The linear profile decomposition for the fourth order Schr\"odinger equation

In this paper, we establish the linear profile decomposition for the one dimensional fourth order Schr\"odinger equation $$iu_t-\mu\Delta u+\Delta^2u=0, t\in\mathbb{R}, x\in\mathbb{R}, u(0,x)=f(x)\in L^2,$$ where $\mu\ge 0$. As an application, we establish a dichotomy result on the existence of extremals to the symmetric Schr\"odinger Strichartz inequality.