Long-time existence for semi-linear Klein-Gordon equations with quadratic potential

We prove that small smooth solutions of semi-linear Klein-Gordon equations with quadratic potential exist over a longer interval than the one given by local existence theory, for almost every value of mass. We use normal form for the Sobolev energy. The difficulty in comparison with some similar results on the sphere comes from the fact that two successive eigenvalues $\lambda, \lambda'$ of $\sqrt{-\Delta+|x|^2}$ may be separated by a distance as small as $\frac{1}{\sqrt{\lambda}}$.