One way to approach such questions is to start with an abstract situation (Hilbert scales).  If $T$ is an unbounded s.a. operator with $T>1$ on a separable Hilbert space (in your case, the Schrödinger operator on the usual $L^2$-space), then it embeds in a natural way into an increasing family $H^\alpha$  (with $\alpha \in ]-\infty,\infty[$) of Hilbert spaces, with limiting spaces
 $H^{\infty}$, their intersection, and $H^{-\infty}$, their union.  These are a Fréchet space and a $DF$ space, resp.  They are both nuclear under natural conditions on the spectrum of $T$.

As indicated above the Sobolev spaces arise from the Schrödinger operator.  I think that this answers your question. For references, just google "Hilbert scales".  Most of the interesting spaces of distributions, test functions and Sobolev-type spaces arise in this manner by choosing the classical operators (Laplace-Beltrami, Legendre, Laguerre ...).