Spectral Theory in Hilbert Space

10. BOUNDARY CONDITIONS 65
Let us now consider the singular case. We will then first consider
the case when one endpoint is regular and the other singular. So,
assume that I = [a, b) with a regular and b possibly singular.
Lemma 10.7. There are elements of D1 which vanish in a neighborhood
of b and have arbitrarily prescribed initial values u(a) and u ′ (a).
Proof. Let c ∈ (a, b) and f ∈ L2 (a, b) vanish in (c, b). Now solve
−u ′′ + qu = f with initial data u(c) = u ′ (c) = 0 so that u vanishes in
(c, b). It follows that u ∈ D1, and we need to show that u(a) and u ′ (a)
can be chosen arbitrarily by selection of f. Note that if −v ′′ + qv = 0
integrating by parts twice shows that
c
〈f, v〉 = (−u ′′ + qu)v = [−u ′ v + uv ′ ] c a = u ′ (a)v(a) − u(a)v ′ (a).
a
If v1 and v2 are solutions of −v ′′ +qv = 0 satisfying v1(a) = 1, v ′ 1(a) = 0
and v2(a) = 0, v ′ 2(a) = −1 respectively, we obtain u(a) = 〈f, v2〉 and
u ′ (a) = 〈f, v1〉. Since v1, v2 are linearly independent we can choose
f to give arbitrary values to this, for example by choosing f as an
appropriate linear combination of v1 and v2 in [a, c].
The fact that T1 and T0 are closed means that their domains are
Hilbert spaces with norm-square u 2 1 = u 2 +T1u 2 . We shall always
view D1 and D0 as spaces in this way. We also note that if u ∈ D1, then
u is continuously differentiable. If K is a compact interval we define
C 1 (K) to be the linear space of continuously differentiable functions
provided with the norm uK = sup K |u|+sup K |u ′ |. Convergence for a
sequence {uj} ∞ 1 in this space therefore means uniform convergence on
K of uj and u ′ j as j → ∞. This space is easily seen to be complete, and
thus a Banach space. As we noted above, if K is a compact subinterval
of I, then the restriction to K of any element of D1 is in C 1 (K). We
will need the following fact.
Lemma 10.8. For every compact subinterval K ⊂ I there exists a
constant CK such that uK ≤ CKu1 for every u ∈ D1. In particular,
the linear forms D1 ∋ u ↦→ u(x) and D1 ∋ u ↦→ u ′ (x) are locally
uniformly bounded in x.
Proof. The restriction map D1 ∋ u ↦→ u ∈ C 1 (K) is linear and
we will show that this map is closed. By the closed graph theorem (see
Appendix A) it then follows that this map is bounded, which is the
statement of the lemma.
To show that the map is closed we must show that if uj → u in
D1 and the restrictions to K of uj converge to ũ in C 1 (K), then the
restriction to K of u equals ũ. But this is clear, since if uj converges
in L 2 (I) to u, then their restrictions to K converge in L 2 (K) to the
restriction of u to K. At the same time the restrictions to K converge