From Classical Mechanics to Quantum Field Theory

114 From Classical Mechanics to Quantum Field Theory. A Tutorial

2.2.6 Spectral decomposition andrepresentation theorems

We are in a position to state the fundamental result of the spectral theory of
selfadjoint operators, which extend the expansion (2.4) to an integral formula valid
also in the infinite dimensional case, and where the set of eigenvalues is replaced
by the full spectrum of the selfadjoint operator.

∫ To state the theorem, we preventively notice that (2.49) implies that
f(λ)dP(λ)isselfadjointiffis real: The idea of the theorem is to prove that
every selfadjoint operator can be written this way for a specificfand with respect
to a PVM onRassociated with the operator itself.

Notation 2.2.54. From now on,B(T) denotes the Borelσ-algebra on the topo-
logical spaceT.

Theorem 2.2.55(Spectral decomposition theorem for selfadjoint operators).Let
Abe a selfadjoint operator in the complex Hilbert spaceH.

(a)There is a unique PVM,P(A):B(R)→L(H), such that

A=

∫

R

λdP(A)(λ).

In particularD(A)=Δid,whereid:Rλ→λ.

(b)Defining thesupportofP(A),supp(P(A)), as the complement inRof the

union of all open setsO⊂CwithPO(A)=0it results

supp(P(A))=σ(A)

so that

P(A)(E)=P(A)(E∩σ(A)), ∀E∈B(R). (2.55)

(c)λ∈σp(A)if and only ifP(A)({λ})=0, this happens in particular ifλis

an isolated point ofσ(A).

(d)λ∈σc(A)if and only ifP(A)({λ})=0butP(A)(E)=0ifEλis an

open set ofR.

The proof can be found, e.g., in[8; 5; 9; 6].

Remark 2.2.56.Theorem 2.2.55 is a particular case, of a more general theorem
(see[8; 5; 6]and especially[ 9 ])validwhenAis a (densely defined closed) normal
operator. The general statement is identical, it is sufficient to replace everywhere
RforC. A particular case is the one ofAunitary. In this case, the statement
can be rephrased replacing everywhereRforTsince it includes the spectrum ofA
in this case ((d) remark 2.2.42).