The linear profile decomposition for the fourth order Schrödinger equation

In this paper, we establish the linear profile decomposition for the one-dimensional fourth order Schrödinger equation{iut−μΔu+Δ2u=0,t∈R,x∈R,u(0,x)=f(x)∈L2, where μ⩾0μ⩾0. As an application, we establish a dichotomy result on the existence of extremals to the symmetric Schrödinger Strichartz inequality.