  We prove global asymptotic bifurcation for a very general class of
asymptotically linear Schr\"odinger equations \begin{equation}\label{1}
\{{array}{lr} \D u + f(x,u)u = \lam u \quad \text{in} \ {\mathbb R}^N, u \in
H^1({\mathbb R}^N)\setmimus\{0\}, \quad N \ge 1. {array}. \end{equation} The
method is topological, based on recent developments of degree theory. We use
the inversion $u\to v:= u/\Vert u\Vert_X^2$ in an appropriate Sobolev space
$X=W^{2,p}({\mathbb R}^N)$, and we first obtain bifurcation from the line of
trivial solutions for an auxiliary problem in the variables $(\lambda,v) \in
{\mathbb R} \x X$. This problem has a lack of compactness and of regularity,
requiring a truncation procedure. Going back to the original problem, we obtain
global branches of positive/negative solutions 'bifurcating from infinity'. We
believe that, for the values of $\lambda$ covered by our bifurcation approach,
the existence result we obtain for positive solutions of \eqref{1} is the most
general so far
