Spectral Theory in Hilbert Space

numbers A and B ≥ 0 such that
m(λ) = A + Bλ +
11. STURM-LIOUVILLE EQUATIONS 75
∞
−∞
( 1 t
−
t − λ t2 ) dρ(t).
+ 1
The spectral measure dρ gives rise to a Hilbert space L 2 ρ, which
consists of those functions û which are measurable with respect to dρ
and for which û 2 ρ = ∞
−∞ |û|2 is finite. Alternatively, we may think of
L 2 ρ as the completion in this norm of compactly supported, continuous
functions. These alternative definitions give the same space, but we
will not prove this here. We denote the scalar product in L 2 ρ by 〈·, ·〉ρ.
The main result of this chapter is the following.
Theorem 11.8.
(1) If u ∈ L 2 (a, b) the integral x
0 uϕ(·, t) converges in L2 ρ as x → b.
The limit is called the generalized Fourier transform of u and
is denoted by F(u) or û. We write this as û(t) = 〈u, ϕ(·, t)〉,
although the integral may not converge pointwise.
(2) The mapping u ↦→ û is unitary between L 2 (a, b) and L 2 ρ so that
the Parseval formula 〈u, v〉 = 〈û, ˆv〉ρ is valid if u, v ∈ L 2 (a, b).
(3) The integral
K û(t)ϕ(x, t) dρ(t) converges in L2 (a, b) as K →
R through compact intervals. If û = F(u) the limit is u, so
the integral is the inverse of the generalized Fourier transform.
Again, we write u(x) = 〈û, ϕ(x, ·)〉ρ for u ∈ L2 (a, b), although
the integral may not converge pointwise.
(4) Let E∆ denote the spectral projector of T for the interval ∆.
Then E∆u(x) =
∆
ûϕ(x, ·) dρ.
(5) 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 ).
Before we prove this theorem, let us interpret it in terms of the
spectral theorem. If the interval ∆ shrinks to a point t, then E∆ tends
to zero, unless t is an eigenvalue, in which case we obtain the projection
on the eigenspace. By (4) this means that eigenvalues are precisely
those points at which the function ρ has a (jump) discontinuity; continuous
spectrum thus corresponds to points where ρ is continuous, but
which are still points of increase for ρ, i.e., there is no neighborhood of
the point where ρ is constant. In terms of measure theory, this means
that the atomic part of the measure dρ determines the eigenvalues, and
the diffuse part of dρ determines the continuous spectrum.
We will prove Theorem 11.8 through a long (but finite!) sequence
of lemmas. First note that for u ∈ L 2 (a, b) with compact support in
[a, b) the function û(λ) = 〈u, ϕ(·, λ)〉 is an entire function of λ since
ϕ(x, λ) is entire, locally uniformly in x, according to Theorem 10.1.
Lemma 11.9. The function 〈Rλu, v〉 − m(λ)û(λ)ˆv(λ) is entire for
all u, v ∈ L 2 (a, b) with compact supports in [a, b).