I added links to those articles which are currently open access (only two more, sorry)

I think the problem with this subject is, that there are many different attempts to give a rigorous framework for rigged Hilbert spaces in physics. I don't know if there is a generally accepted useful version. Hence it's not surprising that there are now good free online resources about this subject.

When I collected the references I was interested to have as much rigor as possible. If I remember correctly all of those are written in a usual mathematical style. In Madrid's thesis there are many examples concerning quantum mechanics. For a more general approach I would look at M.Gadella and F.Gómez. A unified mathematical formalism for the dirac formulation of quantum mechanics. Foundations of Physics, 32:815--869, 2002 In this paper they tried to unify some (perhaps most) versions of rigorous frameworks for rigged Hilbert space in view of quantum physics.

@Willie Wong: Yes, it is clear that one can convert a higher order differential equation to a lower order one, but I don't want to enlarge the space where my ODE lives. For example I want to know if there are typical behaviour which a solution of third order ODE on $\mathbb{R}^3$ can have, but no solution of a second order ODE on $\mathbb{R}^3$, i.e. on the same space.

I don't know an answer to your question. But as a side note: There is an interesting paper by Lieb and Ynvason concerning your motivation: arxiv.org/abs/cond-mat/9708200. Perhaps the references therein could be helpful to answer your question.