SchmidtHilbert theory

INTEGRAL EQUATIONS
Neumann series, which converge only if the condition (23.38) is satisfied. Thus the
Fredholm method leads to a unique, non-singular solution, provided that d(λ) = 0.
In fact, as we might suspect, the solutions of d(λ) = 0 give the eigenvalues of the
homogeneous equation corresponding to (23.34), i.e. with f(x) ≡ 0.
Use Fredholm theory to solve the integral equation (23.39).
Using (23.36) and (23.41), the solution to (23.39) can be written in the form
1
1
D(x, z; λ)
y(x) = x + λ
R(x, z; λ)z dz = x + λ
z dz.
d(λ)
0
0
(23.47)
In order to find the form of the resolvent kernel R(x, z; λ), we begin by setting
D0 (x, z) = K(x, z) = xz
and
d0 = 1
and use the recurrence relations (23.45) and (23.46) to obtain
1
1
1
D0 (x, x) dx =
x2 dx = ,
d1 =
3
0
0
3 1
1
xz
xz
z
D1 (x, z) =
xz12 z dz1 =
−
− xz 1 = 0.
3
3
3 0
0
Applying the recurrence relations again we find that dn = 0 and Dn (x, z) = 0 for n > 1.
Thus, from (23.42) and (23.43), the numerator and denominator of the resolvent respectively
are given by
λ
D(x, z; λ) = xz
and
d(λ) = 1 − .
3
Substituting these expressions into (23.47), we find that the solution to (23.39) is given
by
1
xz 2
dz
1
−
λ/3
0
1
λx
z3
3x
x
=x+
=
,
=x+λ
1 − λ/3 3 0
3−λ
3−λ
y(x) = x + λ
which, as expected, is the same as the solution (23.40) found by constructing a Neumann
series. 23.7 Schmidt–Hilbert theory
The Schmidt–Hilbert (SH) theory of integral equations may be considered as
analogous to the Sturm–Liouville (SL) theory of differential equations, discussed
in chapter 17, and is concerned with the properties of integral equations with
Hermitian kernels. An Hermitian kernel enjoys the property
K(x, z) = K ∗ (z, x),
(23.48)
and it is clear that a special case of (23.48) occurs for a real kernel that is also
symmetric with respect to its two arguments.
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23.7 SCHMIDT–HILBERT THEORY
Let us begin by considering the homogeneous integral equation
y = λKy,
where the integral operator K has an Hermitian kernel. As discussed in section 23.3, in general this equation will have solutions only for λ = λi , where the λi
are the eigenvalues of the integral equation, the corresponding solutions yi being
the eigenfunctions of the equation.
By following similar arguments to those presented in chapter 17 for SL theory,
it may be shown that the eigenvalues λi of an Hermitian kernel are real and
that the corresponding eigenfunctions yi belonging to different eigenvalues are
orthogonal and form a complete set. If the eigenfunctions are suitably normalised,
we have
b
yi |yj =
a
yi∗ (x)yj (x) dx = δij .
(23.49)
If an eigenvalue is degenerate then the eigenfunctions corresponding to that
eigenvalue can be made orthogonal by the Gram–Schmidt procedure, in a similar
way to that discussed in chapter 17 in the context of SL theory.
Like SL theory, SH theory does not provide a method of obtaining the eigenvalues and eigenfunctions of any particular homogeneous integral equation with
an Hermitian kernel; for this we have to turn to the methods discussed in the
previous sections of this chapter. Rather, SH theory is concerned with the general properties of the solutions to such equations. Where SH theory becomes
applicable, however, is in the solution of inhomogeneous integral equations with
Hermitian kernels for which the eigenvalues and eigenfunctions of the corresponding homogeneous equation are already known.
Let us consider the inhomogeneous equation
y = f + λKy,
(23.50)
where K = K† and for which we know the eigenvalues λi and normalised
eigenfunctions yi of the corresponding homogeneous problem. The function f
may or may not be expressible solely in terms of the eigenfunctions yi , and to
accommodate this situation we write the unknown solution y as y = f + i ai yi ,
where the ai are expansion coefficients to be determined.
Substituting this into (23.50), we obtain
f+
ai yi = f + λ
i
ai yi
i
λi
+ λKf,
(23.51)
where we have used the fact that yi = λi Kyi . Forming the inner product of both
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INTEGRAL EQUATIONS
sides of (23.51) with yj , we find
ai
ai yj |yi = λ
yj |yi + λyj |Kf.
λi
i
i
(23.52)
Since the eigenfunctions are orthonormal and K is an Hermitian operator,
we have that both yj |yi = δij and yj |Kf = Kyj |f = λ−1
j yj |f. Thus the
coefficients aj are given by
aj =
λλ−1
j yj |f
1 − λλ−1
j
=
λyj |f
,
λj − λ
(23.53)
yi |f
yi .
λi − λ
i
(23.54)
and the solution is
y=f+
ai yi = f + λ
i
This also shows, incidentally, that a formal representation for the resolvent kernel
is
yi (x)y ∗ (z)
i
.
(23.55)
R(x, z; λ) =
λ
−
λ
i
i
If f can be expressed as a linear superposition of the yi , i.e. f = i bi yi , then
bi = yi |f and the solution can be written more briefly as
bi
yi .
(23.56)
y=
1
−
λλ−1
i
i
We see from (23.54) that the inhomogeneous equation (23.50) has a unique
solution provided λ = λi , i.e. when λ is not equal to one of the eigenvalues of
the corresponding homogeneous equation. However, if λ does equal one of the
eigenvalues λj then, in general, the coefficients aj become singular and no (finite)
solution exists.
Returning to (23.53), we notice that even if λ = λj a non-singular solution to
the integral equation is still possible provided that the function f is orthogonal
to every eigenfunction corresponding to the eigenvalue λj , i.e.
b
yj∗ (x)f(x) dx = 0.
yj |f =
a
The following worked example illustrates the case in which f can be expressed in
terms of the yi . One in which it cannot is considered in exercise 23.14.
Use Schmidt–Hilbert theory to solve the integral equation
π
y(x) = sin(x + α) + λ
sin(x + z)y(z) dz.
(23.57)
0
It is clear that the kernel K(x, z) = sin(x + z) is real and symmetric in x and z and is
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