Spectral Theory in Hilbert Space

CHAPTER 10
Boundary conditions
A simple example of a formally symmetric differential equation is
given by the general Sturm-Liouville equation
(10.1) −(pu ′ ) ′ + qu = wf.
Here the coefficients p, q and w are given real-valued functions in a
given interval I. Standard existence and uniqueness theorems for the
initial value problem are valid if 1/p, q and w are all in Lloc(I). There
are (at least) two Hermitian forms naturally associated with this equa-
tion, namely
I (pu′ v ′ + quv) and
I
uvw. Under appropriate positivity
conditions either of these forms is a suitable choice of scalar product for
a Hilbert space in which to study (10.1). The corresponding problems
are then called left definite and right definite respectively. We will not
discuss left definite problems in these lectures.
If p is not differentiable it is most convenient to interpret (10.1) as
a first order system
0 1
U
−1 0
′
q 0
+
0 − 1
w 0
U = V .
p 0 0
u
This equation becomes equivalent to (10.1) on setting U =
−pu ′
and letting the first component of V be f. It is a special case of a
fairly general first order system
(10.2) Ju ′ + Qu = W v
where J is a constant n × n matrix which is invertible and skew-
Hermitian (i.e., J ∗ = −J) and the coefficients Q and W are n × n
matrix-valued functions which are locally integrable on I. In addition
Q is assumed Hermitian and W positive semi-definite, and u, v are
n × 1 matrix-valued functions. We shall study such systems in Chapters
13–15.
Here we shall just deal with the case of the simple inhomogeneous
scalar Sturm-Liouville equation
(10.3) −u ′′ + qu = λu + f
and the corresponding homogeneous eigenvalue problem −u ′′ +qu = λu.
The latter is often called the one-dimensional Schrödinger equation. In
later chapters we shall then see that with minor additional technical
complications we may deal with the first order system (10.2) in much
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