Spectral Theory in Hilbert Space

74 11. STURM-LIOUVILLE EQUATIONS
Taking the imaginary part of this and using the fact that ψ satisfies
the boundary condition at b so that Im(ψ ′ ψ) → 0 at b we obtain
(11.2) 0 ≤
b
a
|ψ(·, λ)| 2 =
Im m(λ)
Im λ ,
since a simple calculation shows that Im(ψ ′ (a, λ)ψ(a, λ)) = Im m(λ). It
follows that m has all the required properties of a Nevanlinna function.
Before we proceed, we note the following corollary, which completes
our results for the case of a discrete spectrum.
Corollary 11.7. Suppose both endpoints of I are either regular or
in the limit circle condition. Then for any selfadjoint realization T the
resolvent is compact. Thus there is a complete orthonormal sequence
of eigenfunctions.
Proof. By Theorem 8.4 it is enough to prove the corollary when
T is given by separated boundary conditions. But as in the proof
of Theorem 11.6 we can then find non-trivial solutions ψ−(·, λ) and
ψ+(·, λ) of −v ′′ + qv = λv satisfying the boundary conditions to the
left and right respectively. If Im λ = 0 the solutions ψ±(·, λ) can not be
linearly dependent, since this would give a non-real eigenvalue for T .
We may therefore assume [ψ+, ψ−] = 1 by multiplying ψ−, if necessary,
by a constant. But then it is seen that ψ−(min(x, y), λ)ψ+(max(x, y), λ)
is Green’s function for T just as in the proof of Theorem 11.6.
It is clear that the assumption implies that deficiency indices equal
2, so that ψ± are in L 2 (I). However, an easy calculation now shows
that
I×I
|g(x, y, λ)| 2 dxdy ≤ 2ψ− 2 ψ+ 2 < ∞.
Thus, according to Theorem 8.7, the resolvent is a Hilbert-Schmidt
operator, so that it is compact.
If at least one of the interval endpoints is singular and in the limit
point condition the resolvent may not be compact (but it can be!). In
this case the only boundary condition will be a separated boundary
condition at the other endpoint, unless this is also in the limit point
condition, when no boundary conditions at all are required.
We now return to the situation treated in Theorem 11.6 when I =
[a, b) with a regular, and T is given by the separated condition (10.7)
at a, and another separated condition at b if needed. Since the mcoefficient
is a Nevanlinna function there is a unique increasing and
left-continuous matrix-valued function ρ with ρ(0) = 0 and unique real