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

Abstract

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

Theorems & Definitions (23)

• Theorem A: Theorem 1 in EKPV_Duke
• Theorem B: Hardy's uncertainty principle in Hardy
• Theorem C: Corollary in Hor
• Theorem 1.1: Linear unique continuation
• Theorem 1.2: Nonlinear unique continuation
• Remark 1.3: Decay notation
• Remark 1.4: Sharpness of the result and discussion on assumptions
• Lemma 3.1
• Remark 3.2: A formal proof
• proof : Proof of Lemma \ref{['lem con']}