ABSTRACT
Functionals, as a special form of mapping between vector spaces and scalar fields, play a very important role in scattering theory. For this reason they deserve a concise analysis which we shall present here. For a given finite or infinite-dimensional abstract space XF defined on the scalar field F we shall introduce a linear functional. This is a linear form f : XF −→ F of the type of a linear operator, which performs correspondence between vectors from XF and scalars from F viz
f(ψ1 + ψ2) = f(ψ1) + f(ψ2) ∀ψ1, ψ2 ∈ XF (4.1)
f(λψ) = λf(ψ) ∀ψ ∈ XF ∀λ ∈ F (4.2)
or in a more compact form
f(λψ1 + µψ2) = λf(ψ1) + µf(ψ2)
∀ψ1, ψ2 ∈ XF ∀λ, µ ∈ F
. (4.3) Of course, the definition of the linear functional (4.3) can be automatically extended to an arbitrary number of independent variables (n < ∞) in the evident form
f(λ1ψ1 + λ2ψ2 + · · ·+ λnψn) = λ1f(ψ1) + λ2f(ψ2) + · · ·+ λnf(ψn) (4.4)
for ∀ψk ∈ XF and ∀λk ∈ F (1 ≤ k ≤ n). The functional f(ψ) is bounded if it holds
|f(ψ)| ≤ c||ψ|| ∀ψ ∈ XF (4.5)
where c is a finite positive real number (c < ∞). Here, it is understood that the norm ||ψ|| is introduced via the following axioms that are related to the
ION-ATOM
scalar product
||ψ|| ≥ 0 ||ψ|| = 0 ⇐⇒ ψ = ∅
||µψ|| = |µ| · ||ψ|| ||ψ + φ|| ≤ ||ψ||+ ||φ||
∀ψ, φ ∈ XF µ ∈ F
. (4.6) An obvious example of a functional is the usual scalar or inner asymmetric product (or Hermitean-symmetric) 〈φ|ψ〉 defined by means of the standard five axioms
• Axiom 1: 〈φ|αψ1 + βψ2〉 = α〈φ|ψ1〉+ β〈φ|ψ2〉 • Axiom 2: 〈αφ1 + βφ2, ψ〉 = α∗〈φ1|ψ〉+ β∗〈φ2|ψ〉 • Axiom 3: 〈φ|ψ〉 = 〈ψ|φ〉∗ • Axiom 4: 〈ψ|ψ〉 ≥ 0 • Axiom 5: 〈ψ|ψ〉 = 0 ⇐⇒ ψ = ∅ (4.7)
where φ, ψ, ψ1, ψ2 ∈ H , α, β ∈ C and, as usual, the star superscript denotes complex conjugation. It then follows that the scalar product 〈φ|ψ〉 is antilinear and linear functional relative to the first (φ) and the second (ψ) term, respectively. On the other hand, a linear functional f is, by definition, in the class of linear operators. Therefore, it is clear that the set of all the images of such an operator i.e. the range Rf will be conveniently determined if we know how f acts on the elements of a given basis bβ ≡ {β1, β2, ..., βn}. This is indeed the case, as can be shown by referring to the expansion theorem in a finite-dimensional space
ψ = n∑
ψβkβk ψ β k ∈ F (4.8)
where ψβk ≡ ψ(βk) are the expansion coefficients i.e. the coordinates of the vector ψ with respect to the basis bβ . Employing (4.4) we arrive at the following formula
[f ]ψ = [f ] n∑
ψβkβk = n∑
[f ]ψβkβk = n∑
ψβk [f ]βk
∴ [f ]ψ = n∑
ψβk [f ]βk (4.9)
where the square brackets around f as indicated by [f ] symbolize the action of the functional f in the analogy with the standard notion of a function
[f ]ψ ≡ f(ψ). (4.10)
In this way, the expression (4.8) becomes
f(ψ) = n∑
ψβk f(βk). (4.11)
This shows that if the set bβ = {βk}nk=1 is a basis in the space XnF , then the general and explicit functional f with the feature (4.4) is given by the linear combination
f(ψ) = f(ψβ1 β1 + ψ β 2 β2 + · · ·+ ψβnβn)
= ψβ1 f(β1) + ψ β 2 f(β2) + · · ·+ ψβnf(βn). (4.12)
Notice the concreteness of the expression (4.12) with respect to the general property (4.4). In the expression (4.12) the quantities β1, β2, ..., βn are the elements of the basis bβ ⊆ XF and ψβ1 , ψβ2 , ..., ψβn are the expansion coefficients, whereas ψ1, ψ2, ..., ψn are arbitrary vectors from XnF and λ1, λ2, ..., λn ∈ F . Further, it clearly follows from (4.12) that the image f(ψ) of the functional of any vector ψ ∈ Df is fully determined by the knowledge of the elements that are obtained from the action of the same functional f(βk) (1 ≤ k ≤ n) on the basis functions bβ . The obtained general form (4.12) of the linear functional formally resembles the usual way of presenting the scalar product
〈φ|ψ〉 = ∑ k
φe∗k ψ e k. (4.13)
Nevertheless, it must be stressed that the derivation of (4.12) is carried out for a general vector space XnF which does not need to be unitary. However, in the special case of unitary spaces YnF that are of great importance for scattering theory, we shall show that (4.12) indeed coincides with the standard expression (4.13) of the scalar product 〈φ|ψ〉 which is, as mentioned before, a linear functional with respect to its second term ψ. This constitutes the content of the Ries-Freshe theorem.
