How would we treat an operator of the form $ \hat{\mathbf{A}} \propto \hat{\mathbf{r}} \cdot \hat{{\mathbf{p}}} $ ?
Would it have eigenstates that are also eigenfunctions of position and/or momentum?
EDIT:
I am interested in knowing if there exists a common set of eigenstates of both r and p, which normally would not commute
If you keep the order when expanding the inner product you get terms of the form $x^ip_i$ which are not self-adjoint, due to the fact that $x^i$ doesn't commute with $p_i$. Hence the spectrum of such operator won't lie in $\mathbb R$, nor there is any hope in finding simultaneous eigenstates for both $x^i$ and $p_i$.
Expression like this are usually quantised by symmetrization: some people would simply write $$\widehat{\mathbf r\cdot\mathbf p} = \frac12(\hat{\mathbf r}\cdot\hat{\mathbf p} + \hat{\mathbf p}\cdot\hat{\mathbf r})$$ which is now self-adjoint.
