On uniqueness properties of solutions of the generalized fourth-order Schr\"odinger equations

Abstract

In this paper, we study uniqueness properties of solutions to the generalized fourth-order Schr\"odinger equations in any dimension $d$ of the following forms, $$i \partial_t u + \sum_{j=1}^d \partial_{x_j}^{\, 4} u = V(t, x) u, \quad \text{and} \quad i \partial_t u + \sum_{j=1}^d \partial_{x_j}^{\, 4} u + F (u, \bar{u}) = 0.$$ 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 $u_1 \equiv u_2$.