  We investigate existence and asymptotic completeness of the wave operators
for nonlinear Klein-Gordon and Schr\"odinger equations with a defocusing
exponential nonlinearity in two space dimensions. A certain threshold is
defined based on the value of the conserved Hamiltonian, below which the
exponential potential energy is dominated by the kinetic energy via a
Trudinger-Moser type inequality.
  We prove that if the energy is below or equal to the critical value, then the
solution approaches a free Klein-Gordon solution at the time infinity. The
interesting feature in the critical case is that the Strichartz estimate
together with Sobolev-type inequalities can not control the nonlinear term
uniformly on each time interval, but with constants depending on how much the
solution is concentrated. Thus we have to trace concentration of the energy
along time, in order to set up favorable nonlinear estimates, and then to
implement Bourgain's induction argument. We show the same result for the
"subcritical" nonlinear Schr\"odinger equation.
