  Time-dependent processes accompanied by vortex creations and annihilations
are investigated in terms of the eigenstates in conjugate spaces of Gel'fand
triplets in 2-dimensions. Creations and annihilations of vortices are described
by the insertions of unstable eigenstates with complex-energy eigenvalues into
stable states written by the superposition of eigenstates with zero-energy
eigenvalues. Some concrete examples are presented in terms of the
eigenfunctions of the 2-dimensional parabolic potential barrier, i.e., $-m
\gamma^2 (x^2+y^2)/2$. We show that the processes accompanied by vortex
creations and annihilations can be analyzed in terms of the eigenfunctions in
the conjugate spaces of Gel'fand triplets. Throughout these examinations we
point out three interesting properties of the zero-energy flows. (i) Mechanisms
using the zero-energy flows are absolutely economical from the viewpoint of
energy consumption. (ii) An enormous amount of informations can be
discriminated in terms of the infinite variety of the zero-energy flows. (iii)
The zero-energy flow patterns are absolutely stable in any disturbance by
inserting arbitrary decaying flows with complex-energy eigenvalues.
