Spectral Theory in Hilbert Space

EXERCISES FOR CHAPTER 15 115
boundary conditions (cf. Chapters 10 and 13). This will be a condition
at a, and if the boundary form does not vanish at b, also a condition
at b. Choose the point c = 0 and the fundamental matrix F such
that its first column satisfies the boundary condition at a. Show that
M(λ) =
m(λ) 1
2
1
2
0
, where the Titchmarsh-Weyl function m(λ) is a
scalar-valued Nevanlinna function.
ϕ θ
Now write F = −pϕ ′ −pθ ′
. Show that there is a scalar Green’s
function for the operator given by
ϕ(x, λ)ψ(y, λ), x < y,
g(x, y, λ) =
ψ(x, λ)ϕ(y, λ), y < x,
where ψ(x, λ) = θ(x, λ) + m(λ)ϕ(x, λ), with the property that the
solution of −(pu ′ ) ′ + qu = λwu + wv which is in L 2 w and satisfies the
boundary conditions is given by u(x) = Rλv(x) = ∞
g(x, y, λ)v(y) dy.
0
Show also that the spectral matrix P =
ρ 0
0 0 , where the spectral
function ρ is the function in the representation (6.1) for the function
m(λ), and that
b
Im m(λ) = Im λ |ψ(x, λ)| 2 dx.
Finally show that the generalized Fourier transform of ψ is always given
by ˆ ψ(t, λ) = 1/(t − λ).
Thus the spectral theory for the general Sturm-Liouville equation
has precisely the same basic features as for the simple case treated in
Chapter 11.
a