Spectral Theory in Hilbert Space

38 6. NEVANLINNA FUNCTIONS
Proof. By absolute convergence we may change the order of integration
in the last integral. The inner integral is then easily calculated
to be
1
(arctan((x − t)/ε) − arctan((y − t)/ε)).
π
This is bounded by 1, and also by a constant multiple of 1/t2 if ε is
bounded (verify this!). Furthermore it converges pointwise to 0 outside
[y, x], and to 1 in (y, x) (and to 1 for t = x and t = y). The theorem
2
follows by dominated convergence.
Proof of Theorem 6.1. The uniqueness of ρ follows immediately
on applying the Stieltjes inversion formula to the imaginary part
of (6.1) for λ = s + iε.
We obtain (6.1) from the Riesz-Herglotz theorem by a change of
variable. The mapping z = 1+iλ maps the upper half plane bijectively
1−iλ
to the unit disk, so G(z) = −iF (λ) is defined for z in the unit disk
and has positive real part. Applying Theorem 6.3 we obtain, after
simplification,
F (λ) = Re F (i) + 1
2π
π
−π
1 + λ tan(θ/2)
tan(θ/2) − λ
dσ(θ) .
Setting t = tan(θ/2) maps the open interval (−π, π) onto the real axis.
For θ = ±π the integrand equals λ, so any mass of σ at ±π gives rise
to a term βλ with β ≥ 0. After the change of variable we get
F (λ) = α + βλ +
∞
−∞
1 + tλ
t − λ
dτ(t) ,
where we have set α = Re F (i) and τ(t) = σ(θ)/(2π). Since
1 + tλ
t − λ =
1 t
−
t − λ 1 + t2
(1 + t 2 )
we now obtain (6.1) by setting ρ(t) = t
0 (1 + s2 ) dτ(s).
It remains to show the uniqueness of α and β. However, setting
λ = i, it is clear that α = Re F (i), and since we already know that ρ
is unique, so is β.
Actually one can calculate β directly from F since by dominated
convergence Im F (iν)/ν → β as ν → ∞. It is usual to refer to β as the
‘mass at infinity’, an expression explained by our proof. Note, however,
that it is the mass of τ at infinity and not that of ρ!