Master Dissertation

Chapter 1
Introductory Theory
1.1 Notation
I will mostly adopt the notation used in [4], but for convenience there will be
some differences in the notation of a distribution. Whereas we in [4] used to
write a distribution u acting on a test-function φ in the functional manner
u(φ) I will in this project use the “inner product” notation 〈u, φ〉.
Also note that I will use the definition of the fourier transform found in
[4] rather than the one found in [6]. This will only result in some differences
in the factors.
I will occasionally use Einstein notation along with the convention that
indices from the Greek alphabet take the values 0, 1, 2, 3 and indices from
the Latin alphabet take the values 1, 2, 3.
By a t-dependent operator O(t) : E → F we mean an indexed family of
operators {O(t)}t∈R all from E to F .
Given a t-dependent operator O(t) : E → F . We write O = s- limt→∞ O(t)
if it converges in the strong operator topology to an operator O. That is, if
Oφ = lim
t→∞ (O(t)φ) for all φ ∈ E.
Beware of the many meanings of the symbol δ. Usually it means the
Dirac delta-distribution, but it is also used as a small real, the Kronecker
symbol δi,j and the diagonal map. It should be transparent from the context
which meaning the symbol has.
For convenience I have listed some general notation on the next page:
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