Spectral Theory in Hilbert Space

u(c) = 0 is given by
(13.2) u(x) = F (x, λ)J −1
13. FIRST ORDER SYSTEMS 93
c
x
F ∗ (y, λ)W (y)v(y) dy .
Proof. We have (F ∗ (x, λ)JF (x, λ)) ′ = −(JF ′ (x, λ)) ∗ F (x, λ) +
F ∗ (x, λ)JF ′ (x, λ) = 0 using the differential equation. It follows that
F ∗ (x, λ)JF (x, λ) is constant. Since it equals J for x = c this is its
value for all x ∈ I. It follows that J −1 F ∗ (x, λ) is the inverse matrix of
JF (x, λ). Straightforward differentiation now shows that (13.2) solves
the equation.
Corollary 13.5. If v ∈ L2 W with compact support in I then (13.1)
has a solution u with compact support in I if and only if
I v∗W u0 = 0
for all solutions u0 of the homogeneous equation (13.1) with v = 0.
Proof. If we choose c to the left of the support of v, then by
Lemma 13.4 the function u(x) = F (x)J −1 x
c F ∗W v is the only solution
of (13.1) which vanishes to the left of c. Since F (x)J −1 is invertible
(13.1) has a solution of compact support if and only if
I F ∗W v = 0.
But the columns of F are linearly independent so they are a basis for
the solutions of the homogeneous equation. The corollary follows.
Lemma 13.6. Suppose (u, v) ∈ T ∗ 0 . Then there is a representative
of the equivalence class u, also denoted by u, which is absolutely continuous
and satisfies Ju ′ + Qu = W v. Conversely, if this holds, then
(u, v) ∈ T ∗ 0 .
Proof. Let u1 be a solution of Ju ′ 1 + Qu1 = W v and assume
(u0, v0) ∈ T0 has compact support. Integrating by parts we get
v ∗
W u0 = (Ju ′ 1 + Qu1) ∗
u0 = u ∗ 1(Ju ′
0 + Qu0) = u ∗ 1W v0.
I
I
This proves the converse part of the lemma. We also have 0 = 〈u0, v〉−
〈v0, u〉 = 〈v0, u1 − u〉. Here v0 is an arbitrary compactly supported
element of L2 W for which there exists a compactly supported element
u0 ∈ L2 W satisfying Ju′ 0 + Qu0 = W v0. By Corollary13.5 it follows that
u1 − u solves the homogeneous equation, i.e., u solves (13.1).
It now follows that T0 is symmetric and that its adjoint is given
by the maximal relation T1 consisting of all pairs (u, v) in L2 W × L2W such that u is (the equivalence class of) a locally absolutely continuous
function for which Ju ′ + Qu = W v. We can now apply the theory of
Chapter 9.2. The deficiency indices of T0 are accordingly the number
of solutions of Ju ′ + Qu = iW u and Ju ′ + Qu = −iW u respectively
which are linearly independent in L2 W . Since there are altogether only
I
I