Unique continuation properties for one dimensional higher order Schr\"{o}dinger equations

TL;DR

This paper investigates unique continuation properties for higher order Schrödinger equations, establishing conditions under which solutions must be trivial, including exponential decay and half-space vanishing, with results primarily for one spatial dimension.

Contribution

It proves sharp unique continuation results for one-dimensional higher order Schrödinger equations and discusses challenges in extending these results to higher dimensions.

Findings

Exponential decay at two times implies solution is zero in 1D.

Vanishing in a half-space implies solution is zero in 1D.

Partial results and discussions for higher dimensions.

Abstract

We study two types of unique continuation properties for the higher order Schr\"{o}dinger equation with potential $$ i\partial_tu=(-\Delta_x)^mu+V(t,x)u,\quad(t,x)\in\mathbb{R}^{1+n},\,2\leq m\in\mathbb{N}_+. $$ The first one says if $u$ has certain exponential decay at two times, then $u\equiv0$, and this result is sharp by constructing critical non-trivial solutions. The second one says if $u\equiv0$ in an arbitrary half-space of $\mathbb{R}^{1+n}$, then $u\equiv0$ 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.