Spectral Theory in Hilbert Space

CHAPTER 13
First order systems
We shall here study the spectral theory of general first order system
(13.1) 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. As we shall see,
these properties ensure the proper symmetry of the differential expression.
The functions u and v are n×1 matrix-valued on I. In the special
case when n is even and J =
0 I
−I 0 , I being the unit matrix of order
n/2, systems of the form (13.1) are usually called Hamiltonian systems.
The following existence and uniqueness theorem is fundamental
Theorem 13.1. Suppose A is an n×n matrix-valued function with
locally integrable entries in an interval I, and that B is an n×1 matrixvalued
function, also locally integrable in I. Assume further that c ∈ I
and C is an n × 1 matrix. Then the initial value problem
u ′ + Au = B in I,
u(c) = C,
has a unique n × 1 matrix-valued solution u with locally absolutely continuous
entries defined in I.
The theorem has the following immediate consequence.
Corollary 13.2. The set of solutions to u ′ + Au = 0 in I is an
n-dimensional linear space.
Proofs for Theorem 13.1 and Corollary 13.2 are given in Appendix
C. We will apply them for A = J −1 (Q − λW ), where λ ∈ C, and
B = J −1W v.
We shall study (13.1) in the Hilbert space L 2 W
of equivalence classes
of n × 1 matrix-valued Lebesgue measurable functions u for which
u∗ W u is integrable over I. In this space the scalar product is 〈u, u〉 =
I v∗W u. Two functions u and ũ are considered equivalent if the integral
I (u − ũ)∗W (u − ũ) = 0. Note that this means that they can be
very different pointwise. For example, in the case of the system equivalent
to (10.1) the second component of an element of L2 W is completely
undetermined.
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