Spectral Theory in Hilbert Space

80 11. STURM-LIOUVILLE EQUATIONS
Theorem 11.17. Suppose u ∈ D(T ). Then the inverse transform
〈û, ϕ(x, ·)〉ρ converges locally uniformly to u(x).
Proof. The proof is very similar to that of Corollary 11.4. Put
v = (T − i)u so that u = Riv. Let K be a compact interval, and put
uK(x) =
K û(t)ϕ(x, t) dP (t) = F −1 (χû)(x), where χ is the characteristic
function for K. Define vK similarly. Then by Lemma 11.14
RivK = F −1 ( χ(t)ˆv(t)
) = F
t − i
−1 (χû) = uK.
Since vK → v in L2 (a, b) as K → R, it follows from Theorem 11.1
that uK → u in C1 (L) as K → R, for any compact subinterval L of
[a, b).
Example 11.18 (Sine and cosine transforms). Let us interpret Theorem
11.8 for the case of the equation −u ′′ = λu on the interval [0, ∞).
We shall look at the cases when the boundary condition at 0 is either
a Dirichlet condition (α = 0 in (10.7)) or a Neumann condition (α =
π/2). The general solution of the equation is u(x) = Ae √ −λx +Be − √ −λx .
Let the root be the principal branch, i.e., the branch where the real
part is ≥ 0. Then the only solutions in L 2 (0, ∞) are, unless λ ≥ 0,
the multiples of e −√ −λx = cos(i √ −λx) + i sin(i √ −λx). It follows that
the equation is in the limit point condition at infinity (this is also a
consequence of Theorem 10.10).
With a Dirichlet condition at 0 we have θ(x, λ) = cos(i √ −λx)
and ϕ(x, λ) = −i sin(i √ −λx)/ √ −λ. It follows that the m-function is
mD(λ) = − √ −λ. Similarly, the m-function in the case of a Neumann
condition at 0 is mN(λ) = 1/ √ −λ, using again the principal branch of
the root.
Using the Stieltjes inversion formula Lemma 6.5 we √see that the
t dt for t ≥
corresponding spectral measures are given by dρD(t) = 1
π
0, dρD = 0 in (−∞, 0), respectively dρN(t) = dt
π √ t for t ≥ 0, dρN = 0
in (−∞, 0). If u ∈ L2 (0, ∞) and we define û(t) = ∞
0 u(x) sin(√tx) √ dx,
t
as a generalized integral converging in L2 ρD , then the inversion formula
reads u(x) = 1
∞
π 0 û(t) sin(√tx) dt.
In this case one usually changes variable in the transform and
defines the sine transform S(u)(ξ) = ∞
0 u(x) sin(ξx) dx = ξû(ξ2 ).
Changing variable to ξ = √ t in the inversion formula above then shows
that u(x) = 2
∞
S(u)(ξ) sin(ξx) dξ.
π 0
Similarly, if we set û(t) = ∞
0 u(x) cos(√tx) dx the inversion for-
dt. In this case it is again
mula obtained is u(x) = 1
π
∞
0 û(t) cos(√ tx)
√ t
common to use ξ = √ t as the transform variable, so one defines
the cosine transform C(u)(ξ) = ∞
u(x) cos(ξx) dx. Changing vari-
0
ables in the inversion formula above then gives the inversion formula
C(u)(ξ) cos(ξx) dξ for the cosine transform.
u(x) = 2
π
∞
0