Spectral Theory in Hilbert Space

72 11. STURM-LIOUVILLE EQUATIONS
obtain a cosine series f(x) = a0
2 + ∞ and ak = 〈f(x), cos kx〉.
k=1 ak cos(kx), where a0 = 〈f, 1〉
We have thus retrieved some of the classical versions of Fourier
series, but is clear that many other variants are obtained by simply
varying the boundary conditions, and that many more examples are
obtained by choosing a non-zero q in (10.3).
We now have a satisfactory eigenfunction expansion theory for regular
boundary value problems, so we turn next to singular problems.
We then need to take a much closer look at Green’s function. We shall
here primarily look at the case of separated boundary conditions for
I = [a, b) where a is a regular endpoint and b possibly singular, and
refer the reader to the theory of Chapter 15 for the general case. With
this assumption Green’s function has a particularly simple structure.
Assume that ϕ, θ are solutions of −u ′′ + qu = λu with initial data
ϕ(a, λ) = − sin α, ϕ ′ (a, λ) = cos α and θ(a, λ) = cos α, θ ′ (a, λ) = sin α.
Theorem 11.6. Suppose I = [a, b) with a regular, and that T is
given by the separated condition (10.7) at a, and another separated condition
at b if needed, i.e., if b is regular or in the limit circle condition.
If Im λ = 0, then g(x, y, λ) = ϕ(min(x, y), λ)ψ(max(x, y), λ) where ψ is
called the Weyl solution and is given by ψ(x, λ) = θ(x, λ)+m(λ)ϕ(x, λ).
Here m(λ) is called the Weyl-Titchmarsh m-coefficient and is a Nevanlinna
function in the sense of Chapter 6. The kernel g1 is g1(x, y, λ) =
ϕ ′ (x, λ)ψ(y, λ) if x < y and g1(x, y, λ) = ϕ(y, λ)ψ ′ (x, λ) if x > y.
Proof. It is easily verified that [θ, ϕ] = 1. Now ϕ satisfies the
boundary condition at a and can therefore only satisfy the boundary
condition at b if λ is an eigenvalue and thus real. On the other hand,
there will be a solution in L2 (a, b) satisfying the boundary condition
at b, since if deficiency indices are 1 there is no condition at b, and if
deficiency indices are 2, then the condition at b is a linear, homogeneous
condition on a two-dimensional space, which leaves a space of dimension
1. Thus we may find a unique m(λ) so that ψ = θ + mϕ satisfies the
boundary condition at b. It follows that [ψ, ϕ] = [θ, ϕ] + m[ϕ, ϕ] = 1.
Now setting v(x) = 〈u, g(x, ·, λ)〉 and assuming that u ∈ L2 (a, b)
has compact support we obtain
x
b
v(x) = ψ(x, λ) uϕ(·, λ) + ϕ(x, λ) uψ(·, λ),
a
so that v(a) = − sin α b
uψ(·, λ). Differentiating we obtain
a
(11.1) v ′ (x) = ψ ′ x
(x, λ) uϕ(·, λ) + ϕ ′ b
(x, λ) uψ(·, λ),
a
x
x