Spectral Theory in Hilbert Space

11. STURM-LIOUVILLE EQUATIONS 77
for µ = c, d. However, Lemma 11.10 ensures the absolute convergence
of these integrals. Changing the order of integration gives
Γ
û(λ)û(λ)m(λ) dλ =
∞
−∞
Γ
û(λ)û(λ)( 1 t
−
t − λ t2 ) dλ dρ(t)
+ 1
= −2πi
c
d
|û(t)| 2 dρ(t)
since for c < t < d the residue of the inner integral is −|û(t)| 2 dρ(t)
whereas t = c, d do not carry any mass and the inner integrand is
regular for t < c and t > d.
Similarly we have
Γ
〈Rλu, u〉 dλ =
∞
−∞
d〈Etu, u〉
Γ
dλ
t − λ
= −2πi
c
d
d〈Etu, u〉
which completes the proof.
Lemma 11.12. If u ∈ L2 (a, b) the generalized Fourier transform
uϕ(·, t) as x → b. Furthermore,
û ∈ L 2 ρ exists as the L 2 ρ-limit of x
a
〈Etu, v〉 =
t
−∞
ûˆv dρ .
In particular, 〈u, v〉 = 〈û, ˆv〉ρ if u and v ∈ L 2 (a, b).
Proof. If u has compact support Lemma 11.11 shows that (11.3)
holds for a dense set of values c, d since functions of bounded variation
are a.e. differentiable. Since both Et and ρ are left-continuous we
obtain, by letting d ↑ t, c → −∞ through such values,
〈Etu, v〉 =
t
−∞
ûˆv(t) dρ(t)
when u, v have compact supports; first for u = v and then in general
by polarization. As t → ∞ we also obtain that 〈u, v〉 = 〈û, ˆv〉ρ when u
and v have compact supports.
For arbitrary u ∈ L 2 (a, b) we set, for c ∈ (a, b),
uc(x) =
u(x) for x < c
0 otherwise
and obtain a transform ûc. If also d ∈ (a, b) it follows that ûc −ûdρ =
uc − ud, and since uc → u in L 2 (a, b) as c → b, Cauchy’s convergence
principle shows that ûc converges to an element û ∈ L 2 ρ as c → b. The
lemma now follows in full generality by continuity.