100 From Classical Mechanics to Quantum Field Theory. A Tutorial
BothFandF−preserve the scalar product
〈Ff,Fg〉=〈f,g〉, 〈F−f,F−g〉=〈f,g〉, ∀f,g∈S(Rn)and thus they also preserve the norm of L^2 (Rn,dnx), in particular, ||F|| =
||F−||= 1. As a consequence of Proposition 2.2.11, using the fact thatS(Rn)
is dense inL^2 (Rn,dnx), one easily proves thatFandF−uniquely continuously
extend to unitary operators, respectively,Fˆ:L^2 (Rn,dnx)→L^2 (Rn,dnk)and
Fˆ−:L^2 (Rn,dnk)→L^2 (Rn,dnx) such thatFˆ†=Fˆ−^1 =Fˆ−.ThemapFˆis the
Fourier-Plancherel(unitary)operator.
2.2.3 Criteria for (essential) selfadjointness
Let us briefly introduce, without proofs (see [ 5 ]), some commonly used math-
ematical tecnology to study (essential) selfadjointness of symmetric operators.
IfAis a densely defined symmetric operator in the complex Hilbert spaceH,
define the deficiency indices, n± := dimH± (cardinal numbers in general)
whereH± are the (closed) subspaces of the solutions of (A†±iI)x± =0[8;
5; 9].
Proposition 2.2.33.IfAis a densely defined symmetric operator in the complex
Hilbert spaceHthe following holds.
(a)Ais essentially selfadjoint (thus it admits an unique selfadjoint extension)
ifn±=0, that isH±={ 0 }.
(b)Aadmits selfadjoint extensions if and only ifn+=n−and these extension
are labelled by means ofn+parameters.Remark 2.2.34. IfA is symmetric, an easy sufficient condition, due to von
Neumann, forn+=n−is thatCA⊂ACwhereC:H→His aconjugation
that is an isometric surjectiveantilinear^7 map withCC=I.
Indeed, using the definition of A†andD(A†)and observing that (from the
polarization identity 2.26)〈Cy|Cx〉=〈y|x〉, the condition
CA⊂AC implies the condition CA†⊂A†C.ThereforeA†x=±ixif and only ifA†Cx=C(±ix)=∓iCx.SinceCpreserves
normality and norm of vectors, we conclude thatn+=n−.
Taking Cas the standard conjugation of functions inL^2 (Rn,dnx),thisresult
proves in particular that all operators in QM of theSch ̈ordingerform as (2.24)
admit selfadjoint extensions when defined on dense domains.
(^7) In other wordsC(αx+βy)=αCx+βCyifα, β∈Candx, y∈H.