m=−∞

n−m

= µX

. (7.5.16)

We try

= ξ

. (7.5.17)

Then we obtain

M(ξ) = µ. (7.5.18)

The roots of Eq. (7.5.18) provide the solutions to Eq. (7.5.16). For µ = 0, we obtain

= (ξ

)

, (7.5.19)

where ξ

is a zero of M(ξ ).

Case B: Semi-inﬁnite Toeplitz matrix.

The system of the semi-inﬁnite inhomogeneous linear algebraic equations (7.5.1)

becomes

∞



m=0

n−m

= f

, n ≥ 0, (7.5.20a)

which is the inhomogeneous Wiener–Hopf sum equation.

We let

M(ξ) ≡

∞



n=−∞

,0≤ arg ξ ≤ 2π , (7.5.21)

be such that M(exp[iθ ]) = 0for0≤ θ ≤ 2π. We assume that

∞



n=0



< ∞, (7.5.22)

and

< (1 +ε)

−m

, m ≥ 0, ε>0. (7.5.23)

We deﬁne

≡

∞



m=0

n−m

for n ≤−1andy

≡ 0forn ≥ 0, (7.5.24a)

7.5 Toeplitz Matrix and Wiener–Hopf Sum Equation 275

≡ 0forn ≤−1, (7.5.24b)

X(ξ ) ≡

∞



n=0

, (7.5.25)

f (ξ ) ≡

∞



n=0

, (7.5.26)

Y(ξ) ≡

−1



n=−∞

. (7.5.27)

We look for the solution which satisﬁes

∞



n=0

< ∞. (7.5.28)

Then Eq. (7.5.20a) is rewritten as

∞



m=−∞

n−m

= f

+ y

for −∞< n < ∞. (7.5.20b)

We note that

∞



n=−∞



∞



n=−∞



∞



m=0

n−m



∞



n=−∞

∞



m=0

< ∞,

where the interchange of the order of the summation is justiﬁed since the ﬁnal

expression is ﬁnite. Multiplying by exp[inθ ] on both sides of Eq. (7.5.20b) and

summing over n,weobtain

M(ξ)X(ξ) = f (ξ ) +Y(ξ ). (7.5.29)

Homogeneous problem :Weset

= 0orf (ξ ) = 0.

The problem we will solve ﬁrst is

M(ξ)X(ξ) = Y(ξ), (7.5.30)

276 7 Wiener–Hopf Method and Wiener–Hopf Integral Equation

where

X(ξ )analyticfor

< 1 +ε,

Y(ξ)analyticfor

> 1.

(7.5.31)

We deﬁne the index ν of M(ξ) in the counter-clockwise direction by

ν ≡

2πi

ln[M(exp[iθ ])]



θ=2π

θ=0

. (7.5.32)

Suppose that M(ξ ) has been factored into the following form:

M(ξ) = N

(ξ)/N

out

(ξ), (7.5.33a)

where



(ξ)analyticfor

< 1 +ε, and continuous for

≤ 1,

out

(ξ)analyticfor

> 1, and continuous for

≥ 1.

(7.5.33b)

Then Eq. (7.5.30) is rewritten as

(ξ)X(ξ) = N

out

(ξ)Y(ξ ) ≡ G(ξ ), (7.5.34)

where G(ξ) is entire in the complex ξ plane. The form of G(ξ)isexamined.

Case 1:theindex ν = 0.

By the now familiar formula, Eq. (7.3.25), we have

ln[M(ξ



)]



− ξ

dξ



2πi



−



ln[M(ξ



)]



− ξ

dξ



2πi



, (7.5.35)

where the integration contours, C

and C

, are displayed in Figure 7.16.

Thus we have

(ξ) = exp



ln[M(ξ



)]



−ξ

dξ



2πi



→ 1as

→∞, (7.5.36a)

out

(ξ) = exp



ln[M(ξ



)]



− ξ

dξ



2πi



→ 1as

→∞. (7.5.36b)

We ﬁnd by Liouville’s theorem,

G(ξ) = 0, (7.5.37)

and hence we have no nontrivial homogeneous solution when ν = 0.