Spectral Theory in Hilbert Space

106 15. SINGULAR PROBLEMS
We obtain
0 = u(x) = Rλ(v − λu)(x) = 〈v − λu, G ∗ (x, ·, λ)〉W
= 〈v, G ∗ (x, ·, λ)〉W − 〈u, λG ∗ (x, ·, λ)〉W .
But according to Lemma 13.6 this means that each column of y ↦→
G ∗ (x, y, λ) is in the domain of the maximal relation for (13.1) on the
intervals I ∩ (−∞, x) and I ∩ (x, ∞) and satisfies the equation Ju ′ +
Qu = λW u on these intervals, so (2) follows. It also follows that we
have
G ∗ (x, y, λ) =
F (y, λ)P ∗ +(x, λ), y < x
F (y, λ)P ∗ −(x, λ), y > x,
for some n × n matrix-valued functions P+ and P−.
If u is compactly supported and in L2 W we have, for x outside the
convex hull of the support of u,
(15.2) Rλu(x) = P±(x, λ)〈u, F (·, λ)〉W .
The function v = Rλu satisfies the equation Jv ′ + Qv = λW v + W u,
so we may write P±(x, λ) = F (x, λ)H±(λ), and since Rλu ∈ D(T )
it certainly satisfies the boundary conditions determining T . If the
support of u is large enough the scalar product in (15.2) can be any
column vector, in view of Assumption 13.7, so for every y each column
of x ↦→ G(x, y, λ) also satisfies the boundary conditions determining T .
This proves (3). If the endpoints of I are a and b respectively we now
have
x
Rλu(x) = F (x, λ)
Differentiating this we obtain
JRλu ′ + (Q − λW )Rλu
a
H+(λ)F ∗ (·, λ)W u +
b
x
H−(λ)F ∗ (·, λ)W u .
= JF (x, λ)(H−(λ) − H+(λ))F ∗ (x, λ)W (x)u(x),
so JF (x, λ)(H−(λ) − H+(λ))F ∗ (x, λ) should be1 the unit matrix. In
view of the fact that JF (x, λ) is the inverse of J −1F ∗ (x, λ) this means
that H−(λ) − H+(λ) = J −1 . If we define M(λ) = (H−(λ) + H+(λ))/2
we now obtain (15.1).
If now u and v both have compact supports we have
〈Rλu, v〉W = v ∗ (x)W (x)G(x, y, λ)W (y)u(y) dxdy,
1 Actually, one must again argue using Assumption 13.7. We leave the details
to the reader.