Spectral Theory in Hilbert Space

CHAPTER 12
Inverse spectral theory
In this chapter we continue to study the simple Sturm-Liouville
equation −u ′′ + qu = λu, on an interval with at least one regular
endpoint. Our aim is to give some results on inverse spectral theory,
i.e., questions related to the determination of the equation, in this
case the potential q from spectral data, such as eigenvalues, spectral
measures or similar things. Our object of study is the eigen-value
problem
(12.1)
(12.2)
− u ′′ + qu = λu on [0, b),
u(0) cos α + u ′ (0) sin α = 0.
Here α is an arbitrary, fixed number in [0, π), so that the boundary
condition is an arbitrary separated boundary condition. We assume q ∈
L1 loc [0, b), i.e., q integrable on any interval [0, c] with c ∈ (0, b), so that
0 is a regular endpoint for the equation. The other endpoint b may be
infinite or finite, in the latter case singular or regular. If the deficiency
indices for the equation in L2 (0, b) are (1, 1) the operator corresponding
to (12.1), (12.2) is selfadjoint; if they are (2, 2) a boundary condition
at b is required to obtain a selfadjoint operator. We assume that, if
necessary, a choice of boundary condition at b is made, so that we are
dealing with a self-adjoint operator which we will call T .
If the deficiency indices are (2, 2) we know the spectrum is discrete
(Theorem 11.7), but when the deficiency indices are (1, 1) the spectrum
can be of any type. As in Chapter 11, let ϕ and θ be solutions of (12.1)
satisfying initial conditions
ϕ(0, λ) = − sin α
(12.3)
ϕ ′ (0, λ) = cos α ,
θ(0, λ) = cos α
θ ′ (0, λ) = sin α .
Then Green’s function for T is given by
g(x, ·, λ) = ϕ(min(x, y), λ)ψ(max(x, y), λ)
where ψ(x, λ) = θ(x, λ) + m(λ)ϕ(x, λ) and the Titchmarsh-Weyl mfunction
m(λ) is determined so that ψ satisfies the boundary condition
at b. In particular ψ ∈ L 2 (0, b). Let the Nevanlinna representation of
m be
m(λ) = A + Bλ +
∞
−∞
1 t
−
t − λ t2
dρ(t),
+ 1
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