  This paper is concerned with the bifurcation from infinity of the nonlinear
Schr\"odinger equation $$-\Delta u+V(x)u=\lambda u+f(x,u),\hspace{0.4cm} x\in
\mathbb{R}^N.$$ We treat this problem in the framework of dynamical systems by
considering the corresponding parabolic equation on unbounded domains. Firstly,
we establish a global invariant manifold for the parabolic equation on
$\mathbb{R}^N$. Then, we restrict the parabolic equation to this invariant
manifold, which generates a system of finite dimension. Finally, we use the
Conley index theory and the shape theory of attractors to establish some new
results on bifurcations from infinity and multiplicity of solutions of the
Schr\"odinger equation under an appropriate Landesman-Lazer type condition.
