Scattering for the one-dimensional Klein-Gordon equation with exponential nonlinearity

We consider the asymptotic behavior of solutions to the Cauchy problem for the defocusing nonlinear Klein-Gordon equation (NLKG) with exponential nonlinearity in the one spatial dimension with data in the energy space H-1 (R) x L-2 (R). We prove that any energy solution has a global bound of the L-t,x(6) space-time norm, and hence, scatters in H-1 (R) x L-2 (R) as t -> +/-infinity. The proof is based on the argument by Killip-Stovall-Visan (Trans. Amer. Math. Soc. 364(3) (2012) 1571-1631). However, since well-posedness in H-1/2 (R) x H-1/2 (R) for NLKG with the exponential nonlinearity holds only for small initial data, we use the (LtWx8-1/2,6)-W-6-norm for some s > 1/2 instead of the L-t,x(6)-norm, where W-x(8,p) denotes the sth order L-p-based Sobolev space.