Masujima M. Applied Mathematical Methods in Theoretical Physics
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2.6 Scalar Field Interacting with Static Source 65
with the initial condition, U(0) = 1.
In what follows, we solve (2.6.6) explicitly and show that U(t)andv are related
through
U
−1
(−∞) = v.
To solve (2.6.6), we set
U(t) = exp[iG(t)]U
1
(t), (2.6.7)
where G(t)andU
1
(t) are unknown. Substituting (2.6.7) into (2.6.6), we obtain
i
d
dt
U
1
(t)
=
&
exp[−iG(t)]H
int
(t)exp[iG(t)]
− i exp[−iG(t)]
d
dt
exp[iG(t)]
'
U
1
(t).
Using the formula
exp[−iG(t)]
d
dt
exp[iG(t)] = i
˙
G(t) +
i
2
2!
[
˙
G(t), G(t)] +
i
3
3!
[[
˙
G(t), G(t)], G(t)] +···,
we obtain
H
int
(t) ≡ exp[−iG(t)]H
int
(t)exp[iG(t)] −i exp[−iG(t)]
d
dt
exp[iG(t)]
= H
int
(t) +i[H
int
(t), G(t)] +
i
2
2!
[[H
int
(t), G(t)], G(t)] +···
− i{i
˙
G(t) +
i
2
2!
[
˙
G(t), G(t)] +
i
3
3!
[[
˙
G(t), G(t)], G(t)] +···}.
So we set
G(t) ≡−
1
t
0
dt
1
H
int
(t
1
).
Since, in this model, we have
[H
int
(t
1
), H
int
(t
2
)] = 2i
k
λ
k
2
sin ω
k
(t
2
− t
1
) = c-number,
the third terms onward and many others in each infinite series vanish. So we have
H
int
(t) = i[H
int
(t), G(t)] +i
2
[
˙
G(t), G(t)]
=−
i
2
t
0
dt
1
[H
int
(t), H
int
(t
1
)]
66 2 Integral Equations and Green’s Functions
=
1
t
0
dt
1
k
λ
k
2
sin ω
k
(t
1
− t)
=
k
λ
k
2
ω
k
(cos ω
k
t − 1).
Hence, we immediately obtain U
1
(t)as
U
1
(t) = exp
−
i
t
0
dt
1
k
λ
k
2
ω
k
(cos ω
k
t
1
− 1)
= exp
−
i
k
λ
k
2
ω
2
k
sin ω
k
t
exp
i
k
λ
k
2
ω
k
t
.
Substituting this into the decomposition for U(t), we obtain
U(t) = exp
−
i
t
0
dt
1
H
int
(t
1
) −
i
k
λ
k
2
ω
2
k
sin ω
k
t +
i
k
λ
k
2
ω
k
t
= exp
−
i
t
0
dt
1
k
(λ
k
a
k
exp[−iω
k
t
1
] +λ
∗
k
a
†
k
exp[iω
k
t
1
])
+
k
λ
k
2
ω
2
k
sin ω
k
t
= exp
k
1
ω
k
{λ
k
a
k
(exp[−iω
k
t] −1) −λ
∗
k
a
†
k
(exp[iω
k
t] −1)}
− i
λ
k
2
2
ω
2
k
sin ω
k
t
.
Sincewehavechosenµ to be nonzero constant, ω
k
never becomes zero. For each
oscillatory term in the exponent, we invoke the following identity and its complex
conjugate:
1
ω
k
exp[iω
k
t] =−i lim
ε→0
+
∞
t
dt
1
exp[iω
k
t
1
]exp[−ε
|
t
1
|
] →
t→−∞
−2πiδ(ω
k
) = 0.
Thus we finally obtain
U
−1
(−∞) = exp
1
ω
k
k
{λ
k
a
k
− λ
∗
k
a
†
k
}
= v.
2.7 Problems for Chapter 2 67
2.7
Problems for Chapter 2
2.1. (due to H. C.). Solve
φ(x) = 1 +λ
1
0
(xy + x
2
y
2
)φ(y)dy .
(a) Show that this is equivalent to a 2 ×2 matrix equation.
(b) Find the eigenvalues and the corresponding eigenvectors of the
kernel.
(c) Find the solution of the inhomogeneous equation if
λ = eigenvalues.
(d) Solve the corresponding Fredholm integral equation of the
first kind.
2.2. (due to H. C.). Solve
φ(x) = 1 +λ
x
0
xyφ(y)dy.
Discuss the solution of the homogeneous equation.
2.3. (due to H. C.). Consider the integral equation,
φ(x) = f (x) +λ
+∞
−∞
e
−(x
2
+y
2
)
φ(y)dy, −∞ < x < ∞.
(a) Solve this equation for
f (x) = 0.
For what values of λ, does it have nontrivial solutions?
(b) Solve this equation for
f (x) = x
m
,withm = 0, 1, 2, ....
Does this inhomogeneous equation have any solutions when λ is equal
to an eigenvalue of the kernel?
68 2 Integral Equations and Green’s Functions
Hint: You may express your results in terms of the Gamma function,
(
z
)
=
∞
0
t
z−1
e
−t
dt,Rez > 0.
2.4. (due to H. C.). Solve the following integral equation,
u(θ) = 1 +λ
2π
0
sin(φ −θ)u(φ)dφ,0≤ θ<2π,
where u(θ) is periodic with period 2π. Does the kernel of this equation have
any real eigenvalues?
Hint: Note
sin(φ − θ) = sin φ cos θ −cos φ sin θ.
2.5. (due to D. M.) Consider the integral equation
φ(x) = 1 +λ
1
0
x
n
− y
n
x − y
φ(y)dy,0≤ x ≤ 1.
(a) Solve this equation for n = 2. For what values of λ, does the equation
have no solutions?
(b) Discuss how you would solve this integral equation for arbitrary
positive integer n.
2.6. (due to D. M.) Solve the integral equation,
φ(x) = 1 +
1
0
(1 +x + y + xy)
ν
φ(y)dy,0≤ x ≤ 1, ν : real.
Hint: Note that the kernel
(1 +x + y + xy)
ν
,
can be factorized.
2.7. In the Fredholm integral equation of the second kind, if the kernel is given
by
K(x, y) =
N
n=1
g
n
(x)h
n
(y),
show that the integral equation is equivalent to an N × N matrix equation.
2.7 Problems for Chapter 2 69
2.8. (due to H. C.). Consider the motion of an harmonic oscillator with a
time-dependent spring constant,
d
2
dt
2
x + ω
2
x =−A(t)x,
where ω is a constant and A(t) is a complicated function of t. Transform this
differential equation together with the boundary conditions
x (T
i
) = x
i
and x(T
f
) = x
f
,
to an integral equation.
Hint: Construct a Green’s function G(t, t
) satisfying
G(T
i
, t
) = G(T
f
, t
) = 0.
2.9. Generalize the discussion of Section 2.4 to the case of three spatial
dimensions and transform the Schr
¨
odinger equation with the initial
condition,
lim
t→−∞
e
iωt
ψ(
x, t) = e
ikz
,
to an integral equation.
Hint: Construct a Green’s function G(t, t
) satisfying
i
∂
∂t
+
∇
2
G(
x , t;
x
, t
) = δ(t − t
)δ
3
(
x −
x
),
G(
x , t;
x
, t
) = 0fort < t
.
2.10. (due to H. C.). Consider the equation
−
∂
2
∂t
2
+
∂
2
∂x
2
−m
2
φ(x, t) = U(x, t)φ(x, t).
If the initial and final conditions are
φ(x, −T) = f (x)andφ(x, T) = g(x),
transform the equation to an integral equation.
Hint: Consider Green’s function
−
∂
2
∂t
2
+
∂
2
∂x
2
−m
2
G(x, t;x
, t
) = δ(x − x
)δ(t − t
).
70 2 Integral Equations and Green’s Functions
2.11. (due to D. M.) The time-independent Schr
¨
odinger equation with the
periodic potential, V(x) =−(a
2
+k
2
cos
2
x), reads
d
2
dx
2
ψ(x) +(a
2
+ k
2
cos
2
x )ψ (x) = 0.
Show directly that even periodic solutions of this equation, which are even
Mathieu functions, satisfy the homogeneous integral equation,
ψ(x) = λ
π
−π
exp[k cos x cos y]ψ(y)dy.
Hint: Show that φ(x)definedby
φ(x) ≡
π
−π
exp[k cos x cos y]ψ(y)dy
is even and periodic, and satisfies the above time-independent Schr
¨
odinger
equation. Thus, ψ(x) is the constant multiple of φ(x),
ψ(x) = λφ(x).
2.12. (due to H. C.). Consider the differential equation,
d
2
dt
2
φ(t) = λe
−t
φ(t), 0 ≤ t < ∞, λ = constant,
together with the initial conditions,
φ(0) = 0andφ
(0) = 1.
(a) Find the partial differential equation for Green’s function G(t, t
).
Determine the form of G(t, t
)whent = t
.
(b) Transform the differential equation for φ(t) together with the initial
conditions to an integral equation. Determine the conditions on G(t, t
).
(c) Determine G(t, t
).
(d) Substitute your answer for G(t, t
) into the integral equation and verify
explicitly that the integral equation is equivalent to the differential
equation together with the initial conditions.
(e) Does the initial value problem have a solution for all λ?Ifso,isthe
solution unique?
2.13. In the Volterra integral equation of the second kind, if the kernel is given by
K(x, y) =
N
n=1
g
n
(x)h
n
(y),
show that the integral equation can be reduced to an ordinary differential
equation of the Nth order.
2.7 Problems for Chapter 2 71
2.14. Consider the partial differential equation of the form
∂
2
∂t
2
−
∂
2
∂x
2
φ(x, t) = p(x, t) −λ
∂
2
∂x
2
φ(x, t) ·
∂
∂x
φ(x, t),
where
−∞ < x < ∞, t ≥ 0,
and λ is a constant, with the initial conditions specified by
φ(x,0)= a(x),
and
∂
∂t
φ(x,0)= b(x).
This partial differential equation describes the displacement of a vibrating
string under the distributed load p(x, t).
(a) Find Green’s function for this partial differential equation.
(b) Express this initial value problem in terms of an integral equation
using Green’s function found in (a). Explain how you would find an
approximate solution if λ were small.
Hint: By applying the Fourier transform in x,findafunctionφ
0
(x, t)which
satisfies the wave equation,
∂
2
∂t
2
−
∂
2
∂x
2
φ
0
(x, t) = 0,
and the given initial conditions.
2.15. Show that Green’s function G(
r;
r
) for the Poisson equation in three spatial
dimensions,
∇
2
G(
r;
r
) =−4πδ(
r −
r
),
is given by
G(
r;
r
) =
1
r −
r
=
∞
l=0
r
l
<
r
l+1
>
P
l
(cos θ ),
where θ is the angle between
r and
r
, P
l
(cos θ )isthelth-order Legendre
polynomial of the first kind, and r
<
(r
>
) is the smaller (the larger) of the
lengths,
r
and
r
,
r
<
=
1
2
!
r
+
r
"
−
1
2
r
−
r
and
r
>
=
1
2
!
r
+
r
"
+
1
2
r
−
r
.
72 2 Integral Equations and Green’s Functions
2.16. Show that Green’s function G(
r;
r
) for the Helmholtz equation in three
spatial dimension,
(
∇
2
+ k
2
)G(
r;
r
) =−4πδ(
r −
r
),
is given by
G(
r;
r
) =
exp[ik
r −
r
]
r −
r
= k
∞
l=0
(2l + 1)j
l
(kr
<
)h
(1)
l
(kr
>
)P
l
(cos θ ),
where j
l
(kr)isthelth-order spherical Bessel function and h
(1)
l
(kr)isthe
lth-order spherical Hankel function of the first kind. Show that Green’s
function G(
r;
r
) which is even in k for the Helmholtz equation is given by
G(
r;
r
) =
cos k
r −
r
r −
r
= k
∞
l=0
(2l + 1)j
l
(kr
<
)n
l
(kr
>
)P
l
(cos θ ),
where n
l
(kr)isthelth-order spherical Neumann function,
n
l
(kr) = Re h
(1)
l
(kr).
2.17. Consider the differential equation for spherical Bessel functions,
−
d
2
dr
2
−
2
r
d
dr
+
l(l + 1)
r
2
− k
2
h
(1)
l
(kr)
j
l
(kr)
= 0,
with the boundary conditions
h
(1)
l
(kr)
j
l
(kr)
→
(1/i
l+1
)(exp[ikr]/kr)
sin(kr − lπ/2)/kr
as kr →∞.
(a) We define
w
l
(kr)
u
l
(kr)
=
krh
(1)
l
(kr),
krj
l
(kr).
Show that u
l
(kr)andw
l
(kr) satisfy the following differential equation:
−
d
2
dx
2
+
l(l + 1)
x
2
− 1
w
l
(x)
u
l
(x)
= 0,
2.7 Problems for Chapter 2 73
with the boundary conditions
w
l
(x →∞)
u
l
(0)
=
exp[ix ]
0
and the normalization determined by
w
l
(x)
u
l
(x)
→
(1/i
l+1
)(exp[ix])
sin(x − lπ/2)
as x →∞.
(b) Define the operators H
l
, A
+
l
,andA
−
l
by
H
l
≡
−
d
2
dx
2
+
l(l + 1)
x
2
,
and
A
+
l
≡−
d
dx
+
l
x
, A
−
l
≡
d
dx
+
l
x
.
Show that
A
+
l
A
−
l
= H
l
and A
−
l
A
+
l
= H
l−1
.
(c) If
H
l
ψ
l
= ψ
l
,
show that
A
+
l
A
−
l
ψ
l
= ψ
l
and A
−
l
A
+
l
A
−
l
ψ
l
= A
−
l
ψ
l
,
and hence
H
l−1
(A
−
l
ψ
l
) = (A
−
l
ψ
l
).
A
−
l
is a lowering operator which takes ψ
l
into ψ
l−1
. Similarly, since
A
−
l+1
A
+
l+1
= H
l
,
show that
A
−
l+1
A
+
l+1
ψ
l
= ψ
l
and A
+
l+1
A
−
l+1
A
+
l+1
ψ
l
= A
+
l+1
ψ
l
,
and hence
H
l+1
(A
+
l+1
ψ
l
) = (A
+
l+1
ψ
l
).
A
+
l+1
is a raising operator which takes ψ
l
into ψ
l+1
.
74 2 Integral Equations and Green’s Functions
(d) As x →∞,byobserving
A
+
→−
d
dx
, A
−
→
d
dx
,
show that
A
+
w
l
→ exp[ix]/i
l+2
,
A
−
w
l
→ exp[ix ]/i
l
,
and
A
+
u
l
→ sin(x − (l + 1)π/2),
A
−
u
l
→ sin(x − (l − 1)π/2).
Namely, A
+
and A
−
correctly maintain the asymptotic forms of w
l
and
u
l
and hence the normalizations of w
l
and u
l
.
(e) Starting from the solutions for l = 0,
w
0
= exp[ix]/i and u
0
= sin x,
obtain w
l
and u
l
and hence h
(1)
l
and j
l
.
2.18. Consider the scattering off a spherically symmetric potential V(r).
(a) Prove that the free Green’s function in the spherical polar coordinate is
given by
2
2m
(
x
1
E −
ˆ
H
0
+ iε
x
)
=−ik
l
m
Y
l,m
(
ˆ
r)Y
∗
l,m
(
ˆ
r
)j
l
(kr
<
)h
(1)
l
(kr
>
),
where r
<
(r
>
) stands for the smaller (larger) of r and r
.
(b) The Lippmann–Schwinger equation can be written for spherical waves
as
Elm(+)
%
=
Elm
%
+
1
E −
ˆ
H
0
+ iε
V
Elm(+)
%
.
Using a), show that this equation, written in the
x-representation, leads to
an integral equation for the radial function, A
l
(k;r), as follows:
A
l
(k;r) = j
l
(kr) −(
2mik
2
)
∞
0
j
l
(kr
<
)h
(1)
l
(kr
>
)V(r
)A
l
(k;r
)r
2
dr
,
where the radial function, A
l
(k;r), is defined by
*
x
Elm(+)
%
= c
l
A
l
(k;r)Y
l,m
(
ˆ
r),