On $1$d quadratic Klein-Gordon equations with a potential and symmetries

This paper is a continuation of a previous work Germain-Pusateri (2020) by the first two authors. We focus on $1$ dimensional quadratic Klein-Gordon equations with a potential, under some assumptions that are less general than Germain-Pusateri (2020), but allow us to present some simplifications in the proof of global existence with decay for small solutions. In particular, we can propagate a stronger control on a basic $L^2$-weighted type norm while providing some shorter and less technical proofs for some of the arguments.