Spectral Theory in Hilbert Space

11. STURM-LIOUVILLE EQUATIONS 79
Proof. By Lemma 11.12, 〈Etu, v〉 = t
−∞
〈Rλu, v〉 =
∞
−∞
d〈Etu, v〉
t − λ =
∞
−∞
By properties of the resolvent
Rλu 2 =
1
2i Im λ 〈Rλu − R λ u, u〉 =
û(t)ˆv(t) dρ(t)
t − λ
∞
−∞
ûˆv dρ, so that
= 〈û(t)/(t − λ), ˆv(t)〉ρ.
d〈Etu, u〉
|t − λ| 2 = û(t)/(t − λ)2 ρ.
Setting v = Rλu and using Lemma 11.12, it therefore follows that
û(t)/(t−λ) 2 ρ = 〈û(t)/(t−λ), F(Rλu)〉ρ = F(Rλu) 2 ρ. It follows that
we have û(t)/(t − λ) − F(Rλu)ρ = 0, which was to be proved.
Lemma 11.15. The generalized Fourier transform is unitary from
L 2 (a, b) to L 2 ρ and the inverse transform is the inverse of this map.
Proof. According to Lemma 11.13 we need only show that if
û ∈ L2 ρ has inverse transform 0, then û = 0. Now, according to
Lemma 11.14, F(v)(t)/(t − λ) is a transform for all v ∈ L2 (a, b) and
non-real λ. Thus we have 〈û(t)/(t − λ), F(v)(t)〉ρ = 0 for all non-real λ
if û is orthogonal to all transforms. But we can view this scalar product
as the Stieltjes-transform of the measure t
ûF(v) dρ, so applying the
−∞
inversion formula Lemma 6.5 we have
ûF(v) dρ = 0 for all compact
K
intervals K, and all v ∈ L2 (a, b). Thus the cutoff of û, which equals
û in K and 0 outside, is also orthogonal to all transforms, i.e., has
inverse transform 0 according to Lemma 11.13. It follows that
v(x) = û(t)ϕ(x, t) dρ(t)
K
is the zero-element of L2 (a, b) for any compact interval K. Differentiating
under the integral sign we also see that v ′ (x) =
K ûϕ′ (x, ·) dρ
is the zero element of L2 (a, b). But these functions are continuous, so
they are pointwise 0. Now 0 = v ′ (a) cos α − v(a) sin α =
û dρ. Thus
K
û dρ is the zero measure, so that û = 0 as an element of L2 ρ.
Lemma 11.16. If u ∈ D(T ), then F(T u)(t) = tû(t). Conversely, if
û and tû(t) are in L 2 ρ, then F −1 (û) ∈ D(T ).
Proof. We have u ∈ D(T ) if and only if u = Rλ(T u − λu), which
holds if and only if û(t) = (F(T u)(t) − λû(t))/(t − λ), i.e., F(T u)(t) =
tû(t), according to Lemmas 11.14 and 11.15.
This completes the proof of Theorem 11.8. We also have the following
analogue of Corollary 11.4.