Scattering of solutions for critical and subcritical nonlinear Klein-Gordon equations in $H^s$

We study the scattering theory for nonlinear Klein-Gordon equations $u_{t t} + (m^2-\Delta)u = f_1(u) + f_2(u)$. We show that the scattering operator carries a band in $H^s \times H^{s-1}$ into $H^s \times H^{s-1}$ for all $s\in [1/2,\ \infty)$ if $f_i(u)\ (i = 1,\ 2)$ have $H^s$-critical or $H^s$-subcritical powers.