Spectral Theory in Hilbert Space

14. EIGENFUNCTION EXPANSIONS 103
Proof. Rλuj(x) = 〈uj, G∗ (x, ·, λ)〉W → 0 since the columns of
y ↦→ G∗ (x, y, λ) are in L2 W for any x ∈ I. Now let K be a compact
subinterval of I. A weakly convergent sequence in L2 W is bounded, so
since Rλ maps L2 W boundedly into C(K), it follows that Rλuj(x) is
bounded independently of j and x for x ∈ K.
Corollary 14.4. If the interval I is compact, then any selfadjoint
restriction T of T1 has compact resolvent. Hence T has a complete
orthonormal sequence of eigenfunctions in HT .
Proof. Suppose uj ⇀ 0 weakly in L2 W . If I is compact, then
Lemma 14.3 implies that Rλuj → 0 pointwise and boundedly in I,
and hence by dominated convergence Rλuj → 0 in L2 W . Thus Rλ is
compact. The last statement follows from Theorem 8.3.
If T has compact resolvent, then the generalized Fourier series of
any u ∈ HT converges to u in L2 W ; if we just have u ∈ L2W the series
converges to the projection of u onto HT . For functions in the domain
of T much stronger convergence is obtained.
Corollary 14.5. Suppose T has a complete orthonormal sequence
of eigenfunctions in HT . If u ∈ D(T ), then the generalized Fourier series
of u converges locally uniformly in I. In particular, if I is compact,
the convergence is uniform in I.
Proof. Suppose u ∈ D(T ) = D( ˜ T ), i.e., ˜ T u = v for some v ∈ HT ,
and let ˜v = v − iu, so that u = Ri˜v. If e is an eigenfunction of
T with eigenvalue λ we have ˜ T e = λe or ( ˜ T + i)e = (λ + i)e so
that R−ie = e/(λ + i). It follows that 〈u, e〉W e = 〈Ri˜v, e〉W e =
〈˜v, R−ie〉W e = 1
λ−i 〈˜v, e〉W e = 〈˜v, e〉Rie. If sNu denotes the N:th partial
sum of the Fourier series for u it follows that sNu = RisN ˜v, where sN ˜v
is the N:th partial sum for ˜v. Since sN ˜v → ˜v in HT , it follows from
Theorem 14.2 and the remark after it that sNu → u in C(K), for any
compact subinterval K of I.
The convergence is actually even better than the corollary shows,
since it is absolute and uniform (see Exercise 14.2).
Example 14.6. Consider the operator of Example 4.8, which is
−i d
dx considered in L2 (−π, π), with the boundary condition u(−π) =
u(π). This is a regular, selfadjoint realization of (13.1) for n = 1,
J = −i, Q = 0 and W = 1, and it is clear that H∞ = {0}. Hence there
is a complete orthonormal sequence of eigenfunctions in L 2 (−π, π).
The solutions of −iu ′ = λu are the multiples of e iλx , and the bound-
ary condition implies that λ is an integer. We obtain the classical
(complex) Fourier series expansion u(x) = ∞ k=−∞ ûkeikx , where ûk
=
1 π
2π −π u(x)e−ikx dx. According to our results, the series converges in
L 2 (−π, π) for any u ∈ L 2 (−π, π), and uniformly if u is absolutely continuous
with derivative in L 2 (−π, π).