Spectral Theory in Hilbert Space

114 15. SINGULAR PROBLEMS
Since vK → v in L2 W as K → R, it follows from Theorem 14.2 that
uK → u in C(L) as K → R, for any compact subinterval L of I.
Example 15.14. Let us interpret Theorem 15.5 for the case of
the operator of Example 4.6, Green’s function of which is given in
Example 8.8. Comparing (8.2) with (15.1), we see that M(λ) = i/2 for
λ in the upper half plane. By Lemma 6.5 the corresponding spectral
measure is P (t) = limε→0 1
t
t
Im M(µ + iε) dµ = . This means that
π 0 2π
if f ∈ L2 (R), then as a, b → ∞ the integral b
−a f(x) e−ixt dt converges
in the sense of L2 (R) to a function ˆ f ∈ L2 (R). Furthermore the integral
1 b
2π −a ˆ f(t) eixt dt converges in the same sense to f as a and b → ∞.
We also conclude that ∞
−∞ |f|2 = 1
∞
2π −∞ | ˆ f| 2 . Finally, if f is locally
absolutely continuous and together with its derivative in L2 (R), then
the transform of −if ′ is t ˆ f(t) and conversely, if ˆ f and t ˆ f(t) are both in
L2 (R), then the inverse transform of ˆ f is locally absolutely continuous,
and its derivative is in L2 (R) and is the inverse transform of it ˆ f(t). We
also get from Theorem 15.13 that if f has these properties, then the
inverse transform of ˆ f converges absolutely and locally uniformly to f.
Actually, it is here easy to see that the convergence is uniform on the
whole axis, but nevertheless it is clear that we have retrieved all the
basic properties of the classical Fourier transform.
Exercises for Chapter 15
Exercise 15.1. Use, e.g., estimates in the variation of constants
formula Lemma 13.4 for v = (λ − µ)u to show that all columns of
F (x, µ) are in L2 W , then so are those of F (x, λ).
Exercise 15.2. Show that the two definitions of L2 P given in the
text are equivalent. What needs to be proved is that any measurable
n × 1 matrix-valued function with finite norm can be approximated in
norm by a similar function which is C∞ 0 .
Hint: Use a cut off and convolution with a C ∞ 0 -function of small support.
Exercise 15.3. In Lemma 15.9 is claimed that for every compact
interval K the integral
K F (x, t) dP (t) û(t) ∈ HT , but this is never
proved; or is it? Clarify this point!
Exercise 15.4. Consider, as in the beginning of Chapter 10, the
first order system corresponding to a general Sturm-Liouville equation
−(pu ′ ) ′ + qu = λwu on [a, b),
where 1/p, q and w are integrable on any interval [a, x], x ∈ (a, b).
Also assume that p and q are real-valued functions and w ≥ 0 and not
a.e. equal to 0. Consider a selfadjoint realization given by separated