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

TL;DR

This paper investigates the uniqueness of solutions to generalized fourth-order Schrödinger equations, demonstrating that solutions with rapid decay at two times must be trivial, and that differences of solutions with similar decay are identical.

Contribution

It establishes new uniqueness results for solutions of generalized fourth-order Schrödinger equations with decay conditions at two times.

Findings

Solutions with fast decay at two times are trivial.

Differences of solutions with sufficient decay at two times are identical.

Results apply to both linear and nonlinear generalized fourth-order Schrödinger 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 $u_1 \equiv u_2$.