Classification of two dimensional split trianguline representations of $p$-adic fields

The aim of this paper is to classify two dimensional split trianguline representations of $p$-adic fields. This is a generalization of a result of Colmez who classified two dimensional split trianguline representations of $\mathrm{G}\mathrm{a}\mathrm{l}({\bar{\mathbb{Q}}}_p\mathrm{/}{\mathbb{Q}}_p)$ by using $(\varphi ,\mathrm{\Gamma })$-modules over Robba ring. In this paper, we classify two dimensional split trianguline representations of $\mathrm{G}\mathrm{a}\mathrm{l}(\bar{K}\mathrm{/}K)$ for general $p$-adic field $K$ by using $B$-pairs defined by Berger.