Spectral Theory in Hilbert Space

88 12. INVERSE SPECTRAL THEORY
every ε > 0 as λ → ∞ along some non-real ray. Conversely, if α = ˜α
and q = ˜q on (0, a), then (m(λ) − ˜m(λ))e 2(a−ε) Re √ −λ → 0 for every
ε > 0 as λ → ∞ along any non-real ray.
We will prove both theorems by the same method, the crucial point
of which is the following lemma.
Lemma 12.6. For any fixed x ∈ (0, b) holds ϕ(x, λ)ψ(x, λ) → 0 as
λ → ∞ along a non-real ray.
Note that ϕ(x, λ)ψ(x, λ) is Green’s function on the diagonal x = y.
We shall postpone the proof a moment and see how the theorem follows
from it. We first have a corollary.
Corollary 12.7. Suppose α = ˜α = 0 or α = 0 = ˜α. Then both
˜ϕ(x, λ)ψ(x, λ) and ϕ(x, λ) ˜ ψ(x, λ) tend to 0 as λ → ∞ along a non-real
ray, locally uniformly in x.
Proof. Clearly (12.4) implies that for fixed x and ˜α = 0 we have
ϕ(x, λ)/ ˜ϕ(x, λ) → sin α/ sin ˜α as λ → ∞
along a non-real ray. If α = ˜α = 0 we instead obtain the limit 1, so the
corollary follows from Lemma 12.6.
We shall also need a standard theorem from complex analysis, which
is a slight elaboration of the maximum principle.
Theorem 12.8 (Phragmén-Lindelöf). Suppose f is analytic in a
closed sector bounded by two rays from the origin, that it is bounded on
the rays, and that |f(z)| ≤ AeB|z|1/2 in the sector, for some constants
A and B. Then f is bounded in the sector.
This is just one of the simplest versions of a general class of theorems,
which are all known under the names of Phragmén and Lindelöf.
Proofs are given in many textbooks on complex analysis, but for the
reader’s convenience we also give a proof here.
Proof. We may without loss of generality assume that the rays
are given by the angles ±β. Let ε > 0 and F (z) = e−εzγ f(z), where
1/2 < γ < π/(2β) and the branch of z γ is chosen to be positive real for
positive real z. Now, for z = re ±iβ we have |F (z)| = e −εrγ cos(βγ) |f(z)|,
where cos(βγ) > 0. Let M be a bound for f on the rays. Then we
have |F (z)| ≤ M on the rays.
For z = Re iδ with |δ| ≤ β we have
|F (z)| ≤ A exp(BR 1/2 − εR γ cos(βγ))
which tends to 0 as R → ∞. Thus, on all circular sectors bounded
by the rays we have |F (z)| ≤ M on the boundary if the radius R is
sufficiently large. By the maximum principle this also holds in the
interior of the circular sector. Since R can be chosen arbitrarily large,