Unique continuation through hyperplane for higher order parabolic and Schrödinger equations

Consider the higher order parabolic operator $\partial_t+(-Δ_x)^m$ and the higher order Schrödinger operator $i^{-1}\partial_t+(-Δ_x)^m$ 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.

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