On uniqueness properties of solutions of the generalized fourth-order Schrödinger equations

In this paper, we study uniqueness properties of solutions to the generalized fourth-order Schrödinger equations in any dimension $d$ of the following forms, $$i{\mathrm{\partial }}_{t}u+\sum _{j=1}^d{\mathrm{\partial }}_{x_j}^{4}u=V(t,x)u,\text{and}i{\mathrm{\partial }}_{t}u+\sum _{j=1}^d{\mathrm{\partial }}_{x_j}^{4}u+F(u,\bar{u})=0.$$ We show that a linear solution $u$ with fast enough decay in certain Sobolev spaces at two different times has to be trivial. Consequently, if the difference between two nonlinear solutions $u_1$ and $u_2$ decays sufficiently fast at two different times, it implies that $u_1\equiv u_2$.