Unique continuation through hyperplane for higher order parabolic and Schr\"odinger equations

TL;DR

This paper establishes unique continuation properties across hyperplanes for higher order parabolic and Schrödinger equations, demonstrating that solutions vanishing on one side must vanish entirely under certain conditions.

Contribution

It proves new unique continuation results for higher order parabolic and Schrödinger equations, extending previous understanding to more complex operators and higher dimensions.

Findings

Solutions vanish on one side imply global vanishing under conditions

Results apply to operators of arbitrary order m and dimension n

Includes both global and local unique continuation results

Abstract

Consider the higher order parabolic operator $\partial_t+(-\Delta_x)^m$ and the higher order Schr\"{o}dinger operator $i^{-1}\partial_t+(-\Delta_x)^m$ in $X=\{(t,x)\in\mathbb{R}^{1+n};~|t|<A,|x_n|<B\}$, where $m$ and $n$ are any positive integers. Under certain lower order and regularity assumptions, we prove that if solutions to the linear problems vanish when $x_n>0$, then the solutions vanish in $X$. Such results are global if $n>1$, and we also prove some relevant local results.