Spectral Theory in Hilbert Space

2. UNIQUENESS THEOREMS 89
the bound is valid in the entire domain bounded by the rays. It follows
that if z is in this domain, then |f(z)| ≤ Meε|z|γ , and letting ε → 0 we
obtain the desired result.
Proof of Theorem 12.4. According to the Nevanlinna representation
formula for m and ˜m their difference is constant = C, since
the linear term Bλ is always absent by the asymptotic formulas of
Theorem 12.2. In particular, since Dirichlet m-functions are always
unbounded near ∞ on a non-real ray and all others are bounded, we
must have either α = ˜α or α = 0 = ˜α if dρ = d˜ρ. Thus, according to
Corollary 12.7, the difference ˜ϕ(x, λ)ψ(x, λ) − ϕ(x, λ) ˜ ψ(x, λ) tends to
0 as λ → ∞ along a non-real ray. This difference is
˜ϕ(x, λ)θ(x, λ) − ϕ(x, λ) ˜ θ(x, λ) + Cϕ(x, λ) ˜ϕ(x, λ),
which is an entire function of λ tending to 0 along non-real rays, and it
may be bounded by a multiple of eB|λ|1/2 for some constant B according
to (12.4). By Theorem 12.8 such a function is bounded in the entire
plane, and therefore constant by Liouville’s theorem, hence identically
0 since the limit is zero along the rays. It follows that
θ(x, λ)/ϕ(x, λ) = ˜ θ(x, λ)/ ˜ϕ(x, λ) + C
for all x, λ. Differentiating with respect to x, using the fact that
θ ′ ϕ − θϕ ′ = 1, we obtain ϕ 2 (x, λ) = ˜ϕ 2 (x, λ). Taking the logarith-
mic derivative of this we obtain ϕ′ (x,λ)
ϕ(x,λ) = ˜ϕ′ (x,λ)
˜ϕ(x,λ) .
For x = 0 this gives α = ˜α, and thus that m and ˜m are asymptotically
the same. Thus C = 0, so that m = ˜m. Differentiating once more
we obtain ϕ ′′ /ϕ = ˜ϕ ′′ / ˜ϕ which means that q = ˜q on min(b, ˜ b). From
this follows that ϕ = ˜ϕ and θ = ˜ θ, and thus also ψ = ˜ ψ, on min(b, ˜ b).
This implies that b = ˜ b, since otherwise ψ (or ˜ ψ) would satisfy selfadjoint
boundary conditions both at b and ˜ b, so that ψ would be an
eigenfunction to a non-real eigen-value for a selfadjoint operator. Since
ψ = ˜ ψ also the boundary conditions at b = ˜ b (if any) are the same. It
follows that T = ˜ T .
Proof of Theorem 12.5. Our starting point is that if α = ˜α
the functions ˜ϕ(x, λ)ψ(x, λ) and ϕ(x, λ) ˜ ψ(x, λ) tend to 0 as λ → ∞
along a non-real ray. Their difference is
(12.7) ˜ϕ(x, λ)θ(x, λ) − ϕ(x, λ) ˜ θ(x, λ) + (m(λ) − ˜m(λ))ϕ(x, λ) ˜ϕ(x, λ).
Suppose first that α = ˜α and q = ˜q on (0, a). Then the first two
terms cancel on (0, a), so that (m(λ) − ˜m(λ))ϕ(x, λ) ˜ϕ(x, λ) → 0 as
λ → ∞ along non-real rays if x ∈ (0, a). By (12.4) this implies that
(m(λ) − ˜m(λ))e 2(a−ε) Re √ −λ ) → 0 as λ → ∞ along any non-real ray.
Conversely, the estimate for m − ˜m implies first that α = ˜α and
then that for 0 < x < a the last term of (12.7) tends to 0 according
to assumption and (12.4), so that the entire function ˜ϕ(x, λ)θ(x, λ) −