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 ∂ t u + ∑ j = 1 d ∂ x j 4 u = V t , x u , and i ∂ t u + ∑ j = 1 d ∂ x j 4 u + F u , 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 ≡ u 2 .