Spectral Theory in Hilbert Space

84 12. INVERSE SPECTRAL THEORY
where A ∈ R, B ≥ 0 and ρ increases (dρ is a positive measure) and
∞
−∞
dρ(t)
t 2 +1 < ∞. The transform space L2 ρ consists of those functions û,
measurable with respect to dρ, for which û 2 ρ = ∞
−∞ |û|2 dρ is finite.
The generalized Fourier transform of u ∈ L 2 (0, b) is
û(t) =
b
0
u(x)ϕ(x, t) dx,
converging in L 2 ρ, and with inverse given by
u(x) =
∞
−∞
û(t)ϕ(x, t) dρ(t),
which converges in L 2 (0, b). Furthermore, u = ûρ (Parseval) and
u ∈ D(T ) if and only if û and tû(t) ∈ L 2 (0, b), and then T u(t) = tû(t).
In the case when one has a discrete spectrum, which means that the
spectrum consists of isolated eigenvalues (of finite multiplicity), the
function ρ is a step function, with a step at each eigenvalue. Suppose
the eigenvalues are λ1, λ2, . . . and that the size of the step is cj =
limε↓0(ρ(λj + ε) − ρ(λj − ε)). Then the inverse transform takes the
form
u(x) =
∞
û(λj)ϕ(x, λj)cj,
j=1
where û(λj) = 〈u, ϕ(·, λj〉. For u = ϕ(·, λj) the expansion becomes
ϕ(x, λj) = ϕ(·, λj) 2 ϕ(x, λj)cj. It follows that cj = ϕ(·, λj) −2 . Note
that ϕ(·, λj) is an eigenfunction associated with λj, so the jump cj of ρ
at λj is the so called normalization constant for the eigenfunction. The
name comes from the fact that a normalized eigenfunction is given by
ej = √ cj ϕ(·, λj). We have shown the following proposition.
Proposition 12.1. In the case of a discrete spectrum knowledge
of the spectral function ρ is equivalent to knowing the eigenvalues and
the corresponding normalization constants.
1. Asymptotics of the m-function
In order to discuss some results in inverse spectral theory we need
a few results on the asymptotic behavior of the m-function for large λ.
We denote by mα(λ) the m-function for the boundary condition (12.2)
and some fixed boundary condition at b. The following theorem is a
simplified version of a result from [3].
Theorem 12.2. We have
m0(λ) = − √ −λ + o(|λ| 1/2 )