80 From Classical Mechanics to Quantum Field Theory. A Tutorial
even ifψ∈L^2 (R,dx). To fix the problem, we can simply restrict the domain of
Xto the linear subspace ofL^2 (R,dx)
D(X):=
{
ψ∈L^2 (R,dx)∣∣
∣∣
∫
R|xψ(x)|^2 dx <+∞}
. (2.15)
Though it holds
〈Xψ,φ〉=〈ψ,Xφ〉 for allψ,φ∈D(X), (2.16)we cannot say thatXis selfadjoint simply because we have not yet given the
definition of adjoint operator of an operator defined in a non-maximal domain in
an infinite dimensional Hilbert space. In this general case, the identity (2.2) does
not define a (unique) operatorX†without further technical requirements. We
just say here, to comfort the reader, thatXis truly selfadjoint with respect to a
general definition we shall give in the next section, when its domain is (2.15).
Like (2.3) in the finite dimensional case, the identity (2.16) implies that all
eigenvalues ofXmust be real if any. Unfortunately, for every fixedx 0 ∈Rthere is
noψ∈L^2 (R,dx) withXψ=x 0 ψandψ= 0. (A functionψsatisfyingXψ=x 0 ψ
must also satisfyψ(x)=0ifx=x 0 , due to the definition ofX. Henceψ=0,as
an element ofL^2 (R,dx)justbecause{x 0 }has zero Lebesgue measure!) All that
seems to prevent the existenceof a spectral decomposition ofXlike the one in
(2.4), sinceXdoes not admit eigenvectors inL^2 (R,dx)(anda fortioriinD(X)).
The definition ofP suffers from similar troubles. The domain ofP cannot be
the wholeL^2 (R,dx) but should be restricted to a subset of (weakly) differentiable
functions with derivative inL^2 (R,dx). The simplest definition is
D(P):=
{
ψ∈L^2 (R,dx)∣∣
∣∣
∣∃w-dψ(x)
dx,
∫
R∣∣
∣∣w-dψ(x)
dx∣∣
∣∣
2
dx <+∞}
. (2.17)
Above w-dψdx(x) denotes theweak derivative ofψ^2. As a matter of factD(P)
coincides with theSobolev spaceH^1 (R).
Again, without a precise definition of adjoint operator in an infinite dimensional
Hilbert space (with non-maximal domain) we cannot say anything more precise
about the selfadjointness ofPwith that domain. We say however thatP turns
out to be selfadjoint with respect to the general definition we shall give in the next
section provided its domain is (2.17).
∫^2 f:R→C, defined up to zero-measure set, is the weak derivative ofg∈L^2 (R,dx)ifitholds
Rgdhdxdx=−
∫
Rfhdxfor everyh∈C 0 ∞(R). Ifgis differentiable, its standard derivative coincide
with the weak one.