Spectral Theory in Hilbert Space

CHAPTER 11
Sturm-Liouville equations
The spectral theorem we proved in Chapter 7 is very powerful, but
sometimes its abstract nature is a drawback, and one needs a more explicit
expansion, analogous to Fourier series or Fourier transforms. A
general theorem of this type was proved by von Neumann in 1949, but
it is still of a fairly abstract nature. It can be applied to elliptic partial
differential equations (G˚arding around 1952), but gives more satisfactory
results when applied to ordinary differential equations. How to
do this was described by G˚arding in an appendix to John, Bers and
Schechter: Partial Differential Equations, (1964). A slightly more general
situation was treated in [2]. For Sturm-Liouville equations one
can, however, as easily obtain an expansion theorem directly. We will
do that in this chapter.
As in our proof of the spectral theorem, we will deduce our results
from properties of the resolvent, but now need to have a more explicit
description of the resolvent operator. The first step is to prove that the
resolvent is actually an integral operator. First note that all elements
of D1 are continuously differentiable with locally absolutely continuous
derivative, and according to Lemma 10.8 point evaluations of elements
of D1 (and their derivatives) are locally uniformly bounded linear forms
on D1.
If T is a selfadjoint realization of (10.3) in L 2 (I) its resolvent Rλ is a
bounded operator on L 2 (I) for every λ in the resolvent set. If E denotes
the identity on L 2 (I) we have (T − λ)Rλ = E so that T Rλ = E + λRλ.
Thus T Rλ ≤ 1 + |λ|Rλ. Since Rλu is in the domain of T we
may also view the resolvent as an operator Rλ : L 2 (I) → D1, where
D1 is viewed as a Hilbert space provided with the graph norm, as
on page 65. This operator is bounded since Rλu 2 1 = Rλu 2 +
T Rλu 2 ≤ (Rλ 2 + (|λ|Rλ + 1) 2 )u 2 . It is also clear that the
analyticity of Rλ implies the analyticity of T Rλ = E + λRλ, and therefore
the analyticity of Rλ : L 2 (I) → D1. We obtain the following
theorem.
Theorem 11.1. Suppose I is an interval, and that T is a selfadjoint
realization in L 2 (I) of the equation (10.3). Then the resolvent Rλ of T
may be viewed as a bounded linear map from L 2 (I) to C 1 (K), for any
compact subinterval K of I, which depends analytically on λ ∈ ρ(T ),
in the uniform operator topology. Furthermore, there exists Green’s
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