Spectral Theory in Hilbert Space

60 10. BOUNDARY CONDITIONS
the same way. This will of course include the more general Sturm-
Liouville equation (10.1).
We shall study (10.3) in the Hilbert space L 2 (I) where I is an
interval and the function q is real-valued and locally integrable in I,
i.e., integrable on every compact subinterval of I. In L 2 (I) the scalar
product is 〈u, v〉 =
I uv.
Before we begin we need to quote a few standard facts about Sturm-
Liouville equations. Basic for what follows is the following existence
and uniqueness theorem.
Theorem 10.1. Suppose q is locally integrable in an interval I and
that c ∈ I. Then, for any locally integrable function f and arbitrary
complex constants A, B and λ the initial value problem
−u ′′ + qu = λu + f in I,
u(c) = A, u ′ (c) = B
has a unique, continuously differentiable solution u with locally absolutely
continuous derivative defined in I. If A, B are independent of λ
the solution u(x, λ) and its x-derivative will be entire functions of λ,
locally uniformly in x.
We shall use this only if f is actually locally square integrable. The
theorem has the following immediate consequence.
Corollary 10.2. Let q, λ and I be as in Theorem 10.1. Then the
set of solutions to −u ′′ + qu = λu in I is a 2-dimensional linear space.
If one rewrites −u ′′ + qu = λu + f as a first order system according
to the prescription before (10.1), then Theorem 10.1 and Corollary 10.2
become special cases of the theorems for first order systems given in
Appendix C.
In order to get a spectral theory for (10.3) we need to define a minimal
operator, show that it is densely defined and symmetric, calculate
its adjoint and find the selfadjoint restrictions of the adjoint.
We define Tc to be the operator u ↦→ −u ′′ +qu with domain consisting
of those continuously differentiable functions u which have compact
support, i.e., they are zero outside some compact subinterval of the interior
of I, and which are such that u ′ is locally absolutely continuous
with −u ′′ + qu ∈ L 2 (I).
We will show that Tc is densely defined and symmetric and then
calculate its adjoint, but first need some preparation. If u, v are differentiable
functions we define [u, v] = u(x)v ′ (x) − u ′ (x)v(x). This is
called the Wronskian of u and v. It is clear that [u, v] = −[v, u], in
particular [u, u] = 0. The following elementary fact is very important.
Proposition 10.3. If u and v are linearly independent solutions of
−v ′′ + qv = λv on I, then the Wronskian [u, v] is a non-zero constant
on I.