Mathematical Foundations of Quantum Mechanics 105(2) the first identity in (2.41) is a well-known recurrence relation of the Hilbert ba-
sis ofL^2 (R,dx) made of Schwartz’ functions known asHermite functions{Hn}n∈N,
andψ 0 (x)=H 0 (x).
Exploiting Nelson’s criterium, we conclude that the symmetric operatorH 0 is
essentially selfadjoint inD(H 0 )=S(R)andH:=H 0 =H† 0 , becauseH 0 admits a
Hilbert basis of eigenvectors with corresponding eigenvaluesω(n+^12 ).
2.2.4 Spectrumofanoperator
Our goal is to extend (2.4) to a formula valid in the infinite dimensional case. As
we shall see shortly, passing to the infinite dimensional case, the sum is replaced by
an integral andσ(A) must be enlarged with respect to the pure set of eigenvalues
ofA. This is because, as already noticed inthe first section, there are operators
which should be decomposed with the prescription (2.4) but they do not have
eigenvalues, though they play a crucial role in QM.
Notation 2.2.40.IfA:D(A)→His injective,A−^1 indicates its inverse when the
co-domain ofAis restricted toRan(A). In other words,A−^1 :Ran(A)→D(A).
The definition ofspectrumof the operatorA:D(A)→Hextends the notion of set
of eigenvalues. The eigenvalues ofAare the numbersλ∈Csuch that (A−λI)−^1
does not exist. When passing infinite dimensions, topological issues take place.
As a matter of fact, even if (A−λI)−^1 exists, it may be bounded or unbounded
and its domainRan(A−λI) may or may not be dense. These features permit us
to define a suitable extension of the notion of a set of eigenvalues.
Definition 2.2.41.LetAbe an operator in the complex Hilbert spaceH.The
resolvent setofAis the subset ofC,
ρ(A):={λ∈C|(A−λI) is injective,Ran(A−λI)=H,(A−λI)−^1 is bounded}
ThespectrumofAis the complementσ(A):=C\ρ(A) and it is given by the
union of the following pairwise disjoint three parts:
(i) thepoint spectrum,σp(A), whereA−λInot injective (σp(A)istheset
ofeigenvaluesofA);
(ii) the continuous spectrum, σc(A), where A − λI injective,
Ran(A−λI)=Hand (A−λI)−^1 not bounded;
(iii) the residual spectrum, σr(A), where A − λI injective and
Ran(A−λI)=H.