Expectation values of unbounded operator

[Ravi Mohan]

I am reading an intriguing article on rigged Hilbert space
http://arxiv.org/abs/quant-ph/0502053

On page 8, the author describes the need for rigged Hilbert space. For that, he considers an unbounded operator $A$, corresponding to some observable in space of square integrable functions $\mathcal{H}$, with the domain $\mathcal{D}(A)$. The author states that in general, $\mathcal{D}(A)$ does not remain invariant under the action of $A$.

Now the author claims that such non-invariance makes expectation values ill-defined on the whole Hilbert space $\mathcal{H}$.

I am not able to understand the claim.

Let us consider $\phi\in\mathcal{D}(A)$. Due to invariance, $\psi=A\phi$ may not belong to $\mathcal{D}(A)$, but it remains in $\mathcal{H}$. Thus the expectation value $(\phi,A\phi)=(\phi,\psi)$ should be well defined (or am I doing something wrong?).

 

[DrDu]

You are right. I suppose he means expectation values of $A^2$ as appearing in the next equation.

 

[Ravi Mohan]

Thank you for clearing that.