Spectral Theory in Hilbert Space

APPENDIX C
Linear first order systems
In this appendix we will prove some standard results about linear
first order systems of differential equations which are used in the text.
We will prove no more than we actually need, although the theorems
have easy generalizations to non-linear equations, more complicated parameter
dependence, etc. The first result is the standard existence and
uniqueness theorem, Theorem 13.1, which also implies Theorem 10.1.
Theorem. 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, locally integrable in I. Assume further that c ∈ I and
C is an n × 1 matrix. Then the initial value problem
(C.1)
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.
Corollaries 13.2 and 10.2 are immediate consequences of the theorem.
Corollary. Let A and I be as in the previous theorem. Then the
set of solutions to u ′ = Au in I is an n-dimensional linear space.
Proof. It is clear that any linear combination of solutions is also
a solution, so the set of solutions is a linear space. We must show
that it has dimension n. Let uk solve the initial value problem with
uk(c) equal to the k:th column of the n × n unit matrix. If u is any
solution of the equation, and the components of u(c) are x1, . . . , xn,
then the function x1u1+· · ·+xnun is also a solution with the same initial
data. It therefore coincides with u, and it is clear that no other linear
combination of u1, . . . , un has the same initial data as u. It follows
that u1, . . . , un is a basis for the space of solutions, which therefore is
n-dimensional.
Finally we shall prove Theorem 15.1.
Theorem. A solution u(x, λ) of Ju ′ + Qu = λW u with initial
data independent of λ is an entire function of λ, locally uniformly with
respect to x.
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