Spectral Theory in Hilbert Space

11. STURM-LIOUVILLE EQUATIONS 73
since the other two terms obtained cancel. Thus v ′ (a) = cos α b
uψ(·, λ)
a
so v satisfies the boundary condition at a. If x is to the right of the support
of u we obtain v(x) = ψ(x, λ) b
uϕ(·, λ) so that v also satisfies the
a
boundary condition at b, being a multiple of ψ near b. Differentiating
again we obtain
−v ′′ (x) + (q(x) − λ)v(x) = [ψ, ϕ]u(x) = u(x).
It follows that v = Rλu and, since compactly supported functions are
dense in L2 (a, b), that g(x, y, λ) is Green’s function for our operator.
From (11.1) now follows that the kernel g1 is as stated.
It remains to show that m is a Nevanlinna function. If u and v both
have compact supports in I we have
〈Rλu, v〉 = g(x, y, λ)u(y)v(x) dxdy,
the double integral being absolutely convergent. Similarly
〈u, Rλv〉 = g(y, x, λ)u(y)v(x) dxdy,
and since the integrals are equal for all u, v by Theorem 5.2.2 we obtain
g(x, y, λ) = g(y, x, λ) or, if x < y,
ϕ(x, λ)θ(y, λ) + ϕ(x, λ)ϕ(y, λ)m(λ)
= ϕ(x, λ)θ(y, λ) + ϕ(x, λ)ϕ(y, λ)m(λ),
since ϕ(·, λ) = ϕ(·, λ) and similarly for θ. Since ϕ(x, λ) = 0 for non-real
λ (why?) it follows that m(λ) = m(λ). Now λ ↦→ Rλu(x) is analytic
for non-real λ and for compactly supported u
Rλu(x) = θ(x, λ)
a
x
uϕ(·, λ) + ϕ(x, λ)
x
b
uθ(·, λ)
+ m(λ)ϕ(x, λ)
a
b
uϕ(·, λ).
The first two terms on the right are obviously entire functions according
to Theorem 10.1, as is the coefficient of m(λ), and since by choice
of u we may always assume that this coefficient is non-zero in a neighborhood
of any given λ it follows that m(λ) is analytic for non-real
λ.
Finally, integration by parts shows that
λ
a
x
|ψ| 2 = −ψ ′ ψ x
a +
x
a
(|ψ ′ | 2 + q|ψ| 2 ).