Unique continuation properties for one dimensional higher order Schrödinger equations

Abstract

We study two types of unique continuation properties for the higher order Schrödinger equation with potential $$i{\partial }_{t}u=(-{\Delta }_x{)}^{m}u+V(t,x)u,(t,x)\in {\mathbb{R}}^{1+n},2\le m\in {\mathbb{N}}_{+}.$$ The first one says if $u$ has certain exponential decay at two times, then $u\equiv 0$, and this result is sharp by constructing critical non-trivial solutions. The second one says if $u\equiv 0$ in an arbitrary half-space of ${\mathbb{R}}^{1+n}$, then $u\equiv 0$ identically. The uniqueness theorems are given when $n=1$, but we also prove partial results when $n\in {\mathbb{N}}_{+}$ for their own interests. Possibility or obstacles to proving these unique continuation properties in higher spatial dimensions are also discussed.

We have refined the whole paper to be more comprehensive, mainly the introduction. We also correct some mistakes concerning Examples 1.2

Cited by in corpus (2)

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